SARS-CoV-2 infection and immune responses

Rakhi Harne; Brittany Williams; Hazem F. M. Abdelaal; Susan L. Baldwin; Rhea N. Coler; Rakhi Harne; Brittany Williams; Hazem F. M. Abdelaal; Susan L. Baldwin; Rhea N. Coler

doi:10.3934/microbiol.2023015

AIMS Microbiology

2023, Volume 9, Issue 2: 245-276. doi: 10.3934/microbiol.2023015

Previous Article Next Article

Review Topical Sections

SARS-CoV-2 infection and immune responses

1.
Seattle Children's Research Institute, Center for Global Infectious Disease Research, Seattle Children's Hospital, Seattle, Washington, USA
2.
Department of Global Health, University of Washington, Seattle, Washington, USA
3.
Department of Pediatrics, University of Washington School of Medicine, Seattle, Washington, USA

Received: 13 October 2022 Revised: 14 March 2023 Accepted: 21 March 2023 Published: 29 March 2023

The recent pandemic caused by the SARS-CoV-2 virus continues to be an enormous global challenge faced by the healthcare sector. Availability of new vaccines and drugs targeting SARS-CoV-2 and sequelae of COVID-19 has given the world hope in ending the pandemic. However, the emergence of mutations in the SARS-CoV-2 viral genome every couple of months in different parts of world is a persistent danger to public health. Currently there is no single treatment to eradicate the risk of COVID-19. The widespread transmission of SARS-CoV-2 due to the Omicron variant necessitates continued work on the development and implementation of effective vaccines. Moreover, there is evidence that mutations in the receptor domain of the SARS-CoV-2 spike glycoprotein led to the decrease in current vaccine efficacy by escaping antibody recognition. Therefore, it is essential to actively identify the mechanisms by which SARS-CoV-2 evades the host immune system, study the long-lasting effects of COVID-19 and develop therapeutics targeting SARS-CoV-2 infections in humans and preclinical models. In this review, we describe the pathogenic mechanisms of SARS-CoV-2 infection as well as the innate and adaptive host immune responses to infection. We address the ongoing need to develop effective vaccines that provide protection against different variants of SARS-CoV-2, as well as validated endpoint assays to evaluate the immunogenicity of vaccines in the pipeline, medications, anti-viral drug therapies and public health measures, that will be required to successfully end the COVID-19 pandemic.

Keywords:

Citation: Rakhi Harne, Brittany Williams, Hazem F. M. Abdelaal, Susan L. Baldwin, Rhea N. Coler. SARS-CoV-2 infection and immune responses[J]. AIMS Microbiology, 2023, 9(2): 245-276. doi: 10.3934/microbiol.2023015

Related Papers:

[1]	Ruizhi Yang, Dan Jin, Wenlong Wang . A diffusive predator-prey model with generalist predator and time delay. AIMS Mathematics, 2022, 7(3): 4574-4591. doi: 10.3934/math.2022255
[2]	Yingyan Zhao, Changjin Xu, Yiya Xu, Jinting Lin, Yicheng Pang, Zixin Liu, Jianwei Shen . Mathematical exploration on control of bifurcation for a 3D predator-prey model with delay. AIMS Mathematics, 2024, 9(11): 29883-29915. doi: 10.3934/math.20241445
[3]	Sahabuddin Sarwardi, Hasanur Mollah, Aeshah A. Raezah, Fahad Al Basir . Direction and stability of Hopf bifurcation in an eco-epidemic model with disease in prey and predator gestation delay using Crowley-Martin functional response. AIMS Mathematics, 2024, 9(10): 27930-27954. doi: 10.3934/math.20241356
[4]	Xin-You Meng, Fan-Li Meng . Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting. AIMS Mathematics, 2021, 6(6): 5695-5719. doi: 10.3934/math.2021336
[5]	Heping Jiang . Complex dynamics induced by harvesting rate and delay in a diffusive Leslie-Gower predator-prey model. AIMS Mathematics, 2023, 8(9): 20718-20730. doi: 10.3934/math.20231056
[6]	Qinghui Liu, Xin Zhang . Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response. AIMS Mathematics, 2024, 9(9): 24555-24575. doi: 10.3934/math.20241196
[7]	Hairong Li, Yanling Tian, Ting Huang, Pinghua Yang . Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model. AIMS Mathematics, 2024, 9(10): 29608-29632. doi: 10.3934/math.20241434
[8]	Fatao Wang, Ruizhi Yang, Yining Xie, Jing Zhao . Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator. AIMS Mathematics, 2023, 8(8): 17719-17743. doi: 10.3934/math.2023905
[9]	Eric M. Takyi, Charles Ohanian, Margaret Cathcart, Nihal Kumar . Sex-biased predation and predator intraspecific competition effects in a prey mating system. AIMS Mathematics, 2024, 9(1): 2435-2453. doi: 10.3934/math.2024120
[10]	Liye Wang, Wenlong Wang, Ruizhi Yang . Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect. AIMS Mathematics, 2023, 8(4): 9716-9739. doi: 10.3934/math.2023490

Abstract

1. Introduction

In survey sampling, it is well known fact that suitable use of the auxiliary information may improves the precision of an estimator for the unknown population parameters. The auxiliary information can be used either at the design stage or at estimation stage to increase the accuracy of the population parameter estimators. Several authors presented modified different type of estimators for estimating the finite population mean including ^{[4,9,21,22,23,24,25,26,27]}.

The problem of estimation of finite population mean or total in two-stage sampling scheme using the auxiliary information has been well established. The two stage sampling scheme is an improvement over the cluster sampling, when it is not possible or easy to calculate all the units from the selected clusters. One of the main characteristic could be the budget, and it becomes too difficult to collect information from all the units within the selected clusters. To overcome this, one way is to select clusters, called first stage unit (fsus) and from the given population of interest, select a subsample from the selected clusters called the second stage units (ssu). This also benefits to increase the size of the first stage samples which consist of clusters, and assume to be heterogeneous groups. If there is no variation within clusters then might not be possible to collect information from all the units within selected clusters. In many situations, it is not possible to obtain the complete list of ultimate sampling units in large scale sample surveys, while a list of primary units of clusters may be available. In such situations, we select a random sample of first stage units or primary units using certain probability sampling schemes i.e simple random sampling (with or without replacement), systematic sampling and probability proportional to size (PPS), and then we can perform sub-sampling in selected clusters (first stage units). This approach is called two-stage sampling scheme.

Two-stage has a great varaity of applications, which go far beyond the immediate scop of sample survey. Whenever any process involves in chemical, physical, or biological tests that can be performed on a small amount of materail, it is likely to be drawn as a subsample from a larger amount that is itself a sample.

In large scale survey sampling, it is usual to adopt multistage sampling to estimate the population mean or total of the study variable $y$ . ^[13] proposed a general class of estimators of a finite population mean using multi-auxiliary information under two stage sampling scheme. ^[1] proposed an alternative class of estimators in two stage sampling with two auxiliary variables. ^[10] proposed estimators for finite population mean under two-stage sampling using multivariate auxiliary information. ^[12] suggested a detailed note on ratio estimates in multi-stage sampling. ^[6] given some stratagies in two stage sampling using auxiliary information. ^[3] suggested a class of predictive estimaotrs in two stage sampling using auxiliary information. ^[8] gave a generalized method of estimation for two stage sampling using two auxiliary variables. ^[5] suggested chain ratio estimators in two stage sampling. For certain related work, we refer some latest articles, i.e., ^{[14,15,16,17,18,19,20]}.

In this article, we propose an improved generalized class of estimators using two auxiliary variables under two-stage sampling scheme. The biases and mean sqaure errors of the proposed generalized class of estimators are derived up to first order of approximation. Based on the numerical results, the proposed class of estimators are more efficient than their existing counterparts.

2. Symbols and notation

Consider a finite population $U$ = $\{{U}_{1}, {U}_{2}, ..., {U}_{N}\}$ is divided into $N$ first-stage units (fsus) clusters in the population. Let N be the total number of first stage unit in population, n be the number of first stage units selected in the sample, ${M}_{i}$ be the number of second stage units (ssus) belongs to the ${i}^{th}$ first stage units (fsus), (i = 1, 2, …, N), and mi be the number of fsus selected from the ${i}^{th}$ fsu in the sample of n fsus, (i = 1, 2, …, n).

Let ${y}_{ij}$ , ${x}_{ij}$ and ${z}_{ij}$ be values of the study variable $y$ and the auxiliary variables $\left(x\mathrm{a}\mathrm{n}\mathrm{d}z\right)$ respectively, for the ${j}^{th}$ ssus ${U}_{i} = (j = \mathrm{1, 2}, ..., {M}_{i})$ , in the ${i}^{th}$ fsus. The population mean of the study variable $y$ and the auxiliary variables $(x, z)$ are given by:

$\overline {Y} = \frac{1}{N}\sum _{i = 1}^{N}\;{u}_{i}{\overline {Y}}_{i}, \;\;\;\;\overline {X} = \frac{1}{N}\sum _{i = 1}^{N}\;{u}_{i}{\overline {X}}_{i}, \;\;\;\;\overline {Z} = \frac{1}{N}\sum _{i = 1}^{N}\;{u}_{i}{\overline {Z}}_{i} ,$

where

${\overline {Y}}_{i} = \frac{1}{{M}_{i}}\sum\limits_{j = 1}^{{M}_{i}}\;{y}_{ij}, {\overline {X}}_{i} = \frac{1}{{M}_{i}}\sum\limits_{j = 1}^{{M}_{i}}\;{x}_{ij}, {\overline {Z}}_{i} = \frac{1}{{M}_{i}}\sum\limits_{j = 1}^{{M}_{i}}\;{z}_{ij}, \;\;(i = \mathrm{1, 2}, ..., N).$

${u}_{i} = \frac{{M}_{i}}{\overline {M}} , \;{\rm{and}}\; \overline {M} = \frac{M}{N} , \; M = \sum _{i = 1}^{N}\;{M}_{i} ,$

$R = \frac{\overline {Y}}{\overline {X}} , \; {\rm{and}} \; {R}_{i} = \frac{{\overline {Y}}_{i}}{{\overline {X}}_{i}} ,$

${S}_{by}^{2} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Y}}_{i}-\overline {Y}{)}^{2} ,$

${S}_{bx}^{2} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {X}}_{i}-\overline {X}{)}^{2} ,$

${S}_{bz}^{2} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Z}}_{i}-\overline {Z}{)}^{2} ,$

${S}_{byx} = \frac{1}{N-1}\sum _{i = 1}^{N}\;\left({u}_{i}{\overline {Y}}_{i}-\overline {Y}\right)\left({u}_{i}{\overline {X}}_{i}-\overline {X}\right), \;\;\;\;\;\;\;\;{S}_{byz} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Z}}_{i}-\overline {Z})({u}_{i}{\overline {Y}}_{i}-\overline {Y}) ,$

${S}_{bxz} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Z}}_{i}-\overline {Z})({u}_{i}{\overline {X}}_{i}-\overline {X}), \;\;\;\;\;\;\;\;{S}_{iy}^{2} = \frac{1}{{M}_{i}-1}\sum _{j = 1}^{{M}_{i}}\;({y}_{ij}-{\overline {Y}}_{i}{)}^{2} ,$

${S}_{ix}^{2} = \frac{1}{{M}_{i}-1}\sum _{j = 1}^{{M}_{i}}\;({x}_{ij}-{\overline {X}}_{i}{)}^{2}, \;\;\;\;\;\;\;\;{S}_{iz}^{2} = \frac{1}{{M}_{i}-1}\sum _{j = 1}^{{M}_{i}}\;({z}_{ij}-{\overline {Z}}_{i}{)}^{2} ,$

${S}_{iyx} = \frac{1}{{M}_{i}-1}({y}_{ij}-{\overline {Y}}_{i})({x}_{ij}-{\overline {X}}_{i}), \;\;\;\;\;\;\;\;{S}_{iyz} = \frac{1}{{M}_{i}-1}({y}_{ij}-{\overline {Y}}_{i})({z}_{ij}-{\overline {Z}}_{i}) ,$

${S}_{ixz} = \frac{1}{{M}_{i}-1}\sum\limits_{j = 1}^{{M}_{i}}\;({x}_{ij}-{\overline {X}}_{i})({z}_{ij}-{\overline {Z}}_{i}), \;\;\;\;\;\;\;\;(i = \mathrm{1, 2}, ...N).$

Similarly for sample data:

$\overline {y} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {y}}_{i} = {\overline {y}}^{\mathrm{*}}\left(say\right), \;\overline {x} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {x}}_{i} = {\overline {x}}^{\mathrm{*}}\left(say\right), \;\overline {z} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {z}}_{i} = {\overline {z}}^{\mathrm{*}}\left(say\right),$

where

${\overline {y}}_{i} = \frac{1}{{m}_{i}}\sum\limits_{j = 1}^{{m}_{i}}\;{y}_{ij}, \;\;\;\;{\overline {x}}_{i} = \frac{1}{{m}_{i}}\sum\limits_{j = 1}^{{m}_{i}}\;{x}_{ij}, \;\;\;\;{\overline {z}}_{i} = \frac{1}{{m}_{i}}\sum\limits_{j = 1}^{{m}_{i}}\;{z}_{ij},$

${s}_{by}^{2} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {y}}_{i}-\overline {y}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{bx}^{2} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {x}}_{i}-\overline {x}{)}^{2} ,$

${s}_{bz}^{2} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {z}}_{i}-\overline {z}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{byx} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {y}}_{i}-\overline {y}\left)\right({u}_{i}{\overline {x}}_{i}-\overline {x}) ,$

${s}_{byz} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {y}}_{i}-\overline {y})({u}_{i}{\overline {z}}_{i}-\overline {z}), \;\;\;\;\;\;\;\;{s}_{bxz} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {x}}_{i}-\overline {x})({u}_{i}{\overline {z}}_{i}-\overline {z}) ,$

${s}_{iy}^{2} = \frac{1}{{m}_{i}-1}({y}_{ij}-{\overline {y}}_{i}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{ix}^{2} = \frac{1}{{m}_{i}-1}({x}_{ij}-{\overline {x}}_{i}{)}^{2} ,$

