Dynamic behavior analysis of an $ SVIR $ epidemic model with two time delays associated with the COVID-19 booster vaccination time

Zimeng Lv; Xinyu Liu; Yuting Ding; Zimeng Lv; Xinyu Liu; Yuting Ding

doi:10.3934/mbe.2023261

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 6030-6061. doi: 10.3934/mbe.2023261

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Research article Special Issues

Dynamic behavior analysis of an $SVIR$ epidemic model with two time delays associated with the COVID-19 booster vaccination time

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Academic Editor: Xiang-Sheng Wang

Received: 16 November 2022 Revised: 15 January 2023 Accepted: 15 January 2023 Published: 18 January 2023

Since the outbreak of COVID-19, there has been widespread concern in the community, especially on the recent heated debate about when to get the booster vaccination. In order to explore the optimal time for receiving booster shots, here we construct an $SVIR$ model with two time delays based on temporary immunity. Second, we theoretically analyze the existence and stability of equilibrium and further study the dynamic properties of Hopf bifurcation. Then, the statistical analysis is conducted to obtain two groups of parameters based on the official data, and numerical simulations are carried out to verify the theoretical analysis. As a result, we find that the equilibrium is locally asymptotically stable when the booster vaccination time is within the critical value. Moreover, the results of the simulations also exhibit globally stable properties, which might be more beneficial for controlling the outbreak. Finally, we propose the optimal time of booster vaccination and predict when the outbreak can be effectively controlled.

Keywords:

Citation: Zimeng Lv, Xinyu Liu, Yuting Ding. Dynamic behavior analysis of an $SVIR$ epidemic model with two time delays associated with the COVID-19 booster vaccination time[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6030-6061. doi: 10.3934/mbe.2023261

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Abstract

1. Introduction

We consider the following family of nonlinear oscillators

$\begin{equation} y_{zz}+k(y)y_{z}^{3}+h(y)y_{z}^{2}+f(y)y_{z}+g(y) = 0, \end{equation}$

(1.1)

where $k$ , $h$ , $f\neq0$ and $g\neq0$ are arbitrary sufficiently smooth functions. Particular members of (1.1) are used for the description of various processes in physics, mechanics and so on and they also appear as invariant reductions of nonlinear partial differential equations ^[1,2,3].

Integrability of (1.1) was studied in a number of works ^{[4,5,6,7,8,9,10,11,12,13,14,15,16]}. In particular, in ^[15] linearization of (1.1) via the following generalized nonlocal transformations

$\begin{equation} w = F(y), \quad d\zeta = (G_{1}(y)y_{z}+G_{2}(y))dz. \end{equation}$

(1.2)

was considered. However, equivalence problems with respect to transformations (1.2) for (1.1) and its integrable nonlinear subcases have not been studied previously. Therefore, in this work we deal with the equivalence problem for (1.1) and its integrable subcase from the Painlevé-Gambier classification. Namely, we construct an equivalence criterion for (1.1) and a non-canonical form of Ince Ⅶ equation ^[17,18]. As a result, we obtain two new integrable subfamilies of (1.1). What is more, we demonstrate that for any equation from (1.1) that satisfy one of these equivalence criteria one can construct an autonomous first integral in the parametric form. Notice that we use Ince Ⅶ equation because it is one of the simplest integrable members of (1.1) with known general solution and known classification of invariant curves.

Moreover, we show that transformations (1.2) preserve autonomous invariant curves for equations from (1.1). Since the considered non-canonical form of Ince Ⅶ equation admits two irreducible polynomial invariant curves, we obtain that any equation from (1.1), which is equivalent to it, also admits two invariant curves. These invariant curves can be used for constructing an integrating factor for equations from (1.1) that are equivalent to Ince Ⅶ equation. If this integrating factor is Darboux one, then the corresponding equation is Liouvillian integrable ^[19]. This demonstrates the connection between nonlocal equivalence approach and Darboux integrability theory and its generalizations, which has been recently discussed for a less general class of nonlocal transformations in ^[20,21,22].

The rest of this work is organized as follows. In the next Section we present an equivalence criterion for (1.1) and a non-canonical form of the Ince Ⅶ equation. In addition, we show how to construct an autonomous first integral for an equation from (1.1) satisfying this equivalence criterion. We also demonstrate that transformations (1.2) preserve autonomous invariant curves for (1.1). In Section 3 we provide two examples of integrable equations from (1.1) and construct their parametric first integrals, invariant curves and integrating factors. In the last Section we briefly discuss and summarize our results.