${s}_{iz}^{2} = \frac{1}{{m}_{i}-1}({z}_{ij}-{\overline {z}}_{i}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{iyx} = \frac{1}{{m}_{i}-1}\sum _{j = 1}^{{m}_{i}}\;({y}_{ij}-{\overline {y}}_{i}\left)\right({x}_{ij}-{\overline {x}}_{i}) ,$

${s}_{iyz} = \frac{1}{{m}_{i}-1}\sum _{j = 1}^{{m}_{i}}\;\left({y}_{ij}-{\overline {y}}_{i}\right)\left({z}_{ij}-{\overline {z}}_{i}\right), \;\;\;\;\;\;\;\;{s}_{ixz} = \frac{1}{{m}_{i}-1}\sum _{j = 1}^{{m}_{i}}\;({x}_{ij}-{\overline {x}}_{i})({z}_{ij}-{\overline {z}}_{i}) .$

In order to obtain the biases and mean sqaured errors, we consider the following relative error terms:

${e}_{0} = \frac{{\overline {y}}^{\mathrm{*}}-\overline {Y}}{\overline {Y}}, \;{e}_{1} = \frac{{\overline {x}}^{\mathrm{*}}-\overline {X}}{\overline {X}}, \;{e}_{2} = \frac{{\overline {z}}^{\mathrm{*}}-\overline {Z}}{\overline {Z}} ,$

$E\left({e}_{0}^{2}\right) = \lambda {C}_{by}^{2}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iy}^{2} = {V}_{y} ,$

$E\left({e}_{1}^{2}\right) = \lambda {C}_{bx}^{2}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{ix}^{2} = {V}_{x} ,$

$E\left({e}_{2}^{2}\right) = \lambda {C}_{bz}^{2}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iz}^{2} = {V}_{z} ,$

$E\left({e}_{0}{e}_{1}\right) = \lambda {C}_{byx}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iyx} = {V}_{yx},$

$E\left({e}_{0}{e}_{2}\right) = \lambda {C}_{byz}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iyz} = {V}_{yz},$

$E\left({e}_{1}{e}_{2}\right) = \lambda {C}_{bxz}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{ixz} = {V}_{xz} ,$

${C}_{by} = \frac{{S}_{by}}{\overline {Y}}, \;\;\;\;\;\;\;\;{C}_{bx} = \frac{{S}_{bx}}{\overline {X}}, \;\;\;\;\;\;\;\;{C}_{bz} = \frac{{S}_{bz}}{\overline {Z}} ,$

${C}_{byx} = \frac{{S}_{byx}}{\overline {Y}\overline {X}}, \;\;\;\;\;\;\;\;{C}_{byz} = \frac{{S}_{byz}}{\overline {Y}\overline {Z}}, \;\;\;\;\;\;\;\;{C}_{bxz} = \frac{{S}_{bxz}}{\overline {X}\overline {Z}} ,$

${C}_{iyx} = \frac{{S}_{iyx}}{\overline {Y}\overline {X}}, \;\;\;\;\;\;\;\;{C}_{iyz} = \frac{{S}_{byz}}{\overline {Y}\overline {Z}}, \;\;\;\;\;\;\;\;{C}_{ixz} = \frac{{S}_{ixz}}{\overline {X}\overline {Z}} ,$

${C}_{iy} = \frac{{S}_{iy}}{\overline {Y}}, \;\;\;\;\;\;\;\;{C}_{ix} = \frac{{S}_{ix}}{\overline {X}}, \;\;\;\;\;\;\;\;{C}_{iz} = \frac{{S}_{iz}}{\overline {Z}} ,$

where,

${\theta }_{i} = \left(\frac{1}{{m}_{i}}-\frac{1}{{M}_{i}}\right), \lambda = (\frac{1}{n}-\frac{1}{N}) .$

3. Existing estimators

In this section, we consider several estimators of the finite population mean under two-stage sampling that are available in the sampling literature, the properties of all estimators considered here are obtained up-to the first order of approximation.

(ⅰ) The usual mean estimator ${\overline {y}}^{\mathrm{*}} = {\overline {y}}_{0}^{\mathrm{*}}$ and its variance under two-stage sampling are given by:

${\overline {y}}_{0}^{\mathrm{*}} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {y}}_{i} ,$

(1)

and

$V\left({\overline {y}}_{0}^{\mathrm{*}}\right) = {\overline {Y}}^{2}{V}_{y} = MSE\left({\overline {y}}_{0}^{\mathrm{*}}\right) .$

(2)

(ⅱ) The usual ratio estimator under two-stage sampling, is given by:

${\overline {y}}_{R}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right) ,$

(3)

where $\overline {X}$ is the known population mean of $x$ .

The bias and $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{R}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{R}^{\mathrm{*}}\right) = \overline {Y}\left[{V}_{x}-{V}_{yx}\right] ,$

(4)

and

$MSE\left({\overline {y}}_{R}^{\mathrm{*}}\right) = {\overline {Y}}^{2}\left[{V}_{y}+{V}_{x}-2{V}_{yx}\right] .$

(5)

(ⅲ) ^[2] Exponential ratio type estimator under two-stage sampling, is given by:

${\overline {y}}_{E}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}exp\left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}+{\overline {x}}^{\mathrm{*}}}\right) .$

(6)

The bias and $MSE$ of ${\overline {y}}_{E}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{E}^{\mathrm{*}}\right) = \overline {Y}\left[\frac{3}{8}{V}_{x}-\frac{1}{2}{V}_{yx}\right] ,$

(7)

and

$MSE\left({\overline {y}}_{E}^{\mathrm{*}}\right) = {\overline {Y}}^{2}\left[{V}_{y}+\frac{1}{4}{V}_{x}-{V}_{yx}\right] .$

(8)

(ⅳ) The traditional difference estimator under two-stage sampling is given by:

${\overline {y}}_{D}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}+d\left(\overline {X}-{\overline {x}}^{\mathrm{*}}\right) ,$

(9)

where $d$ is the constant.

The minimum variance of ${\overline {y}}_{D}^{\mathrm{*}}$ , is given by:

$V\left({\overline {y}}_{Dmin}^{\mathrm{*}}\right) = {\overline {Y}}^{2}{V}_{y}\left(1-{\rho }^{\mathrm{*}2}\right) = MSE\left({\overline {y}}_{D}^{\mathrm{*}}\right) ,$

(10)

where ${\rho }^{\mathrm{*}} = \frac{{V}_{yx}}{\sqrt{{V}_{y}}\sqrt{{V}_{x}}}$ .

The optimum value of $d$ is ${d}_{opt} = \frac{\overline {Y}{V}_{yx}}{\overline {X}{V}_{x}}$ .

(ⅴ) ^[7] Difference type estimator under two-stage sampling, is given by:

${\overline {y}}_{Rao}^{\mathrm{*}} = {d}_{0}{\overline {y}}^{\mathrm{*}}+{d}_{1}\left(\overline {X}-{\overline {x}}^{\mathrm{*}}\right) ,$

(11)

where ${d}_{0}$ and ${d}_{1}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{Rao}^{\mathrm{*}}$ to first order of approximation, is given by:

$Bias\left({\overline {y}}_{Rao}^{\mathrm{*}}\right) = ({d}_{0}-1)\overline {Y} ,$

(12)

and

$MSE({\overline {y}}_{Rao}^{\mathrm{*}}{)}_{min}\cong \frac{{\overline {Y}}^{2}({V}_{x}{V}_{y}-{V}_{yx}^{2})}{{V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x}} = \frac{{\overline {Y}}^{2}{V}_{y}(1-{\rho }^{\mathrm{*}2})}{1+{V}_{y}(1-{\rho }^{\mathrm{*}2})} .$

(13)

The optimum values of ${d}_{0}$ and ${d}_{1}$ are:

${d}_{0opt} = \frac{{V}_{x}}{{V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x}}$ and ${d}_{1opt} = \frac{\overline {Y}{V}_{yx}}{\overline {X}({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})}$ .

(ⅵ) The difference-in-ratio type estimator under two-stage sampling, is given by:

${\overline {y}}_{DR}^{\mathrm{*}} = \left[{d}_{2}{\overline {y}}^{\mathrm{*}}+{d}_{3}(\overline {X}-{\overline {x}}^{\mathrm{*}})\right]\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right) ,$

(14)

where ${d}_{2}$ and ${d}_{3}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DR}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{DR}^{\mathrm{*}}\right)\cong \overline {Y}({d}_{2}-1)-{d}_{2}\overline {Y}{V}_{yx}+{d}_{3}\overline {X}{V}_{x}+{d}_{2}{V}_{x}\overline {Y} ,$

(15)

and

$MSE({\overline {y}}_{DR}^{\mathrm{*}}{)}_{min}\cong \frac{{\overline {Y}}^{2}({V}_{x}^{2}{V}_{y}-{V}_{x}{V}_{yx}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2})}{({V}_{x}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2}-{V}_{x})} .$

(16)

The optimum values of ${d}_{2}$ and ${d}_{3}$ are:

${d}_{2opt} = \frac{{V}_{x}({V}_{x}-1)}{{V}_{x}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2}-{V}_{x}} ,$

${d}_{3opt} = \frac{-{\overline {Y}}^{2}({V}_{x}^{2}+{V}_{x}{V}_{y}-{V}_{x}{V}_{yx}-{V}_{yx}^{2}-{V}_{x}+{V}_{yx})}{\overline {X}({V}_{x}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2}-{V}_{x})} .$

(ⅶ) The difference-in-exponential ratio type estimator under two-stage sampling, is given by:

${\overline {y}}_{DE}^{\mathrm{*}} = \left[{d}_{4}{\overline {y}}^{\mathrm{*}}+{d}_{5}(\overline {X}-{\overline {x}}^{\mathrm{*}})\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}-{\overline {x}}^{\mathrm{*}}}\right) ,$

(17)

where ${d}_{4}$ and ${d}_{5}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DE}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{DE}^{\mathrm{*}}\right) = ({d}_{4}-1)\overline {Y}-\frac{1}{2}{d}_{4}\overline {Y}{V}_{yx}+\frac{3}{8}{d}_{4}\overline {Y}{V}_{x}+\frac{1}{2}{d}_{5}{V}_{x} ,$

(18)

and

$MSE({\overline {y}}_{DE}^{\mathrm{*}}{)}_{min}\cong \frac{-{\overline {Y}}^{2}\left({V}_{x}^{3}+16{V}_{x}^{2}{V}_{y}-16{V}_{x}{V}_{yx}^{2}-64{V}_{x}{V}_{y}+64{V}_{yx}^{2}\right)}{64{V}_{x}{V}_{y}-64{V}_{yx}^{2}+64{V}_{x}} .$

(19)

The optimum values of ${d}_{4}$ and ${d}_{5}$ are:

${d}_{4opt} = -\frac{1}{8}\frac{{V}_{x}({V}_{x}-8)}{({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})} ,$

${d}_{5opt} = \frac{\overline {Y}\left({V}_{x}^{2}+4{V}_{x}{V}_{y}-{V}_{x}{V}_{yx}-4{V}_{yx}^{2}-4{V}_{x}+8{V}_{yx}\right)}{\frac{1}{8}\overline {X}({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})} .$

(ⅷ) The difference-difference type estimator under two stage sampling, is given by:

${\overline {y}}_{DD}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}+{d}_{6}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{d}_{7}(\overline {Z}-{\overline {z}}^{\mathrm{*}}) ,$

(20)

where ${d}_{6}$ and ${d}_{7}$ are constants.

The minimum variance or M $SE$ of ${\overline {y}}_{DD}^{\mathrm{*}}$ to first order of approximation, is given by:

$MSE({\overline {y}}_{DD}^{\mathrm{*}}{)}_{min}\cong \frac{{\overline {Y}}^{2}({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z})}{{V}_{x}{V}_{z}-{V}_{xz}^{2}} .$

(21)

The optimum values of ${d}_{6}$ and ${d}_{7}$ are:

${d}_{6} = \frac{-\overline {Y}({V}_{xz}{V}_{yz}-{V}_{yx}{V}_{z})}{\overline {X}({V}_{x}{V}_{z}-{V}_{xz}^{2})} ,$

${d}_{7} = \frac{\overline {Y}({V}_{x}{V}_{yz}-{V}_{xz}{V}_{yx})}{\overline {Z}({V}_{x}{V}_{z}-{V}_{xz}^{2})}.$

(ix) The difference-difference type estimator under two stage sampling, is given by:

${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}} = {d}_{8}{\overline {y}}^{\mathrm{*}}+{d}_{9}\left(\overline {X}-{\overline {x}}^{\mathrm{*}}\right)+{d}_{10}\left(\overline {Z}-{\overline {z}}^{\mathrm{*}}\right) ,$

(22)

where ${d}_{8}$ , ${d}_{9}$ and ${d}_{10}$ are constants.

The bias and $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$ to first order of approximation is given by:

$Bias\left({\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}\right) = \overline {Y}\left({d}_{8}-1\right) ,$

(23)

and

$MSE\left({\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}\right)\cong \frac{{\overline {Y}}^{2}\left({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}\right)}{{V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2}} .$

(24)

The optimum values of ${d}_{8}$ , ${d}_{9}$ and ${d}_{10}$ are given by:

${d}_{8} = \frac{{V}_{x}{V}_{z}-{V}_{xz}^{2}}{{V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2}} ,$

${d}_{9} = \frac{-\overline {Y}({V}_{xz}{V}_{yz}-{V}_{yx}{V}_{z})}{\overline {X}({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2})} ,$

${d}_{10} = \frac{\overline {Y}({V}_{x}{V}_{yz}-{V}_{xz}{V}_{yx})}{\overline {Z}({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2})}.$

4. Proposed estimator

The principal advantage of our proposed improved generalized class of estimators under two-stage sampling is that it is more flexible, efficient, than the existing estimators. The mean square errors based on two data sets are minimum and percentage relative efficiency is more than hundred as compared to the existing estimators considered here. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3)$ , $\delta$ and $\gamma$ . On the lines of ^[2,7], we propose the following generalized improved class of estimators under two stage sampling for estimation of finite population mean using two auxiliary varaible as given by:

${\overline {y}}_{G}^{\mathrm{*}} = \left[{w}_{1}{\overline {y}}^{\mathrm{*}}+{w}_{2}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{w}_{3}(\overline {Z}-{\overline {z}}^{\mathrm{*}})\right]\left[\left\{exp\delta \left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}+{\overline {x}}^{\mathrm{*}}}\right)\right\}{\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right)}^{\gamma }\right] ,$

(25)

where ${w}_{i}(i = \mathrm{1, 2}, 3)$ are constants, whose values are to be determined; $\delta$ and $\gamma$ are constants i.e., ( $0\le \delta$ , $\gamma \le 1$ ) and can be used to construct the different estimators.

Using (25), solving ${\overline {y}}_{G}^{\mathrm{*}}$ in terms of errors, we have

${\overline {y}}_{G}^{\mathrm{*}}-\overline {Y} = ({w}_{1}-1)\overline {Y}+{w}_{1}\overline {Y}\left\{{e}_{0}-\frac{1}{2}{\alpha }_{1}{e}_{1}+\frac{1}{8}{\alpha }_{2}{e}_{1}^{2}-\frac{1}{2}{\alpha }_{1}{e}_{0}{e}_{1}\right\}$

$-{w}_{2}\overline {X}\left\{{e}_{1}-\frac{1}{2}{\alpha }_{1}{e}_{1}^{2}\right\}-{w}_{3}\overline {Z}\left\{{e}_{2}-\frac{1}{2}{\alpha }_{1}{e}_{1}{e}_{2}\right\} ,$

where

${\alpha }_{1} = \delta +2\gamma$ and ${\alpha }_{2} = \delta (\delta +2)+4\gamma (\delta +\gamma +1)$ .

The bias and $MSE$ of ${\overline {y}}_{G}^{\mathrm{*}}$ are given by:

$Bias\left({\overline {y}}_{G}^{\mathrm{*}}\right)\cong ({w}_{1}-1)\overline {Y}+{w}_{1}\overline {Y}\left\{\frac{1}{8}{\alpha }_{2}{V}_{x}-\frac{1}{2}{\alpha }_{1}{V}_{yx}\right\}+{w}_{2}\overline {X}{\alpha }_{1}\frac{{V}_{x}}{2}+{w}_{3}\overline {Z}{\alpha }_{1}\frac{{V}_{xz}}{2} ,$

(26)

and

$MSE\left({\overline {y}}_{G}^{\mathrm{*}}\right)\cong ({w}_{1}-1{)}^{2}+{w}_{1}^{2}{\overline {Y}}^{2}A+{w}_{2}^{2}{\overline {X}}^{2}B+{w}_{3}^{2}{\overline {Z}}^{2}C-{w}_{1}{\overline {Y}}^{2}D-{w}_{2}\overline {Y}\overline {X}E$

$-{w}_{3}\overline {Y}\overline {Z}F+2{w}_{1}{w}_{2}\overline {Y}\overline {X}G+2{w}_{1}{w}_{3}\overline {Y}\overline {Z}H+2{w}_{2}{w}_{3}\overline {X}\overline {Z}I ,$

(27)

where

$A = {V}_{y}+\frac{1}{4}{V}_{x}({\alpha }_{1}^{2}+{\alpha }_{2})-2{\alpha }_{1}{V}_{yx}, \;\;\;\;B = {V}_{x}, \;\;\;\;C = {V}_{z},$

$D = \frac{1}{4}{\alpha }_{2}{V}_{x}-{\alpha }_{1}{V}_{yx}, \;\;\;\;\;\;\;\;E = {\alpha }_{1}{V}_{x}, \;\;\;\;F = {\alpha }_{1}{V}_{xz},$

$G = {\alpha }_{1}{V}_{x}-{V}_{yx}, \;\;\;\;\;\;\;\;H = {\alpha }_{1}{V}_{xz}-{V}_{xz}, \;\; I = {V}_{xz} .$

Solving (27), the minimum $MSE$ of ${\overline {y}}_{G}^{\mathrm{*}}$ to first order of approximation are given by:

$MSE({\overline {y}}_{G}^{\mathrm{*}}{)}_{min} = {\overline {Y}}^{2}[1-\frac{{\mathrm{\Omega }}_{2}}{4{\mathrm{\Omega }}_{1}}] ,$

(28)

where

${\mathrm{\Omega }}_{1} = ABC-A{I}^{2}-B{H}^{2}-C{G}^{2}+2GHI+BC-{I}^{2} ,$

and

${\mathrm{\Omega }}_{2} = AB{F}^{2}+AC{E}^{2}-2AEFI+BC{D}^{2}-2BDFH-2CDEG-{D}^{2}{I}^{2}+2DEHI$

$+2DFGI-{E}^{2}{H}^{2}+2EFGH-{F}^{2}{G}^{2}+4BCD+B{F}^{2}-4BFH+C{E}^{2}$

$-4CEG-4D{I}^{2}-2EFI+4EHI+4FGI+4BC+4{I}^{2}.$

The optimum values of ${w}_{i}(i = 1, \;2, \;3)$ are given by:

${w}_{1opt} = \frac{{\mathrm{\Omega }}_{3}}{2{\mathrm{\Omega }}_{1}}, {w}_{2opt} = \frac{\overline {Y}{\mathrm{\Omega }}_{4}}{2\overline {X}{\mathrm{\Omega }}_{1}},$ and ${w}_{3opt} = \frac{\overline {Y}{\mathrm{\Omega }}_{5}}{2\overline {Z}{\mathrm{\Omega }}_{1}}$ ,

where

${\mathrm{\Omega }}_{3} = BCD-BFH-CEG-D{I}^{2}+EHI+FGI+2GI+2BC-2{I}^{2},$

${\mathrm{\Omega }}_{4} = ACE-AFI-CDG+DHI-E{H}^{2}+FGH+CE-2CG-FI+2HI,$

${\mathrm{\Omega }}_{5} = ABF-AEI-BDH+DGI+EGH-F{G}^{2}+BF-2BH-EI+2GI .$

From (28), we produce the following two estimators called ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ . Put $(\delta = 0, \gamma = 1)$ and $(\delta = 1, \gamma = 0)$ in (25), we get the following two estimators respectively:

$\left(i\right)\;\;{\overline {y}}_{G1}^{\mathrm{*}} = \left[{w}_{4}{\overline {y}}^{\mathrm{*}}+{w}_{5}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{w}_{6}(\overline {Z}-{\overline {z}}^{\mathrm{*}})\right]\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right) ,$

$\left(ii\right)\;\;{\overline {y}}_{G2}^{\mathrm{*}} = \left[{w}_{7}{\overline {y}}^{\mathrm{*}}+{w}_{8}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{w}_{9}(\overline {Z}-{\overline {z}}^{\mathrm{*}})\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}+{\overline {x}}^{\mathrm{*}}}\right) ,$

where ${w}_{i}(i = 4, \;5, \;6, \;7, \;8, \;9)$ are constants. Solving ${\overline {y}}_{G1}^{\mathrm{*}}$ , in terms of errors, we have:

$({\overline {y}}_{G1}^{\mathrm{*}}-\overline {Y}) = \left[-\overline {Y}+{w}_{4}\overline {Y}+{w}_{4}\overline {Y}{e}_{o}-{w}_{5}\overline {X}{e}_{1}-{w}_{6}\overline {Z}{e}_{2}\right].\left[1-\frac{1}{2}{e}_{1}+\frac{3}{8}{e}_{1}^{2}\right] ,$

$\left({\overline {y}}_{G1}^{\mathrm{*}}-\overline {Y}\right) = \left[\begin{array}{ll}& {w}_{4}\overline {Y}-{w}_{4}\overline {Y}{e}_{1}+\frac{3}{8}{w}_{4}\overline {Y}{e}_{1}^{2}+{w}_{4}\overline {Y}{e}_{0}-\frac{1}{2}{w}_{4}\overline {Y}{e}_{0}{e}_{1}-\overline {Y}+\frac{1}{2}\overline {Y}{e}_{1}-\frac{3}{8}\overline {Y}{e}_{1}^{2}\\ & -{w}_{5}\overline {X}{e}_{1}+\frac{1}{2}{w}_{5}\overline {X}{e}_{1}^{2}-{w}_{6}\overline {Z}{e}_{2}+\frac{1}{2}{w}_{6}\overline {Z}{e}_{1}{e}_{2}\end{array}\right] .$

(29)

The bias and MSE of ${\overline {y}}_{G1}^{\mathrm{*}}$ , to first order of approximation is given by:

${\rm{Bias}}( {\overline {y}}_{G1}^{\mathrm{*}} ) = \frac{3}{8}{w}_{4}\overline {Y}{V}_{x}^{2} - \frac{1}{2}{w}_{4}\overline {Y}{V}_{yx}-\frac{3}{8}\overline {Y}{V}_{x}^{2}+\frac{1}{2}{w}_{5}\overline {X}{V}_{x}^{2}+\frac{1}{2}{w}_{6}\overline {Z}{V}_{xz} ,$

By squaring and taking expectation of (29), we get the mean square error:

$MSE\left({\overline {y}}_{G1}^{\mathrm{*}}\right) = \left[\begin{array}{ll}& {w}_{6}^{2}{V}_{z}^{2}+{w}_{4}^{2}{V}_{y}^{2}+{w}_{5}^{2}{V}_{x}^{2}-2{w}_{4}^{2}R{V}_{yx}+2{w}_{4}R{V}_{yx}-2R{w}_{5}{V}_{x}^{2}\\ & -2{w}_{4}{R}^{2}{V}_{x}^{2}+{w}_{4}^{2}R{V}_{x}^{2}+2{w}_{4}R{w}_{5}{V}_{x}^{2}-2R{w}_{6}{V}_{xz}+{w}_{4}^{2}{\overline {Y}}^{2}\\ & -2{w}_{4}{\overline {Y}}^{2}+{\overline {Y}}^{2}+{R}^{2}{V}_{x}^{2}-2{w}_{4}{w}_{5}{V}_{yx}-2{w}_{4}{w}_{6}{V}_{yz}+2{w}_{5}{w}_{6}{V}_{xz}\\ & +2{w}_{4}R{w}_{6}{V}_{xz}\end{array}\right] .$

(30)

Differentiate (30) with respect to ${w}_{4}$ , ${w}_{5}$ and ${w}_{6}$ , we get the optimum values of ${w}_{4}$ , ${w}_{5}$ and ${w}_{6}$ i.e.,

${w}_{4opt} = \frac{{\overline {Y}}^{2}({V}_{x}^{2}{V}_{z}^{2}-{V}_{xz}^{2})}{\left[\begin{array}{ll}& -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+R{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & -R{V}_{x}^{2}{V}_{xz}^{2}-{\overline {Y}}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}^{2}-{V}_{yx}^{2}{V}_{z}^{2}+2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} ,$

${w}_{5opt} = \frac{\left[\begin{array}{ll}& -{R}^{3}{V}_{x}^{4}{V}_{z}^{2}-{R}^{2}{V}_{x}^{4}{V}_{z}^{2}-{R}^{3}{V}_{x}^{2}{V}_{xz}^{2}-R{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}\\ & -{\overline {Y}}^{2}{V}_{yx}{V}_{z}^{2}+R{V}_{x}^{2}{V}_{yz}^{2}+R{V}_{xz}^{2}{V}_{y}^{2}+R{V}_{yx}^{2}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{xz}{V}_{yz}-2{R}_{Vxz}{V}_{yx}{V}_{yz}\end{array}\right]}{\left[\begin{array}{ll}& -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+R{V}_{x}^{2}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & -R{V}_{x}^{2}{V}_{xz}^{2}-{\overline {Y}}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}^{2}-{V}_{yx}^{2}{V}_{z}+2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} ,$

${w}_{6opt} = \frac{{\overline {Y}}^{2}({V}_{x}^{2}{V}_{yz}-{V}_{xz}{V}_{yx})}{\left[\begin{array}{ll}& -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+R{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & -R{V}_{x}^{2}{V}_{xz}^{2}-{\overline {Y}}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}^{2}-{V}_{yx}^{2}{V}_{z}^{2}+2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} .$

Substituting the optimum values of ${w}_{4}$ , ${w}_{5}$ and ${w}_{6}$ in (30), we get the minimum mean square error of ${\overline {y}}_{G1}^{\mathrm{*}}$ , given by:

$MSE({\overline {y}}_{G1}^{\mathrm{*}}{)}_{min} = \frac{{\overline {Y}}^{2}\left[\begin{array}{ll}& {R}^{2}{V}_{x}^{4}{V}_{z}^{2}-R{V}_{x}^{4}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & +R{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{yz}^{2}+{V}_{xz}^{2}{V}_{y}^{2}+{V}_{yx}^{2}{V}_{z}^{2}-2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]}{\left[\begin{array}{ll}& {R}^{2}{V}_{x}^{4}{V}_{z}^{2}-R{V}_{x}^{4}{V}_{z}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & +R{V}_{x}^{2}{V}_{xz}^{2}+{\overline {Y}}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{yz}^{2}+{V}_{xz}^{2}{V}_{y}^{2}+{V}_{yx}^{2}{V}_{z}^{2}-2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} .$

(31)

Solving ${\overline {y}}_{G2}^{\mathrm{*}}$ , in terms of errors, we have

$({\overline {y}}_{G2}^{\mathrm{*}}-\overline {Y}) = \left[{w}_{7}\overline {Y}+{w}_{7}\overline {Y}{e}_{0}-\overline {Y}-{w}_{8}\overline {X}{e}_{1}-{w}_{9}\overline {Z}{e}_{2}\right]\left(1-{e}_{1}+{e}_{1}^{2}\right) ,$

$\left({\overline {y}}_{G2}^{\mathrm{*}}-\overline {Y}\right) = \left[\begin{array}{ll}& {w}_{7}\overline {Y}+{w}_{7}\overline {Y}{e}_{0}-\overline {Y}-{w}_{8}\overline {X}{e}_{1}-{w}_{9}\overline {z}{e}_{2}-{w}_{7}\overline {Y}{e}_{1}-{w}_{7}\overline {Y}{e}_{1}+\overline {Y}{e}_{1}\\ & +{w}_{8}\overline {X}{e}_{1}^{2}-{w}_{9}\overline {Z}{e}_{1}{e}_{2}+{w}_{7}\overline {Y}{e}_{1}^{2}-\overline {Y}{e}_{1}^{2}\end{array}\right] .$