2. Main results

We begin with the equivalence criterion between (1.1) and a non-canonical form of the Ince Ⅶ equation, that is ^[17,18]

$\begin{equation} w_{\zeta\zeta}+3w_{\zeta}+\epsilon w^{3}+2w = 0. \end{equation}$

(2.1)

Here $\epsilon\neq0$ is an arbitrary parameter, which can be set, without loss of generality, to be equal to $\pm 1$ .

The general solution of (1.1) is

$\begin{equation} w = {\rm e}^{-(\zeta-\zeta_{0})}\rm{cn}\left\{\sqrt{\epsilon}({\rm e}^{-(\zeta-\zeta_{0})}-C_{1}),\frac{1}{\sqrt{2}}\right\}. \end{equation}$

(2.2)

Here $\zeta_{0}$ and $C_{1}$ are arbitrary constants and $\rm{cn}$ is the Jacobian elliptic cosine. Expression (2.2) will be used below for constructing autonomous parametric first integrals for members of (1.1).

The equivalence criterion between (1.1) and (2.1) can be formulated as follows:

Theorem 2.1. Equation (1.1) is equivalent to (2.1) if and only if either

$\begin{equation} \begin{gathered} (I)\quad 25515lgp^{2}q_{y}+2352980l^{10}+\left(3430q-6667920p^{3}\right) l^{5}\\ -14580qp^{3}-10q^{2}-76545lgqpp_{y} = 0, \end{gathered} \end{equation}$

(2.3)

$\begin{equation} \begin{gathered} (II)\quad 343l^{5}-972p^{3} = 0, \end{gathered} \end{equation}$

(2.4)

holds. Here

$\begin{equation} \begin{gathered} l = 9(fg_{y}-gf_{y}+fgh-3kg^{2})-2f^{3}, \quad p = gl_{y}-3lg_{y}+l(f^{2}-3gh),\\ q = 25515g_{y}lp^{2}-5103lgpp_{y}+686l^{5}-8505p^{2}\left( f^{2}-3gh \right) l+6561p^{3}. \end{gathered} \end{equation}$

(2.5)

The expression for $G_{2}$ in each case is either

$\begin{equation} (I) \quad G_{2} = \frac {126 l^{2}qp^{2}}{470596 l^{10}- \left( 1333584 p^{3}+1372 q \right) l^{5}+q^{2}}, \end{equation}$

(2.6)

$\begin{equation} (II) \quad G_{2}^{2} = -\frac{49l^{3}G_{2}+9p^{2}}{189pl}. \end{equation}$

(2.7)

In all cases the functions $F$ and $G_{1}$ are given by

$\begin{equation} F^{2} = \frac{l}{81\epsilon G_{2}^{3}}, \quad G_{1} = \frac{G_{2}(f-3G_{2})}{3g}. \end{equation}$

(2.8)

Proof. We begin with the necessary conditions. Substituting (1.2) into (2.1) we get

$\begin{equation} y_{zz}+k(y)y_{z}^{3}+h(y)y_{z}^{2}+f(y)y_{z}+g(y) = 0, \end{equation}$

(2.9)

where

$\begin{equation} \begin{gathered} k = \frac{FG_{1}^{3}(\epsilon F^{2}+2)+3G_{1}^{2}F_{y}+G_{1}F_{yy}-F_{y}G_{1,y}}{G_{2}F_{y}}, \\ h = \frac{G_{2}F_{yy}+(6G_{1}G_{2}-G_{2,y})F_{y}+3FG_{2}G_{1}^{2}(\epsilon F^{2}+2)}{G_{2}F_{y}},\\ f = \frac{3G_{2}(F_{y}+FG_{1}(\epsilon F^{2}+2))}{F_{y}}, \quad g = \frac{FG_{2}^{2}(\epsilon F^{2}+2)}{F_{y}}. \end{gathered} \end{equation}$

(2.10)

As a consequence, we obtain that (1.1) can be transformed into (2.1) if it is of the form (2.9) (or (1.1)).

Conversely, if the functions $F$ , $G_{1}$ and $G_{2}$ satisfy (2.10) for some values of $k$ , $h$ , $f$ and $g$ , then (1.1) can be mapped into (2.1) via (1.2). Thus, we see that the compatibility conditions for (2.10) as an overdertmined system of equations for $F$ , $G_{1}$ and $G_{2}$ result in the necessary and sufficient conditions for (1.1) to be equivalent to (2.1) via (1.2).