(32)

The Bias and MSE ${\overline {y}}_{G2}^{\mathrm{*}}$ , to first order of approximation, is given by:

Bias( ${\overline {y}}_{G2}^{\mathrm{*}}$ ) = ${w}_{8}\overline {X}{V}_{x}^{2}-{w}_{9}\overline {Z}{V}_{xz}+{w}_{7}\overline {Y}{V}_{x}^{2}-\overline {Y}{V}_{x}^{2}.$

By squaring and taking expectation of (32), we get the mean square error:

$MSE\left({\overline {y}}_{G2}^{\mathrm{*}}\right) = 4R{w}_{7}{V}_{yx}-4R{V}_{x}^{2}{w}_{8}+{\overline {Y}}^{2}-2{w}_{7}{\overline {Y}}^{2}+3{R}^{2}{V}_{x}^{2}-6{w}_{7}{R}^{2}{V}_{x}^{2}+4{w}_{7}{w}_{9}R{V}_{xz}$

$-2{w}_{7}{w}_{8}{V}_{yx}-2{w}_{7}\overline {Y}{w}_{9}{V}_{yz}+2{w}_{8}{w}_{9}{V}_{xz}-4R{w}_{9}{V}_{xz}-4{w}_{7}^{2}R{V}_{yx}+3{w}_{7}^{3}{R}^{2}{V}_{x}^{2}$

$+{w}_{9}^{2}{V}_{z}^{2}+{w}_{8}^{2}{V}_{x}^{2}+4{w}_{7}{w}_{8}R{V}_{x}^{2}+{w}_{7}^{2}{\overline {Y}}^{2} .$

(33)

Differentiate (33) with respect to ${w}_{7}$ , ${w}_{8}$ and ${w}_{9}$ , we get the optimum values of ${w}_{7}$ , ${w}_{8}$ and ${w}_{9}$ i.e.,

${w}_{7opt} = \frac{({V}_{x}^{2}{V}_{z}^{2}-{V}_{xz}^{2})(-{R}^{2}{V}_{x}^{2}+{\overline {Y}}^{2})}{{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{xz}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz}+{V}_{yx}^{2}{V}_{z}^{2}} ,$

${w}_{8opt} = \frac{\left[\begin{array}{ll}& 2{\overline {Y}}^{2}R{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{R}^{2}{V}_{x}^{2}{V}_{xz}{V}_{yz}+{R}^{2}{V}_{x}^{2}{V}_{yx}{V}_{z}^{2}\\ & +{\overline {Y}}^{3}{V}_{xz}{V}_{yz}-{\overline {Y}}^{2}{V}_{yz}{V}_{z}^{2}-4\overline {Y}R{V}_{xz}{V}_{yx}{V}_{yz}+2R{V}_{yx}^{2}{V}_{z}\end{array}\right]}{\left[\begin{array}{ll}& {R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}\\ & +{\overline {Y}}^{2}{V}_{xz}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz}+{V}_{yx}^{2}{V}_{z}^{2}\end{array}\right]} ,$

${w}_{9opt} = -\frac{-\overline {Y}{R}^{2}{V}_{x}^{4}{V}_{yz}+{\overline {Y}}^{3}{V}_{x}^{2}{V}_{yz}+{R}^{2}{V}_{x}^{2}{V}_{xz}{V}_{yx}-{\overline {Y}}^{2}{V}_{xz}{V}_{yx}}{{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{\overline {Y}}^{2}{V}_{xz}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz}+{V}_{yx}^{2}{V}_{z}^{2}} .$

Substituting the optimum values of ${w}_{7}$ , ${w}_{8}$ and ${w}_{9}$ in (33), we get the minimum mean square error of ${\overline {y}}_{G2}^{\mathrm{*}}$ , given by:

$MSE({\overline {y}}_{G2}^{\mathrm{*}}{)}_{min} = \frac{({R}^{2}{V}_{x}^{2}-{\overline {Y}}^{2})({\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}+{V}_{yx}^{2}{V}_{z}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz})}{\left[\begin{array}{ll}& -{\overline {Y}}^{2}({V}_{x}^{2}{V}_{yz}^{2}-{V}_{x}^{2}{V}_{z}^{2}+{V}_{xz}^{2})+2{\overline {Y}}^{2}{V}_{xz}{V}_{yx}{V}_{yz}\\ & -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}-{V}_{yx}^{2}{V}_{z}^{2}\end{array}\right]} .$

(34)

We can generate the considered and many more estimators from (25), by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3),$ $\delta$ and $\gamma$ , given in Table 1.

Table 1. Members of the proposed generalized family of estimators.

${w}_{1}$	${w}_{2}$	${w}_{3}$	$\sigma$	$\gamma$	Estimator
1	0	0	0	0	${\overline {y}}^{\mathrm{*}}$
1	0	0	0	1	${\overline {y}}_{R}^{\mathrm{*}}$
1	0	0	1	0	${\overline {y}}_{E}^{\mathrm{*}}$
1	$d$	0	0	0	${\overline {y}}_{D}^{\mathrm{*}}$
${d}_{0}$	${d}_{1}$	0	0	0	${\overline {y}}_{Rao}^{\mathrm{*}}$
${d}_{2}$	${d}_{3}$	0	0	1	${\overline {y}}_{DR}^{\mathrm{*}}$
${d}_{4}$	${d}_{5}$	0	1	0	${\overline {y}}_{DE}^{\mathrm{*}}$
0	${d}_{6}$	${d}_{7}$	0	0	${\overline {y}}_{DD}^{\mathrm{*}}$
${d}_{8}$	${d}_{9}$	${d}_{10}$	0	0	${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$
${w}_{4}$	${w}_{5}$	${w}_{6}$	0	1	${\overline {y}}_{G1}^{\mathrm{*}}$
${w}_{7}$	${w}_{8}$	${w}_{9}$	1	0	${\overline {y}}_{G2}^{\mathrm{*}}$

| Show Table

DownLoad: CSV

5. Numerical study

Population 1. [Source: ^[11], Model Assisted Survey Sampling]

There are 124 countries (second stage units) divided into 7 continents (first stage units) according to locations. Continent ${7}^{th}$ consists of only one country therefore, we placed ${7}^{th}$ continent in ${6}^{th}$ continent.

We considered:

$y$ = 1983 import (in millions U.S dollars),

$x$ = 1983 export (in millions U.S dollars),

$z$ = 1982 gross national product (in tens of millions of U.S dollars).

The data are divided into 6 clusters, having $N =$ 6, and $n =$ 3. Also ${\sum }_{i = 1}^{N}\;{M}_{i} =$ 124, $\overline {M} =$ 20.67. In , we show cluster sizes, and population means of the study variable $\left(y\right)$ and the auxiliary variables $(x, z)$ . Tables 3 and 4 give some results.

Table 2. Cluster sizes with population means.

No. of clusters	${M}_{i}$	${m}_{i}$	${u}_{i}$	${\overline {Y}}_{i}$	${\overline {X}}_{i}$	${\overline {Z}}_{i}$
1	38	15	1.8387	2254.6	1901.1	1029.158
2	14	6	0.6774	25533.14	22083.21	25671.57
3	11	4	0.5323	3602.82	5835.455	5028.818
4	33	13	1.5968	12156.79	12438.85	7533.939
5	24	10	1.1613	34226.79	33198	16314.42
6	4	2	0.1936	26392.5	29360.5	43967.75

| Show Table

DownLoad: CSV

Table 3. Statistical computation.

$({u}_{i}{\overline {Y}}_{i}-\overline {Y}{)}^{2}$	$({u}_{i}{\overline {X}}_{i}-\overline {X}{)}^{2}$	$({u}_{i}{\overline {Z}}_{i}-\overline {Z}{)}^{2}$	$\left({u}_{i}{\overline {Y}}_{i}-\overline {Y}\right)$ $({u}_{i}{\overline {X}}_{i}-\overline {X})$	$\left({u}_{i}{\overline {X}}_{i}-\overline {X}\right)$ $({u}_{i}{\overline {Z}}_{i}-\overline {Z})$	$\left({u}_{i}{\overline {Y}}_{i}-\overline {Y}\right)$ $({u}_{i}{\overline {Z}}_{i}-\overline {Z})$
109395354.7	116233397.3	69704363.4	112762554.6	90010971.3	87323155.9
7243544.5	465731.6	51103837.3	1836722	4878593.3	19239878.4
160959577.1	124780258.7	57219949.1	141720068	84498047.6	95969259.7
23109139.4	31199313.33	3200403.1	26851243.6	9992516.3	8599916.4
632160672.8	589329821.2	75771962.7	610369671.8	211316533.3	218860811.8
90158398.4	73831543.1	2989684.1	81587582.8	14857085.7	16417829.8

| Show Table

DownLoad: CSV

Table 4. Statistical computations of variances and covariances.

${S}_{iy}^{2}$	${S}_{ix}^{2}$	${S}_{iz}^{2}$	${S}_{ixy}$	${S}_{ixz}$	${S}_{iyz}$
14634002.89	13229390.42	3667896.461	12035361.66	5676138.848	7031654.09
5199331742	3024354709	6568461403	3920918987	42963119811	5785526585
17474303.56	67544530.07	63348742.76	3322301379	62246450.49	32019714.86
510689624	689903319	440717912.5	586829812.3	522773788	447429378.1
1530618991	1588803380	408376223	1544450491	757258674.4	7559765056
1361248223	1782024492	5663081987	1557362451	3157897870	2755900798

| Show Table

DownLoad: CSV

${S}_{by}^{2} = 204605337.7, \;\;\;\;\;\;\;\;{S}_{bx}^{2} = 187168013, \;\;\;\;\;\;\;\;{S}_{bz}^{2} = 51998039.99,$

${S}_{byx} = 195025568.6, \;\;\;\;\;\;\;\;{S}_{byz} = 89282170.45, \;\;\;\;\;\;\;\;{S}_{bxz} = 83110749.52$

${V}_{y} = 0.27028, \;\;\;\;\;\;\;\;{V}_{x} = 0.25137, \;\;\;\;\;\;\;\;{V}_{z} = 0.30933,$

${V}_{yx} = 0.25723, \;\;\;\;\;\;\;\;{V}_{yz} = 0.24573, \;\;\;\;\;\;\;\;{V}_{xz} = 0.22493.$

$\overline {Y} = 14604.76564, \;\;\;\;\;\;\;\;\overline {X} = 14276.72113, \;\;\;\;\;\;\;\;\overline {Z} = 10241.22672.$

Population 2. [Source: ^[11], Model Assisted Survey Sampling]

Similarly we considered the data as mentioned in Population 1,

$y$ = 1983 import (in millions U.S dollars),

$x$ = 1981 military expenditure (in tens of millions U.S dollars),

$z$ = 1980 population (in millions).

The data are divided into 6 clusters having $N = 6$ , $n = 3$ , ${\sum }_{i = 1}^{N}\;{M}_{i} = 124, \;\overline {M} = 20.67$ .

In , we show cluster sizes, and means of the study variable $\left(y\right)$ and the auxiliary variables $(x, z)$ . Tables 6 and 7 give some computation results.

Table 5. Cluster sizes with population means.

No. of clusters	${M}_{i}$	${m}_{i}$	${u}_{i}$	${\overline {Y}}_{i}$	${\overline {X}}_{i}$	${\overline {Z}}_{i}$
1	38	15	1.8387	13.03684	418.3421	11.88421
2	14	6	0.6774	27.35	10065.21	26.1857
3	11	4	0.5323	23.13636	484.45	21.8818
4	33	13	1.5968	79.65455	3377.75	75.2424
5	24	10	1.1613	20.28333	4929.41	20.9583
6	4	2	0.1936	74.15	30676.25	70.975

| Show Table

DownLoad: CSV

Table 6. Statistical computation.

$({u}_{i}{\overline {Y}}_{i}-\overline {Y}{)}^{2}$	$({u}_{i}{\overline {X}}_{i}-\overline {X}{)}^{2}$	$({u}_{i}{\overline {Z}}_{i}-\overline {Z}{)}^{2}$	$({u}_{i}{\overline {Y}}_{i}-\overline {Y})({u}_{i}{\overline {X}}_{i}-\overline {X})$	$({u}_{i}{\overline {X}}_{i}-\overline {X})({u}_{i}{\overline {Z}}_{i}-\overline {Z})$	$({u}_{i}{\overline {Y}}_{i}-\overline {Y})({u}_{i}{\overline {Z}}_{i}-\overline {Z})$
109395354.7	11504653.07	171.5489	35430628.04	44434.42674	136842.964
7243544.5	7165707.84	296.6622	−461113.24893	7355674.750	−47336.072
160959577.1	15247937.12	544.5471	49517362.59	91132.5364	295949.600
23109139.4	1556060.53	7312.6397	6074534.22	106667.8685	416415.347
632160672.8	2494694.19	112.4143	39914472.69	−16750.3943	−268005.273
90158398.4	3214861.18	451.071	−16997583.03	−38085.13867	201365.558

| Show Table

DownLoad: CSV

Table 7. Statistical computations of variances and covariances.

${S}_{iy}^{2}$	${S}_{ix}^{2}$	${S}_{iz}^{2}$	${S}_{iyx}$	${S}_{ixz}$	${S}_{iyz}$
270.9083357	594166.8257	222.4889331	6380.484353	5806.286629	245.4143812
3906.928847	1281691972	3683.070549	2135135.281	2082979.098	3792.853077
1339.404545	461472.2727	1174.031636	13075.32182	12298.74909	1253.851727
45082.17318	53848774.81	83850.37836	1082424.717	1476243.493	43109.42511
368.9423188	52672480.78	364.7838949	117010.6551	116939.2322	366.7860145
18401.07	3453923758	16855.5025	7970505.317	7628400.308	17611.33833

| Show Table

DownLoad: CSV

${S}_{by}^{2} = 2002.428957, \;\;\;\;\;\;\;\;{S}_{bx}^{2} = 8236782.79, \;\;\;\;\;\;\;\;{S}_{bz}^{2} = 1782.2076,$

${S}_{byx} = 28451.30273, \;\;\;\;\;\;\;\;{S}_{byz} = 1888.920758, \;\;\;\;\;\;\;\;{S}_{xz} = 27939.79,$

${V}_{y} = 0.48633, \;\;\;\;\;\;\;\;{V}_{x} = 0.39654, \;\;\;\;\;\;\;\;{V}_{z} = 0.72000,$

${V}_{yx} = 0.14250, \;\;\;\;\;\;\;\;{V}_{yz} = 0.48726, \;\;\;\;\;\;\;\;{V}_{xz} = 0.16552,$

$\overline {Y} = 36.7702, \;\;\;\;\;\;\;\;\overline {X} = 4163.56, \;\;\;\;\;\;\;\;\overline {Z} = 34.8552.$

The results based on Tables 2–7 are given in and having biasses, mean square errors, and percentage relative efficiencies of the poposed and exisitng estimators w.r.t ${\overline {y}}_{0}^{\mathrm{*}}.$ Tables 8 and 9 show that the proposed estimators perform well as compared to the existing estimators considered here.

Table 8. Biases of different estimators in both data sets.

Estimator	Population 1	Population 2
${\overline {y}}_{0}^{\mathrm{}}$ , ${\overline {y}}_{D}^{\mathrm{}}$ , ${\overline {y}}_{DD}^{\mathrm{*}}$	0	0
${\overline {y}}_{R}^{\mathrm{*}}$	−85.58393	9.341102
${\overline {y}}_{E}^{\mathrm{*}}$	−501.692	2.847944
${\overline {y}}_{Rao}^{\mathrm{*}}$	−102.2916	−11.14854
${\overline {y}}_{DR}^{\mathrm{*}}$	62517.58	−5.729291
${\overline {y}}_{DE}^{\mathrm{*}}$	−1040.227	−4.254352
${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$	−8687.674	0.8665627
${\overline {y}}_{G1}^{\mathrm{*}}$	−911.2082	42.56601
${\overline {y}}_{G2}^{\mathrm{*}}$	4097419	−660.0231

| Show Table

DownLoad: CSV

Table 9.

$\mathrm{M}\mathrm{S}\mathrm{E}$ and

$\mathrm{P}\mathrm{R}\mathrm{E}$ of different estimators w.r.t

${\overline {y}}_{0}^{\mathrm{*}}$ .

	Population 1		Population 2
Estimators	MSE	PRE	MSE	PRE
${\overline {y}}_{0}^{\mathrm{*}}$	57649726.19	100	657.541	100
${\overline {y}}_{R}^{\mathrm{*}}$	1532261.42	3762.39	808.353	81.3433
${\overline {y}}_{E}^{\mathrm{*}}$	16186983.44	356.149	598.912	109.789
${\overline {y}}_{D}^{\mathrm{*}}$	1503085.83	3835.42	588.307	111.768
${\overline {y}}_{Rao}^{\mathrm{*}}$	1492567.93	3862.45	409.935	160.401
${\overline {y}}_{DR}^{\mathrm{*}}$	1489069.32	3871.53	341.831	192.358
${\overline {y}}_{DE}^{\mathrm{*}}$	1189664.93	4845.88	366.982	179.176
${\overline {y}}_{DD}^{\mathrm{*}}$	1025752.55	5620.24	208.189	315.839
${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$	1020843.33	5647.26	180.409	364.472
${\overline {y}}_{G1}^{\mathrm{*}}$	1019205.50	5656.34	165.866	396.429
${\overline {y}}_{G2}^{\mathrm{*}}$	747118.42	7716.28	159.646	411.875

| Show Table

DownLoad: CSV

The following expression is used to obtain the Percent Relative Efficiency (PRE), i.e.,

$PRE = \frac{MSE\left({\overline {y}}_{0}^{\mathrm{*}}\right)}{MSE\left({\overline {y}}_{i}^{\mathrm{*}}\right)}\times 100 ,$

where $i = 0, R, E, D, Rao, DR, DE, DD, DD\left(R\right), {G}_{1}, {G}_{2}.$

6. Discussion

As mentioned above, we used two real data sets to obtain the biases, MSEs or variances and PREs of all estimators under two-stage sampling scheme when using two auxiliary variables. In Tables 2–4 and Tables 5–7, we present the summary statistic of both population. From and , we observed that the proposed class of estimators ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ are more precise than the existing estimators ${\overline {y}}_{0}^{\mathrm{*}}$ , ${\overline {y}}_{R}^{\mathrm{*}}$ , ${\overline {y}}_{E}^{\mathrm{*}}$ , ${\overline {y}}_{D}^{\mathrm{*}}$ , ${\overline {y}}_{Rao}^{\mathrm{*}}$ , ${\overline {y}}_{DR}^{\mathrm{*}}$ , ${\overline {y}}_{DE}^{\mathrm{*}}$ , ${\overline {y}}_{DD}^{\mathrm{*}}$ , ${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$ in terms of MSEs and PREs. It is clear that the proposed improved generalized class of estimators, i.e., performs better than the estimators. As we increase the sample size the mean square error values decreases, and percentage relative efficiency give best results, which are the expected results.

7. Conclusions

In this manuscript, we proposed a generalized class of estimators using two auxiliary variables under two-stage sampling for estimating the finite population mean. In addition, some well-known estimators of population mean like traditional unbiased estimator, usual ratio, exponential ratio type, traditional difference type, Rao difference type, difference-in- ratio type, difference-in-exponential ratio type, difference-in-difference, difference-difference ratio type estimator are created to be members of our suggested improved generalized class of estimators. Expression for the biases and mean squared error have been generated up to the first order of approximation. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3)$ , $\delta$ and $\gamma$ . Both generalized class of estimators ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ perform better as compared to all other considered estimators, although ${\overline {y}}_{G2}^{\mathrm{*}}$ is the best. In Population 2, the performance of ratio estimator $\left({\overline {y}}_{R}^{\mathrm{*}}\right)$ is weak. The gain in Population 1 is more as compared to Population 2.

Acknowledgments

The authors are thankful to the Editor-in-Chief and two anonymous referees for their careful reading of the paper and valuable comments which leads to a significant improvement in article.

Conflict of interest

The authors declare no conflict of interest.

Acknowledgments

Research reported here was supported by the National Institute of Allergy and Infectious Diseases (NIAID) of the National Institutes of Health (NIH) under 1R61AI169207-01-001. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Conflict of interest

The authors declare no conflict of interest.

Author contributions

Conceptualization, R.H and B.W; writing—original draft preparation, R.H, B.W and H.A; writing—review and editing, R.H, B.W, H.A, S.B. and R.N.C.; supervision and funding acquisition, R.N.C. All authors have read and agreed to the published version of the manuscript.