To obtain the compatibility conditions, we simplify system (2.10) as follows. Using the last two equations from (2.10) we find the expression for $G_{1}$ given in (2.8). Then, with the help of this relation, from (2.10) we find that

$\begin{equation} 81\epsilon F^{2}G_{2}^{3}-l = 0, \end{equation}$

(2.11)

and

$\begin{equation} \begin{gathered} 567lG_{2}^{3}+(243lgh-81lf^{2}-81gl_{y}+243lg_{y})G_{2}-7l^{2} = 0,\\ 243lgG_{2,y}+324lG_{2}^{3}-81gl_{y}G_{2}+2l^{2} = 0, \end{gathered} \end{equation}$

(2.12)

Here $l$ is given by (2.5).

As a result, we need to find compatibility conditions only for (2.12). In order to find the generic case of this compatibility conditions, we differentiate the first equation twice and find the expression for $G_{2}^{2}$ and condition (2.3). Differentiating the first equation from (2.12) for the third time, we obtain (2.6). Further differentiation does not lead to any new compatibility conditions. Particular case (2.4) can be treated in the similar way.

Finally, we remark that the cases of $l = 0$ , $p = 0$ and $q = 0$ result in the degeneration of transformations (1.2). This completes the proof.

As an immediate corollary of Theorem 2.1 we get

Corollary 2.1. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then an autonomous first integral of this equation can be presented in the parametric form as follows:

$\begin{equation} y = F^{-1}(w), \quad y_{z} = \frac{G_{2}w_{\zeta}}{F_{y}-G_{1}w_{\zeta}}\,. \end{equation}$

(2.13)

Here $w$ is the general solution of (2.1) given by (2.2). Notice also that, formally, (2.13) contains two arbitrary constants, namely $\zeta_{0}$ and $C_{1}$ . However, without loss of generality, one of them can be set equal to zero.

Now we demonstrate that transformations (1.2) preserve autonomous invariant curves for equations from (1.1).

First, we need to introduce the definition of an invariant curve for (1.1). We recall that Eq (1.1) can be transformed into an equivalent dynamical system

$\begin{equation} y_{z} = P, \quad u_{z} = Q, \quad P = u, \quad Q = -k u^{3}-h u^{2}-f u -g. \end{equation}$

(2.14)

A smooth function $H(y, u)$ is called an invariant curve of (2.14) (or, equivalently, of (1.1)), if it is a nontrivial solution of ^[19]

$\begin{equation} PH_{y}+QH_{u} = \lambda H, \end{equation}$

(2.15)

for some value of the function $\lambda$ , which is called the cofactor of $H$ .

Second, we need to introduce the equation that is equivalent to (1.1) via (1.2). Substituting (1.2) into (1.1) we get

$\begin{equation} w_{\zeta\zeta}+\tilde{k}w_{\zeta}^{3}+\tilde{h}w_{\zeta}^{2}+\tilde{f}w_{\zeta}+\tilde{g} = 0, \end{equation}$

(2.16)

where

$\begin{equation} \begin{gathered} \tilde{k} = \frac{kG_{2}^{3}-gG_{1}^{3}+(G_{1,y}-hG_{1})G_{2}^{2}+(fG_{1}-G_{2,y})G_{1}G_{2}}{F_{y}^{2}G_{2}^{2}},\\ \tilde{h} = \frac{(hF_{y}-F_{yy})G_{2}^{2}-(2fG_{1}-G_{2,y})G_{2}F_{y}+3gG_{1}^{2}F_{y}}{F_{y}^{2}G_{2}^{2}},\\ \tilde{f} = \frac{fG_{2}-3gG_{1}}{G_{2}^{2}}, \quad \tilde{g} = \frac{gF_{y}}{G_{2}^{2}}. \end{gathered} \end{equation}$

(2.17)

An invariant curve for (2.16) can be defined in the same way as that for (1.1). Notice that, further, we will denote $w_{\zeta}$ as $v$ .

Theorem 2.2. Suppose that either (1.1) possess an invariant curve $H(y, u)$ with the cofactor $\lambda(y, u)$ or (2.16) possess an invariant curve $\widetilde{H}(w, v)$ with the cofactor $\tilde{\lambda}(w, v)$ . Then, the other equation also has an invariant curve and the corresponding invariant curves and cofactors are connected via

$\begin{equation} \begin{gathered} H(y,u) = \widetilde{H}\left(F,\frac{F_{y}u}{G_{1}u+G_{2}}\right), \quad \lambda(y,u) = (G_{1}u+G_{2})\tilde{\lambda}\left(F,\frac{F_{y}u}{G_{1}u+G_{2}}\right). \end{gathered} \end{equation}$

(2.18)

Proof. Suppose that $\widetilde{H}(w, v)$ is an invariant curve for (2.16) with the cofactor $\widetilde{\lambda}(w, v)$ . Then it satisfies

$\begin{equation} v\widetilde{H}_{w}+(-\tilde{k} v^{3}-\tilde{h} v^{2}-\tilde{f} v -\tilde{g})\widetilde{H}_{v} = \widetilde{\lambda}\widetilde{H}. \end{equation}$

(2.19)

Substituting (1.2) into (2.19) we get

$\begin{equation} uH_{y}+(-k u^{3}-h u^{2}-f u-g)H = (G_{1}u+G_{2}) \tilde{\lambda}\left(F,\frac{F_{y}u}{G_{1}u+G_{2}}\right) H . \end{equation}$

(2.20)

This completes the proof.