References

[1]	Shaw-Taylor L (2020) An introduction to the history of infectious diseases, epidemics and the early phases of the long-run decline in mortality. Econ Hist Rev 73: E1-e19. https://doi.org/10.1111/ehr.13019
[2]	Mehandru S, Merad M (2022) Pathological sequelae of long-haul COVID. Nat Immunol 23: 194-202. https://doi.org/10.1038/s41590-021-01104-y
[3]	Zhang Q, Wang Y, Qi C, et al. (2020) Clinical trial analysis of 2019-nCoV therapy registered in China. J Med Virol 92: 540-545. https://doi.org/10.1002/jmv.25733
[4]	Andersen KG, Rambaut A, Lipkin WI, et al. (2020) The proximal origin of SARS-CoV-2. Nat Med 26: 450-452. https://doi.org/10.1038/s41591-020-0820-9
[5]	Huang C, Wang Y, Li X, et al. (2020) Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet 395: 497-506. https://doi.org/10.1016/S0140-6736(20)30183-5
[6]	Wu F, Zhao S, Yu B, et al. (2020) A new coronavirus associated with human respiratory disease in China. Nature 579: 265-269. https://doi.org/10.1038/s41586-020-2008-3
[7]	Zhu N, Zhang D, Wang W, et al. (2020) A Novel Coronavirus from Patients with Pneumonia in China, 2019. New Engl J Med 382: 727-733. https://doi.org/10.1056/NEJMoa2001017
[8]	Dong E, Du H, Gardner L (2020) An interactive web-based dashboard to track COVID-19 in real time. Lancet Infect Dis 20: 533-534. https://doi.org/10.1016/S1473-3099(20)30120-1
[9]	WHO coronavirus (COVID-19) dashboard \| WHO coronavirus (COVID-19) dashboard with vaccination data [internet] (2022). [cited 2022 OCT 11]. Available from: https://covid19.who.int/.
[10]	CDC coronavirus COVID-19 Data tracker I CDC coronavirus COVID-19 Data tracker with total deaths [internet] (2022). [cited 2022 OCT 11]. Available from: https://covid.cdc.gov/covid-data-tracker/#datatracker-home.
[11]	Wu X, Liu X, Zhou Y, et al. (2021) 3-month, 6-month, 9-month, and 12-month respiratory outcomes in patients following COVID-19-related hospitalisation: a prospective study. Lancet Respir Med 9: 747-754. https://doi.org/10.1016/S2213-2600(21)00174-0
[12]	Montefusco L, Ben Nasr M, D'Addio F, et al. (2021) Acute and long-term disruption of glycometabolic control after SARS-CoV-2 infection. Nat Metab 3: 774-785. https://doi.org/10.1038/s42255-021-00407-6
[13]	Bridges JP, Vladar EK, Huang H, et al. (2022) Respiratory epithelial cell responses to SARS-CoV-2 in COVID-19. Thorax 77: 203-209. https://doi.org/10.1136/thoraxjnl-2021-217561
[14]	Jamil S, Mark N, Carlos G, et al. (2020) Diagnosis and management of COVID-19 disease. Am J Respir Crit Care Med 201: P19-P20. https://doi.org/10.1164/rccm.2020C1
[15]	Struyf T, Deeks JJ, Dinnes J, et al. (2021) Signs and symptoms to determine if a patient presenting in primary care or hospital outpatient settings has COVID-19. Cochrane Database Syst Rev 7: CD013665. https://doi.org/10.1002/14651858.CD013665.pub2
[16]	Soy M, Keser G, Atagündüz P, et al. (2020) Cytokine storm in COVID-19: pathogenesis and overview of anti-inflammatory agents used in treatment. Clin Rheumatol 39: 2085-2094. https://doi.org/10.1007/s10067-020-05190-5
[17]	Ye Q, Wang B, Mao J (2020) The pathogenesis and treatment of the ‘Cytokine Storm’ in COVID-19. J Infect 80: 607-613. https://doi.org/10.1016/j.jinf.2020.03.037
[18]	Hu B, Huang S, Yin L (2021) The cytokine storm and COVID-19. J Med Virol 93: 250-256. https://doi.org/10.1002/jmv.26232
[19]	Zanza C, Romenskaya T, Manetti AC, et al. (2022) Cytokine storm in COVID-19: immunopathogenesis and therapy. Medicina (Kaunas) 58: 144. https://doi.org/10.3390/medicina58020144
[20]	Sarma A, Christenson SA, Byrne A, et al. (2021) Tracheal aspirate RNA sequencing identifies distinct immunological features of COVID-19 ARDS. Nat Commun 12: 5152. https://doi.org/10.1038/s41467-021-25040-5
[21]	Sinha P, Matthay MA, Calfee CS (2020) Is a “Cytokine Storm” relevant to COVID-19?. JAMA Intern Med 180: 1152-1154. https://doi.org/10.1001/jamainternmed.2020.3313
[22]	Fogarty H, Townsend L, Morrin H, et al. (2021) Persistent endotheliopathy in the pathogenesis of long COVID syndrome. J Thromb Haemost 19: 2546-2553. https://doi.org/10.1111/jth.15490
[23]	Lerum TV, Aaløkken TM, Brønstad E, et al. (2021) Dyspnoea, lung function and CT findings 3 months after hospital admission for COVID-19. Eur Respir J 57: 2003448. https://doi.org/10.1183/13993003.03448-2020
[24]	Montani D, Savale L, Noel N, et al. (2022) Post-acute COVID-19 syndrome. Eur Respir Rev 31: 210185. https://doi.org/10.1183/16000617.0185-2021
[25]	McGroder CF, Zhang D, Choudhury MA, et al. (2021) Pulmonary fibrosis 4 months after COVID-19 is associated with severity of illness and blood leucocyte telomere length. Thorax 76: 1242-1245. https://doi.org/10.1136/thoraxjnl-2021-217031
[26]	Maccio U, Zinkernagel AS, Schuepbach R, et al. (2022) Long-Term Persisting SARS-CoV-2 RNA and Pathological Findings: Lessons Learnt From a Series of 35 COVID-19 Autopsies. Front Med 9: 778489. https://doi.org/10.3389/fmed.2022.778489
[27]	Chun HJ, Coutavas E, Pine A, et al. (2021) Immuno-fibrotic drivers of impaired lung function in post-acute sequelae of SARS-CoV-2 infection (PASC). medRxiv . https://doi.org/10.1101/2021.01.31.21250870
[28]	Long Q, Li J, Hu X, et al. (2021) Follow-Ups on persistent symptoms and pulmonary function among post-acute COVID-19 patients: a systematic review and meta-analysis. Front Med (Lausanne) 8: 702635. https://doi.org/10.3389/fmed.2021.702635
[29]	Cares-Marambio K, Montenegro-Jiménez Y, Torres-Castro R, et al. (2021) Prevalence of potential respiratory symptoms in survivors of hospital admission after coronavirus disease 2019 (COVID-19): A systematic review and meta-analysis. Chron Respir Dis 18: 14799731211002240. https://doi.org/10.1177/14799731211002240
[30]	Sigfrid L, Drake TM, Pauley E, et al. (2021) Long Covid in adults discharged from UK hospitals after Covid-19: A prospective, multicentre cohort study using the ISARIC WHO Clinical Characterisation Protocol. Lancet Reg Health Eur 8: 100186. https://doi.org/10.1016/j.lanepe.2021.100186
[31]	Boehmer TK, Kompaniyets L, Lavery AM, et al. (2021) Association between COVID-19 and myocarditis using hospital-based administrative data—United States, March 2020–January 2021. MMWR Morb Mortal Wekly Rep 70: 1228-1232. https://doi.org/10.15585/mmwr.mm7035e5
[32]	Szarpak L, Pruc M, Filipiak KJ, et al. (2022) Myocarditis: A complication of COVID-19 and long-COVID-19 syndrome as a serious threat in modern cardiology. Cardiol J 29: 178-179. https://doi.org/10.5603/CJ.a2021.0155
[33]	Xie Y, Xu E, Bowe B, et al. (2022) Long-term cardiovascular outcomes of COVID-19. Nat Med 28: 583-590. https://doi.org/10.1038/s41591-022-01689-3
[34]	Soares MN, Eggelbusch M, Naddaf E, et al. (2022) Skeletal muscle alterations in patients with acute Covid-19 and post-acute sequelae of Covid-19. J Cachexia Sarcopenia Muscle 13: 11-22. https://doi.org/10.1002/jcsm.12896
[35]	Blomberg B, Mohn KG-I, Brokstad KA, et al. (2021) Long COVID in a prospective cohort of home-isolated patients. Nat Med 27: 1607-1613. https://doi.org/10.1038/s41591-021-01433-3
[36]	Liu YH, Chen Y, Wang QH, et al. (2022) One-year trajectory of cognitive changes in older survivors of COVID-19 in Wuhan, China. JAMA Neurol 79: 509. https://doi.org/10.1001/jamaneurol.2022.0461
[37]	Najt P, Richards HL, Fortune DG (2021) Brain imaging in patients with COVID-19: A systematic review. Brain Behav Immun Health 16: 100290. https://doi.org/10.1016/j.bbih.2021.100290
[38]	Douaud G, Lee S, Alfaro-Almagro F, et al. (2022) SARS-CoV-2 is associated with changes in brain structure in UK Biobank. Nature 604: 697-707. https://doi.org/10.1038/s41586-022-04569-5
[39]	Loosen SH, Jensen BEO, Tanislav C, et al. (2022) Obesity and lipid metabolism disorders determine the risk for development of long COVID syndrome: a cross-sectional study from 50,402 COVID-19 patients. Infection . https://doi.org/10.1007/s15010-022-01784-0
[40]	Scherer PE, Kirwan JP, Rosen CJ (2022) Post-acute sequelae of COVID-19: A metabolic perspective. eLife 11. https://doi.org/10.7554/eLife.78200
[41]	Paneni F, Patrono C (2022) Increased risk of incident diabetes in patients with long COVID. Eur Heart J 43: 2094-2095. https://doi.org/10.1093/eurheartj/ehac196
[42]	Xie Y, Al-Aly Z (2022) Risks and burdens of incident diabetes in long COVID: a cohort study. Lancet Diabetes Endocrinol 10: 311-321. https://doi.org/10.1016/S2213-8587(22)00044-4
[43]	Mantovani A, Morrone MC, Patrono C, et al. (2022) Long Covid: where we stand and challenges ahead. Cell Death Differ 29: 1891-1900. https://doi.org/10.1038/s41418-022-01052-6
[44]	Su S, Wong G, Shi W, et al. (2016) Epidemiology, Genetic Recombination, and Pathogenesis of Coronaviruses. Trends Microbiol 24: 490-502. https://doi.org/10.1016/j.tim.2016.03.003
[45]	Verma J, Subbarao N (2021) A comparative study of human betacoronavirus spike proteins: structure, function and therapeutics. Arch Virol 166: 697-714. https://doi.org/10.1007/s00705-021-04961-y
[46]	Lu R, Zhao X, Li J, et al. (2020) Genomic characterisation and epidemiology of 2019 novel coronavirus: implications for virus origins and receptor binding. Lancet 395: 565-574. https://doi.org/10.1016/S0140-6736(20)30251-8
[47]	Drosten C, Günther S, Preiser W, et al. (2003) Identification of a Novel Coronavirus in Patients with Severe Acute Respiratory Syndrome. New Engl J Med 348: 1967-1976. https://doi.org/10.1056/NEJMoa030747
[48]	Zaki AM, Van Boheemen S, Bestebroer TM, et al. (2012) Isolation of a Novel Coronavirus from a Man with Pneumonia in Saudi Arabia. New Engl J Med 367: 1814-1820. https://doi.org/10.1056/NEJMoa1211721
[49]	Tracking SARS-CoV-2 Variants [internet] [cited 2022 OCT 11] (2022). Available from: https://www.who.int/activities/tracking-SARS-CoV-2-variants
[50]	Gowrisankar A, Priyanka TMC, Banerjee S (2022) Omicron: a mysterious variant of concern. The Eur Phys J Plus 137. https://doi.org/10.1140/epjp/s13360-021-02321-y
[51]	Mallapaty S (2022) COVID-19: How Omicron overtook Delta in three charts. Nature . https://doi.org/10.1038/d41586-022-00632-3
[52]	Chen Q, Zhang J, Wang P, et al. (2022) The mechanisms of immune response and evasion by the main SARS-CoV-2 variants. iScience : 105044. https://doi.org/10.1016/j.isci.2022.105044
[53]	SARS-CoV-2 Variant Classifications and Definitions from CDC [internet] [cited 2022 OCT 11] (2022). Available from: https://www.cdc.gov/coronavirus/2019-ncov/variants/variant-classifications.html.
[54]	Volz E, Mishra S, Chand M, et al. (2021) Assessing transmissibility of SARS-CoV-2 lineage B.1.1.7 in England. Nature 593: 266-269. https://doi.org/10.1038/s41586-021-03470-x
[55]	Tegally H, Wilkinson E, Giovanetti M, et al. (2021) Detection of a SARS-CoV-2 variant of concern in South Africa. Nature 592: 438-443. https://doi.org/10.1038/s41586-021-03402-9
[56]	Imai M, Halfmann PJ, Yamayoshi S, et al. (2021) Characterization of a new SARS-CoV-2 variant that emerged in Brazil. Proc Natl Acad Sci USA 118: e2106535118. https://doi.org/10.1073/pnas.2106535118
[57]	Li B, Deng A, Li K, et al. (2022) Viral infection and transmission in a large, well-traced outbreak caused by the SARS-CoV-2 Delta variant. Nat Commun 13. https://doi.org/10.1038/s41467-022-28089-y
[58]	Washington State Department of Health, Monitoring COVID-19 variants [internet] [cited 2022 OCT 11] (2022). Available from: https://doh.wa.gov/emergencies/covid-19/variants.
[59]	Annavajhala MK, Mohri H, Wang P, et al. (2021) Emergence and Expansion of the SARS-CoV-2 Variant B.1.526 Identified in New York. Cold Spring Harbor Laboratory . https://doi.org/10.1101/2021.02.23.21252259
[60]	Menni C, Klaser K, May A, et al. (2021) Vaccine side-effects and SARS-CoV-2 infection after vaccination in users of the COVID Symptom Study app in the UK: a prospective observational study. Lancet Infect Dis 21: 939-949. https://doi.org/10.1016/S1473-3099(21)00224-3
[61]	Karim SSA, Karim QA (2021) Omicron SARS-CoV-2 variant: a new chapter in the COVID-19 pandemic. Lancet 398: 2126-2128. https://doi.org/10.1016/S0140-6736(21)02758-6
[62]	Tian F, Tong B, Sun L, et al. (2021) N501Y mutation of spike protein in SARS-CoV-2 strengthens its binding to receptor ACE2. eLife 10. https://doi.org/10.7554/eLife.69091
[63]	Kidd M, Richter A, Best A, et al. (2021) S-variant SARS-CoV-2 lineage B1.1.7 is associated with significantly higher viral load in samples tested by taqpath polymerase chain reaction. J Infect Dis 223: 1666-1670. https://doi.org/10.1093/infdis/jiab082
[64]	Davies NG, Jarvis CI, Edmunds WJ, et al. (2021) Increased mortality in community-tested cases of SARS-CoV-2 lineage B.1.1.7. Cold Spring Harbor Laboratory . https://doi.org/10.1101/2021.02.01.21250959
[65]	Davies NG, Abbott S, Barnard RC, et al. (2021) Estimated transmissibility and impact of SARS-CoV-2 lineage B.1.1.7 in England. Science 372. https://doi.org/10.1126/science.abg3055
[66]	Challen R, Brooks-Pollock E, Read JM, et al. (2021) Risk of mortality in patients infected with SARS-CoV-2 variant of concern 202012/1: matched cohort study. BMJ 372: n579. https://doi.org/10.1136/bmj.n579
[67]	Alenquer M, Ferreira F, Lousa D, et al. (2021) Signatures in SARS-CoV-2 spike protein conferring escape to neutralizing antibodies. PLOS Pathog 17: e1009772. https://doi.org/10.1371/journal.ppat.1009772
[68]	Zhou D, Dejnirattisai W, Supasa P, et al. (2021) Evidence of escape of SARS-CoV-2 variant B.1.351 from natural and vaccine-induced sera. Cell 184: 2348-2361.e2346. https://doi.org/10.1016/j.cell.2021.02.037
[69]	Li Q, Nie J, Wu J, et al. (2021) SARS-CoV-2 501Y.V2 variants lack higher infectivity but do have immune escape. Cell 184: 2362-2371.e2369. https://doi.org/10.1016/j.cell.2021.02.042
[70]	O'Toole A, Hill V, Pybus OG, et al. (2021) Tracking the international spread of SARS-CoV-2 lineages B.1.1.7 and B.1.351/501Y-V2. Wellcome Open Res 6: 121. https://doi.org/10.12688/wellcomeopenres.16661.1
[71]	Faria NR, Mellan TA, Whittaker C, et al. (2021) Genomics and epidemiology of the P.1 SARS-CoV-2 lineage in Manaus, Brazil. Science 372: 815-821. https://doi.org/10.1126/science.abh2644
[72]	Collier DA, Ferreira IATM, Kotagiri P, et al. (2021) Age-related immune response heterogeneity to SARS-CoV-2 vaccine BNT162b2. Nature 596: 417-422. https://doi.org/10.1038/s41586-021-03739-1
[73]	Wheatley AK, Juno JA (2022) COVID-19 vaccines in the age of the delta variant. Lancet Infect Dis 22: 429-430. https://doi.org/10.1016/S1473-3099(21)00688-5
[74]	Baral P, Bhattarai N, Hossen ML, et al. (2021) Mutation-induced changes in the receptor-binding interface of the SARS-CoV-2 Delta variant B.1.617.2 and implications for immune evasion. Biochem Biophys Res Commun 574: 14-19. https://doi.org/10.1016/j.bbrc.2021.08.036
[75]	Grabowski F, Kochańczyk M, Lipniacki T (2022) The spread of SARS-CoV-2 variant Omicron with a doubling time of 2.0–3.3 days can be explained by immune evasion. Viruses 14: 294. https://doi.org/10.3390/v14020294
[76]	Callaway E, Ledford H (2021) How bad is Omicron? What scientists know so far. Nature 600: 197-199. https://www.nature.com/articles/d41586-021-03614-z
[77]	Xia S, Wang L, Zhu Y, et al. (2022) Origin, virological features, immune evasion and intervention of SARS-CoV-2 Omicron sublineages. Signal Transduction Target Ther 7: 241. https://doi.org/10.1038/s41392-022-01105-9
[78]	Viana R, Moyo S, Amoako DG, et al. (2022) Rapid epidemic expansion of the SARS-CoV-2 Omicron variant in southern Africa. Nature 603: 679-686. https://doi.org/10.1038/s41586-022-04411-y
[79]	Yamasoba D, Kimura I, Nasser H, et al. (2022) Virological characteristics of the SARS-CoV-2 Omicron BA.2 spike. Cell 185: 2103-2115.e2119. https://doi.org/10.1016/j.cell.2022.04.035
[80]	Weekly epidemiological update on COVID-19-28 September 2022 (2022). Available from: https://www.who.int/publications/m/item/weekly-epidemiological-update-on-covid-19---28-september-2022
[81]	CDC coronavirus COVID-19 Data tracker I CDC coronavirus COVID-19 Variants and Genomic Surveillance US VOC [internet]. [cited 2022 OCT 11] (2022). Available from: https://covid.cdc.gov/covid-data-tracker/#variant-proportions.
[82]	Shang J, Ye G, Shi K, et al. (2020) Structural basis of receptor recognition by SARS-CoV-2. Nature 581: 221-224. https://doi.org/10.1038/s41586-020-2179-y
[83]	Wrapp D, Wang N, Corbett KS, et al. (2020) Cryo-EM structure of the 2019-nCoV spike in the prefusion conformation. Science 367: 1260-1263. https://doi.org/10.1126/science.abb2507
[84]	Conceicao C, Thakur N, Human S, et al. (2020) The SARS-CoV-2 Spike protein has a broad tropism for mammalian ACE2 proteins. PLOS Biol 18: e3001016. https://doi.org/10.1371/journal.pbio.3001016
[85]	Lu L, Liu X, Jin R, et al. (2020) Potential Roles of the Renin-Angiotensin System in the Pathogenesis and Treatment of COVID-19. BioMed Res Int 2020: 1-7. https://doi.org/10.1155/2020/5204348
[86]	Jackson CB, Farzan M, Chen B, et al. (2022) Mechanisms of SARS-CoV-2 entry into cells. Nat Rev Mol Cell Biol 23: 3-20. https://doi.org/10.1038/s41580-021-00418-x
[87]	Luan B, Wang H, Huynh T (2021) Enhanced binding of the N501Y-mutated SARS-CoV-2 spike protein to the human ACE2 receptor: insights from molecular dynamics simulations. FEBS Lett 595: 1454-1461. https://doi.org/10.1002/1873-3468.14076
[88]	Alipoor SD, Mirsaeidi M (2022) SARS-CoV-2 cell entry beyond the ACE2 receptor. Mol Biol Rep 49: 10715-10727. https://doi.org/10.1007/s11033-022-07700-x
[89]	Russell MW, Moldoveanu Z, Ogra PL, et al. (2020) Mucosal immunity in COVID-19: A neglected but critical aspect of SARS-CoV-2 infection. Front Immunol 11: 611337. https://doi.org/10.3389/fimmu.2020.611337
[90]	Alfi O, Yakirevitch A, Wald O, et al. (2021) Human nasal and lung tissues infected ex vivo with SARS-CoV-2 provide insights into differential tissue-specific and virus-specific innate immune responses in the upper and lower respiratory tract. J Virol 95: e0013021. https://doi.org/10.1128/JVI.00130-21
[91]	Ahn JH, Kim J, Hong SP, et al. (2021) Nasal ciliated cells are primary targets for SARS-CoV-2 replication in the early stage of COVID-19. J Clin Invest 131. https://doi.org/10.1172/JCI148517
[92]	Zhang S, Wang L, Cheng G (2022) The battle between host and SARS-CoV-2: Innate immunity and viral evasion strategies. Mol Ther 30: 1869-1884. https://doi.org/10.1016/j.ymthe.2022.02.014
[93]	Zindel J, Kubes P (2020) DAMPs, PAMPs, and LAMPs in Immunity and Sterile Inflammation. Annu Rev Pathol 15: 493-518. https://doi.org/10.1146/annurev-pathmechdis-012419-032847
[94]	Lowery SA, Sariol A, Perlman S (2021) Innate immune and inflammatory responses to SARS-CoV-2: Implications for COVID-19. Cell Host Microbe 29: 1052-1062. https://doi.org/10.1016/j.chom.2021.05.004
[95]	Martin-Sancho L, Lewinski MK, Pache L, et al. (2021) Functional landscape of SARS-CoV-2 cellular restriction. Mol Cell 81: 2656-2668.e2658. https://doi.org/10.1016/j.molcel.2021.04.008
[96]	Xu D, Biswal M, Neal A, et al. (2021) Review Devil's tools: SARS-CoV-2 antagonists against innate immunity. Curr Res Virol Sci 2: 100013. https://doi.org/10.1016/j.crviro.2021.100013
[97]	Han L, Zhuang MW, Deng J, et al. (2021) SARS-CoV-2 ORF9b antagonizes type I and III interferons by targeting multiple components of the RIG-I/MDA-5–MAVS, TLR3–TRIF, and cGAS–STING signaling pathways. J Med Virol 93: 5376-5389. https://doi.org/10.1002/jmv.27050
[98]	Zheng Y, Zhuang M-W, Han L, et al. (2020) Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) membrane (M) protein inhibits type I and III interferon production by targeting RIG-I/MDA-5 signaling. Signal Transduction Targe Ther 5: 299. https://doi.org/10.1038/s41392-020-00438-7
[99]	Tay MZ, Poh CM, Rénia L, et al. (2020) The trinity of COVID-19: immunity, inflammation and intervention. Nat Rev Immunol 20: 363-374. https://doi.org/10.1038/s41577-020-0311-8
[100]	Cabaro S, D'Esposito V, Di Matola T, et al. (2021) Cytokine signature and COVID-19 prediction models in the two waves of pandemics. Sci Rep 11: 20793. https://doi.org/10.1038/s41598-021-00190-0
[101]	Xu Z, Shi L, Wang Y, et al. (2020) Pathological findings of COVID-19 associated with acute respiratory distress syndrome. Lancet Respir Med 8: 420-422. https://doi.org/10.1016/S2213-2600(20)30076-X
[102]	Liao M, Liu Y, Yuan J, et al. (2020) Single-cell landscape of bronchoalveolar immune cells in patients with COVID-19. Nat Med 26: 842-844. https://doi.org/10.1038/s41591-020-0901-9
[103]	Clinical Characteristics of Covid-19 in China. New Engl J Med (2020) 382: 1859-1862. https://doi.org/10.1056/NEJMc2005203
[104]	Wang Y, He Y, Tong J, et al. (2020) Characterization of an asymptomatic cohort of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infected individuals outside of Wuhan, China. Clin Infect Dis 71: 2132-2138. https://doi.org/10.1093/cid/ciaa629
[105]	Sallusto F, Lanzavecchia A, Araki K, et al. (2010) From vaccines to memory and back. Immunity 33: 451-463. https://doi.org/10.1016/j.immuni.2010.10.008
[106]	Guo X, Guo Z, Duan C, et al. (2020) Long-term persistence of IgG antibodies in SARS-CoV infected healthcare workers. Cold Spring Harbor Laboratory . https://doi.org/10.1101/2020.02.12.20021386
[107]	Li CK-F, Wu H, Yan H, et al. (2008) T cell responses to whole SARS coronavirus in humans. J Immunol 181: 5490-5500. https://doi.org/10.4049/jimmunol.181.8.5490
[108]	Lam JH, Smith FL, Baumgarth N (2020) B cell activation and response regulation during viral infections. Viral Immunol 33: 294-306. https://doi.org/10.1089/vim.2019.0207
[109]	Elsner RA, Shlomchik MJ (2020) Germinal center and extrafollicular B cell responses in vaccination, immunity, and autoimmunity. Immunity 53: 1136-1150. https://doi.org/10.1016/j.immuni.2020.11.006
[110]	Allen CDC, Okada T, Cyster JG (2007) Germinal-center organization and cellular dynamics. Immunity 27: 190-202. https://doi.org/10.1016/j.immuni.2007.07.009
[111]	Palm AKE, Henry C (2019) Remembrance of things past: Long-term B cell memory after infection and vaccination. Front Immunol 10. https://doi.org/10.3389/fimmu.2019.01787
[112]	Traggiai E, Becker S, Subbarao K, et al. (2004) An efficient method to make human monoclonal antibodies from memory B cells: potent neutralization of SARS coronavirus. Nat Med 10: 871-875. https://doi.org/10.1038/nm1080
[113]	Song G, He W-T, Callaghan S, et al. (2021) Cross-reactive serum and memory B-cell responses to spike protein in SARS-CoV-2 and endemic coronavirus infection. Nat Commun 12. https://doi.org/10.1038/s41467-021-23074-3
[114]	Dugan HL, Stamper CT, Li L, et al. (2021) Profiling B cell immunodominance after SARS-CoV-2 infection reveals antibody evolution to non-neutralizing viral targets. Immunity 54: 1290-1303.e1297. https://doi.org/10.1016/j.immuni.2021.05.001
[115]	Suthar MS, Zimmerman MG, Kauffman RC, et al. (2020) Rapid Generation of Neutralizing Antibody Responses in COVID-19 Patients. Cell Rep Med 1: 100040. https://doi.org/10.1016/j.xcrm.2020.100040
[116]	Piccoli L, Park YJ, Tortorici MA, et al. (2020) Mapping Neutralizing and Immunodominant Sites on the SARS-CoV-2 Spike Receptor-Binding Domain by Structure-Guided High-Resolution Serology. Cell 183: 1024-1042.e1021. https://doi.org/10.1016/j.cell.2020.09.037
[117]	Premkumar L, Segovia-Chumbez B, Jadi R, et al. (2020) The receptor binding domain of the viral spike protein is an immunodominant and highly specific target of antibodies in SARS-CoV-2 patients. Sci Immunol 5. https://doi.org/10.1126/sciimmunol.abc8413
[118]	Robbiani DF, Gaebler C, Muecksch F, et al. (2020) Convergent antibody responses to SARS-CoV-2 in convalescent individuals. Nature 584: 437-442. https://doi.org/10.1038/s41586-020-2456-9
[119]	Sakharkar M, Rappazzo CG, Wieland-Alter WF, et al. (2021) Prolonged evolution of the human B cell response to SARS-CoV-2 infection. Sci Immunol 6: eabg6916. https://doi.org/10.1126/sciimmunol.abg6916
[120]	Carvalho T, Krammer F, Iwasaki A (2021) The first 12 months of COVID-19: a timeline of immunological insights. Nat Rev Immunol 21: 245-256. https://doi.org/10.1038/s41577-021-00522-1
[121]	Liu X, Wang J, Xu X, et al. (2020) Patterns of IgG and IgM antibody response in COVID-19 patients. Emerg Microb Infect 9: 1269-1274. https://doi.org/10.1080/22221751.2020.1773324
[122]	Long QX, Liu BZ, Deng HJ, et al. (2020) Antibody responses to SARS-CoV-2 in patients with COVID-19. Nat Med 26: 845-848. https://doi.org/10.1038/s41591-020-0897-1
[123]	Dan JM, Mateus J, Kato Y, et al. (2021) Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection. Science 371: eabf4063. https://doi.org/10.1126/science.abf4063
[124]	Kuri-Cervantes L, Pampena MB, Meng W, et al. (2020) Comprehensive mapping of immune perturbations associated with severe COVID-19. Sci Immunol 5: eabd7114. https://doi.org/10.1126/sciimmunol.abd7114
[125]	Shenoy S (2021) SARS-CoV-2 (COVID-19), viral load and clinical outcomes; lessons learned one year into the pandemic: A systematic review. World J Crit Care Med 10: 132-150. https://doi.org/10.5492/wjccm.v10.i4.132
[126]	Sterlin D, Mathian A, Miyara M, et al. (2021) IgA dominates the early neutralizing antibody response to SARS-CoV-2. Sci Transl Med 13. https://doi.org/10.1126/scitranslmed.abd2223
[127]	Zhao J, Zhao J, Ashutosh, et al. (2016) Airway memory CD4 + T cells mediate protective immunity against emerging respiratory coronaviruses. Immunity 44: 1379-1391. https://doi.org/10.1016/j.immuni.2016.05.006
[128]	McMahan K, Yu J, Mercado NB, et al. (2021) Correlates of protection against SARS-CoV-2 in rhesus macaques. Nature 590: 630-634. https://doi.org/10.1038/s41586-020-03041-6
[129]	Zhao J, Zhao J, Perlman S (2010) T cell responses are required for protection from clinical disease and for virus clearance in severe acute respiratory syndrome coronavirus-infected mice. J Virol 84: 9318-9325. https://doi.org/10.1128/JVI.01049-10
[130]	Wang Z, Yang X, Zhong J, et al. (2021) Exposure to SARS-CoV-2 generates T-cell memory in the absence of a detectable viral infection. Nat Commun 12: 1724. https://doi.org/10.1038/s41467-021-22036-z
[131]	Le Bert N, Clapham HE, Tan AT, et al. (2021) Highly functional virus-specific cellular immune response in asymptomatic SARS-CoV-2 infection. J Exp Med 218. https://doi.org/10.1084/jem.20202617
[132]	Bertoletti A, Le Bert N, Qui M, et al. (2021) SARS-CoV-2-specific T cells in infection and vaccination. Cell Mol Immunol 18: 2307-2312. https://doi.org/10.1038/s41423-021-00743-3
[133]	Wang Z, Yang X, Zhong J, et al. (2021) Exposure to SARS-CoV-2 generates T-cell memory in the absence of a detectable viral infection. Nat Commun 12. https://doi.org/10.1038/s41467-021-22036-z
[134]	Sekine T, Perez-Potti A, Rivera-Ballesteros O, et al. (2020) Robust T cell immunity in convalescent individuals with asymptomatic or mild COVID-19. Cell 183: 158-168.e114. https://doi.org/10.1016/j.cell.2020.08.017
[135]	Tan AT, Linster M, Tan CW, et al. (2021) Early induction of functional SARS-CoV-2-specific T cells associates with rapid viral clearance and mild disease in COVID-19 patients. Cell Rep 34: 108728. https://doi.org/10.1016/j.celrep.2021.108728
[136]	Le Bert N, Tan AT, Kunasegaran K, et al. (2020) SARS-CoV-2-specific T cell immunity in cases of COVID-19 and SARS, and uninfected controls. Nature 584: 457-462. https://doi.org/10.1038/s41586-020-2550-z
[137]	Kingstad-Bakke B, Lee W, Chandrasekar SS, et al. (2022) Vaccine-induced systemic and mucosal T cell immunity to SARS-CoV-2 viral variants. Proc Natl Acad Sci 119. https://doi.org/10.1073/pnas.2118312119
[138]	Le Bert N, Clapham HE, Tan AT, et al. (2021) Highly functional virus-specific cellular immune response in asymptomatic SARS-CoV-2 infection. J Exp Med 218. https://doi.org/10.1084/jem.20202617
[139]	Codo AC, Davanzo GG, Monteiro LDB, et al. (2020) Elevated glucose levels favor SARS-CoV-2 infection and monocyte response through a HIF-1α/Glycolysis-dependent axis. Cell Metabo 32: 437-446.e435. https://doi.org/10.2139/ssrn.3606770
[140]	Moga E, Lynton-Pons E, Domingo P (2022) The robustness of cellular immunity determines the fate of SARS-CoV-2 infection. Front Immunol 13: 904686. https://doi.org/10.3389/fimmu.2022.904686
[141]	Sherina N, Piralla A, Du L, et al. (2021) Persistence of SARS-CoV-2-specific B and T cell responses in convalescent COVID-19 patients 6–8 months after the infection. Med (NY) 2: 281-295.e284. https://doi.org/10.1016/j.medj.2021.02.001
[142]	Shomuradova AS, Vagida MS, Sheetikov SA, et al. (2020) SARS-CoV-2 epitopes are recognized by a public and diverse repertoire of human T cell receptors. Immunity 53: 1245-1257.e1245. https://doi.org/10.1016/j.immuni.2020.11.004
[143]	Cohen KW, Linderman SL, Moodie Z, et al. (2021) Longitudinal analysis shows durable and broad immune memory after SARS-CoV-2 infection with persisting antibody responses and memory B and T cells. Cell Rep Med 2: 100354. https://doi.org/10.1016/j.xcrm.2021.100354
[144]	Grau-Expósito J, Sánchez-Gaona N, Massana N, et al. (2021) Peripheral and lung resident memory T cell responses against SARS-CoV-2. Nat Commun 12. https://doi.org/10.1038/s41467-021-23333-3
[145]	Balcom EF, Nath A, Power C (2021) Acute and chronic neurological disorders in COVID-19: potential mechanisms of disease. Brain 144: 3576-3588. https://doi.org/10.1093/brain/awab302
[146]	Wiech M, Chroscicki P, Swatler J, et al. (2022) Remodeling of T cell dynamics during long COVID is dependent on severity of SARS-CoV-2 infection. Front Immunol 13: 886431. https://doi.org/10.3389/fimmu.2022.886431
[147]	Grifoni A, Weiskopf D, Ramirez SI, et al. (2020) Targets of T cell responses to SARS-CoV-2 coronavirus in humans with COVID-19 disease and unexposed individuals. Cell 181: 1489-1501.e1415. https://doi.org/10.1016/j.cell.2020.05.015
[148]	Nelde A, Bilich T, Heitmann JS, et al. (2021) SARS-CoV-2-derived peptides define heterologous and COVID-19-induced T cell recognition. Nat Immunol 22: 74-85. https://doi.org/10.1038/s41590-020-00808-x
[149]	Schulien I, Kemming J, Oberhardt V, et al. (2021) Characterization of pre-existing and induced SARS-CoV-2-specific CD8+ T cells. Nat Med 27: 78-85. https://doi.org/10.1038/s41591-020-01143-2
[150]	Rydyznski Moderbacher C, Ramirez SI, Dan JM, et al. (2020) Antigen-specific adaptive immunity to SARS-CoV-2 in acute COVID-19 and associations with age and disease severity. Cell 183: 996-1012.e1019. https://doi.org/10.1016/j.cell.2020.09.038
[151]	Szabo PA, Dogra P, Gray JI, et al. (2021) Longitudinal profiling of respiratory and systemic immune responses reveals myeloid cell-driven lung inflammation in severe COVID-19. Immunity 54: 797-814.e796. https://doi.org/10.1016/j.immuni.2021.03.005
[152]	Luangrath MA, Schmidt ME, Hartwig SM, et al. (2021) Tissue-resident memory T cells in the lungs protect against acute respiratory syncytial virus infection. ImmunoHorizons 5: 59-69. https://doi.org/10.4049/immunohorizons.2000067
[153]	Olson MR, Hartwig SM, Varga SM (2008) The number of respiratory syncytial virus (RSV)-specific memory CD8 T cells in the lung is critical for their ability to inhibit RSV vaccine-enhanced pulmonary eosinophilia. J Immunol 181: 7958-7968. https://doi.org/10.4049/jimmunol.181.11.7958
[154]	Masopust D, Soerens AG (2019) Tissue-resident T cells and other resident leukocytes. Annu Rev Immunol 37: 521-546. https://doi.org/10.1146/annurev-immunol-042617-053214
[155]	Worbs T, Hammerschmidt SI, Förster R (2017) Dendritic cell migration in health and disease. Nat Rev Immunol 17: 30-48. https://doi.org/10.1038/nri.2016.116
[156]	Lee JS, Park S, Jeong HW, et al. (2020) Immunophenotyping of COVID-19 and influenza highlights the role of type I interferons in development of severe COVID-19. Sci Immunol 5: eabd1554. https://doi.org/10.1126/sciimmunol.abd1554
[157]	Lan J, Ge J, Yu J, et al. (2020) Structure of the SARS-CoV-2 spike receptor-binding domain bound to the ACE2 receptor. Nature 581: 215-220. https://doi.org/10.1038/s41586-020-2180-5
[158]	Wang K, Chen W, Zhang Z, et al. (2020) CD147-spike protein is a novel route for SARS-CoV-2 infection to host cells. Signal Transduct Target Ther 5: 283.
[159]	Ni L, Ye F, Cheng M-L, et al. (2020) Detection of SARS-CoV-2-specific humoral and cellular immunity in COVID-19 convalescent individuals. Immunity 52: 971-977.e973. https://doi.org/10.1016/j.immuni.2020.04.023
[160]	Xu Z, Shi L, Wang Y, et al. (2020) Pathological findings of COVID-19 associated with acute respiratory distress syndrome. Lancet Respir Med 8: 420-422. https://doi.org/10.1016/S2213-2600(20)30076-X
[161]	Luo XH, Zhu Y, Mao J, et al. (2021) T cell immunobiology and cytokine storm of COVID-19. Scand J Immunol 93: e12989. https://doi.org/10.1111/sji.12989
[162]	Zhang Q, Bastard P, Liu Z, et al. (2020) Inborn errors of type I IFN immunity in patients with life-threatening COVID-19. Science 370: eabd4570. https://doi.org/10.1126/science.abd4570
[163]	Lucas C, Wong P, Klein J, et al. (2020) Longitudinal analyses reveal immunological misfiring in severe COVID-19. Nature 584: 463-469. https://doi.org/10.1038/s41586-020-2588-y
[164]	Martonik D, Parfieniuk-Kowerda A, Rogalska M, et al. (2021) The role of Th17 response in COVID-19. Cells 10: 1550. https://doi.org/10.3390/cells10061550
[165]	Liu L, Wei Q, Lin Q, et al. (2019) Anti–spike IgG causes severe acute lung injury by skewing macrophage responses during acute SARS-CoV infection. JCI Insight 4. https://doi.org/10.1172/jci.insight.123158
[166]	Liu B, Li M, Zhou Z, et al. (2020) Can we use interleukin-6 (IL-6) blockade for coronavirus disease 2019 (COVID-19)-induced cytokine release syndrome (CRS)?. J Autoimmun 111: 102452. https://doi.org/10.1016/j.jaut.2020.102452
[167]	Hasanvand A (2022) COVID-19 and the role of cytokines in this disease. Inflammopharmacology 30: 789-798. https://doi.org/10.1007/s10787-022-00992-2
[168]	Crotty S (2011) Follicular Helper CD4 T Cells (TFH). Annu Rev Immunol 29: 621-663. https://doi.org/10.1146/annurev-immunol-031210-101400
[169]	Ueno H, Banchereau J, Vinuesa CG (2015) Pathophysiology of T follicular helper cells in humans and mice. Nat Immunol 16: 142-152. https://doi.org/10.1038/ni.3054
[170]	Turner JS, O'Halloran JA, Kalaidina E, et al. (2021) SARS-CoV-2 mRNA vaccines induce persistent human germinal centre responses. Nature 596: 109-113. https://doi.org/10.1038/s41586-021-03738-2
[171]	Turner JS, Zhou JQ, Han J, et al. (2020) Human germinal centres engage memory and naive B cells after influenza vaccination. Nature 586: 127-132. https://doi.org/10.1038/s41586-020-2711-0
[172]	Bok K, Sitar S, Graham BS, et al. (2021) Accelerated COVID-19 vaccine development: milestones, lessons, and prospects. Immunity 54: 1636-1651. https://doi.org/10.1016/j.immuni.2021.07.017
[173]	Hou X, Zaks T, Langer R, et al. (2021) Lipid nanoparticles for mRNA delivery. Nat Rev Mater 6: 1078-1094. https://doi.org/10.1038/s41578-021-00358-0
[174]	Blakney AK, McKay PF, Yus BI, et al. (2019) Inside out: optimization of lipid nanoparticle formulations for exterior complexation and in vivo delivery of saRNA. Gene Ther 26: 363-372. https://doi.org/10.1038/s41434-019-0095-2
[175]	Chaudhary N, Weissman D, Whitehead KA (2021) mRNA vaccines for infectious diseases: principles, delivery and clinical translation. Nat Rev Drug Discov 20: 817-838. https://doi.org/10.1038/s41573-021-00283-5
[176]	Seneff S, Nigh G, Kyriakopoulos AM, et al. (2022) Innate immune suppression by SARS-CoV-2 mRNA vaccinations: The role of G-quadruplexes, exosomes, and MicroRNAs. Food Chem Toxicol 164: 113008. https://doi.org/10.1016/j.fct.2022.113008
[177]	Röltgen K, Nielsen SCA, Silva O, et al. (2022) Immune imprinting, breadth of variant recognition, and germinal center response in human SARS-CoV-2 infection and vaccination. Cell 185: 1025-1040.e1014. https://doi.org/10.1016/j.cell.2022.01.018
[178]	Bos R, Rutten L, Van Der Lubbe JEM, et al. (2020) Ad26 vector-based COVID-19 vaccine encoding a prefusion-stabilized SARS-CoV-2 Spike immunogen induces potent humoral and cellular immune responses. npj Vaccines 5. https://doi.org/10.1038/s41541-020-00243-x
[179]	Joe CCD, Jiang J, Linke T, et al. (2022) Manufacturing a chimpanzee adenovirus-vectored SARS-CoV-2 vaccine to meet global needs. Biotechnol Bioeng 119: 48-58. https://doi.org/10.1002/bit.27945
[180]	Dunkle LM, Kotloff KL, Gay CL, et al. (2022) Efficacy and Safety of NVX-CoV2373 in Adults in the United States and Mexico. N Engl J Med 386: 531-543. https://doi.org/10.1056/NEJMoa2116185
[181]	Jacob-Dolan C, Barouch DH (2022) COVID-19 xaccines: adenoviral vectors. Annu Rev Med 73: 41-54. https://doi.org/10.1146/annurev-med-012621-102252
[182]	Mendonça SA, Lorincz R, Boucher P, et al. (2021) Adenoviral vector vaccine platforms in the SARS-CoV-2 pandemic. npj Vaccines 6. https://doi.org/10.1038/s41541-021-00356-x
[183]	Li M, Wang H, Tian L, et al. (2022) COVID-19 vaccine development: milestones, lessons and prospects. Signal Transduct Target Ther 7: 146. https://doi.org/10.1038/s41392-022-00996-y
[184]	Young M, Crook H, Scott J, et al. (2022) Covid-19: virology, variants, and vaccines. BMJ Med 1: e000040. https://doi.org/10.1136/bmjmed-2021-000040
[185]	Dunkle LM, Kotloff KL, Gay CL, et al. (2022) Efficacy and safety of NVX-CoV2373 in adults in the United States and Mexico. New Engl J Med 386: 531-543. https://doi.org/10.1056/NEJMoa2116185
[186]	Li Q, Wang Y, Sun Q, et al. (2022) Immune response in COVID-19: what is next?. Cell Death Differ 29: 1107-1122. https://doi.org/10.1038/s41418-022-01015-x
[187]	De Gier B, Andeweg S, Joosten R, et al. (2021) Vaccine effectiveness against SARS-CoV-2 transmission and infections among household and other close contacts of confirmed cases, the Netherlands, February to May 2021. Euro surveill 26. https://doi.org/10.2807/1560-7917.ES.2021.26.31.2100640
[188]	Prunas O, Warren JL, Crawford FW, et al. (2022) Vaccination with BNT162b2 reduces transmission of SARS-CoV-2 to household contacts in Israel. Science 375: 1151-1154. https://doi.org/10.1126/science.abl4292
[189]	Lipsitch M, Krammer F, Regev-Yochay G, et al. (2022) SARS-CoV-2 breakthrough infections in vaccinated individuals: measurement, causes and impact. Nat Rev Immunol 22: 57-65. https://doi.org/10.1038/s41577-021-00662-4
[190]	Polack FP, Thomas SJ, Kitchin N, et al. (2020) Safety and efficacy of the BNT162b2 mRNA Covid-19 vaccine. New Engl J Med 383: 2603-2615. https://doi.org/10.1056/NEJMoa2034577
[191]	Baden LR, El Sahly HM, Essink B, et al. (2021) Efficacy and safety of the mRNA-1273 SARS-CoV-2 vaccine. New Engl J Med 384: 403-416. https://doi.org/10.1056/NEJMoa2035389
[192]	Sadoff J, Gray G, Vandebosch A, et al. (2022) Final analysis of efficacy and safety of single-dose Ad26.COV2.S. New Engl J Med 386: 847-860. https://doi.org/10.1056/NEJMoa2117608
[193]	Falsey AR, Sobieszczyk ME, Hirsch I, et al. (2021) Phase 3 safety and efficacy of AZD1222 (ChAdOx1 nCoV-19) Covid-19 vaccine. New Engl J Med 385: 2348-2360. https://doi.org/10.1056/NEJMoa2105290
[194]	Zhang Y, Belayachi J, Yang Y, et al. (2022) Real-world study of the effectiveness of BBIBP-CorV (Sinopharm) COVID-19 vaccine in the Kingdom of Morocco. BMC Public Health 22. https://doi.org/10.1186/s12889-022-14016-9
[195]	Tanriover MD, Doğanay HL, Akova M, et al. (2021) Efficacy and safety of an inactivated whole-virion SARS-CoV-2 vaccine (CoronaVac): interim results of a double-blind, randomised, placebo-controlled, phase 3 trial in Turkey. Lancet 398: 213-222. https://doi.org/10.1016/S0140-6736(21)01429-X
[196]	Erasmus JH, Khandhar AP, O'Connor MA, et al. (2020) An Alphavirus-derived replicon RNA vaccine induces SARS-CoV-2 neutralizing antibody and T cell responses in mice and nonhuman primates. Sci Transl Med 12. https://doi.org/10.1126/scitranslmed.abc9396
[197]	Andreasson U, Perret-Liaudet A, van Waalwijk van Doorn LJC, et al. (2015) A Practical Guide to Immunoassay Method Validation. Front Neuro 6. https://doi.org/10.3389/fneur.2015.00179
[198]	Burd Eileen M (2010) Validation of Laboratory-Developed Molecular Assays for Infectious Diseases. Clin Microbiol Rev 23: 550-576. https://doi.org/10.1128/CMR.00074-09
[199]	Larsen SE, Berube BJ, Pecor T, et al. (2021) Qualification of ELISA and neutralization methodologies to measure SARS-CoV-2 humoral immunity using human clinical samples. J Immunol Methods 499: 113160. https://doi.org/10.1016/j.jim.2021.113160

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Microbiology

2.7 7.0

Metrics

Article views(2753) PDF downloads(121) Cited by(10)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Microbiology

SARS-CoV-2 infection and immune responses

Related Papers:

Abstract

1. Introduction

2. Symbols and notation

3. Existing estimators

4. Proposed estimator

5. Numerical study

6. Discussion

7. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Symbols and notation

3. Existing estimators

4. Proposed estimator

5. Numerical study

6. Discussion

7. Conclusions

Acknowledgments

Conflict of interest

References

AIMS Microbiology

SARS-CoV-2 infection and immune responses

Related Papers:

Abstract

1. Introduction

2. Symbols and notation

3. Existing estimators

4. Proposed estimator

5. Numerical study

6. Discussion

7. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Symbols and notation

3. Existing estimators

4. Proposed estimator

5. Numerical study

6. Discussion

7. Conclusions

Acknowledgments

Conflict of interest

References