As an immediate consequence of Theorem 2.2 we have that transformations (1.2) preserve autonomous first integrals admitted by members of (1.1), since they are invariant curves with zero cofactors.

Another corollary of Theorem 2.2 is that any equation from (1.1) that is connected to (2.1) admits two invariant curves that correspond to irreducible polynomial invariant curves of (2.1). This invariant curves of (2.1) and the corresponding cofactors are the following (see, ^[23] formulas (3.18) and (3.19) taking into account scaling transformations)

$\begin{equation} \widetilde{H} = \pm i \sqrt{-2\epsilon}(v+w)+w^{2}, \quad \tilde{\lambda} = \pm \sqrt{-2\epsilon}w-2. \end{equation}$

(2.21)

Therefore, we have that the following statement holds:

Corollary 2.2. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then is admits the following invariant curves with the corresponding cofactors

$\begin{equation} H = \pm i \sqrt{-2\epsilon}\left(\frac{F_{y}u}{G_{1}u+G_{2}}+F\right)+F^{2}, \quad \lambda = \left(G_{1}u+G_{2}\right)\left(\pm \sqrt{-2\epsilon}F-2\right). \end{equation}$

(2.22)

Let us remark that connections between (2.1) and non-autonomous variants of (1.1) can be considered via a non-autonomous generalization of transformations (1.2). However, one of two nonlocally related equations should be autonomous since otherwise nonlocal transformations do not map a differential equation into a differential equation ^[5].

In this Section we have obtained the equivalence criterion between (1.1) and (2.1), that defines two new completely integrable subfamilies of (1.1). We have also demonstrated that members of these subfamilies posses an autonomous parametric first integral and two autonomous invariant curves.

3. Examples

In this Section we provide two examples of integrable equations from (1.1) satisfying integrability conditions from Theorem 2.1.

Example 1. One can show that the coefficients of the following cubic oscillator

$\begin{equation} y_{zz}-\frac{12\epsilon \mu y}{(\epsilon \mu^{2}y^{4}+2)^{2}}y_{z}^{3}-6\mu y y_{z}+2\mu^{2}y^{3}(\epsilon\mu^{2}y^{4}+2) = 0, \end{equation}$

(3.1)

satisfy condition (2.3) from Theorem 2.1. Consequently, Eq (3.1) is completely integrable and its general solution can be obtained from (2.2) by inverting transformations (1.2). However, it is more convenient to use Corollary 2.1 and present the autonomous first integral of (3.1) in the parametric form as follows:

$\begin{equation} y = \pm\sqrt{\frac{w}{\mu}}, \quad y_{z} = \frac{w(\epsilon w^{2}+2) w_{\zeta}}{2w_{\zeta}+w(\epsilon w^{2}+2)}, \end{equation}$

(3.2)

where $w$ is given by (2.2), $\zeta$ is considered as a parameter and $\zeta_{0}$ , without loss of generality, can be set equal to zero. As a result, we see that (3.1) is integrable since it has an autonomous first integral.

Moreover, using Corollary 2.2 one can find invariant curves admitted by (3.1)

$\begin{equation} \begin{gathered} H_{1,2} = \frac{y^{4}\left[(\sqrt{2}\pm \sqrt{-\epsilon}\mu y^{2})^{2}(\sqrt{2}\mp\sqrt{-\epsilon}\mu y^{2})+2(\epsilon\mu y^{2}\mp\sqrt{-2\epsilon})u\right]}{2\mu^{2}y^{2}(\epsilon\mu^{2}y^{4}+2)-4u},\\ \lambda_{1,2} = \pm\frac{2(\mu y^{2}(\epsilon\mu^{2}y^{4}+2)-2u)(\sqrt{-2\epsilon}\mu y^{2}\mp 2)}{y(\epsilon\mu^{2}y^{4}+2)} \end{gathered} \end{equation}$

(3.3)

With the help of the standard technique of the Darboux integrability theory ^[19], it is easy to find the corresponding Darboux integrating factor of (3.1)

$\begin{equation} M = \frac{(\epsilon \mu^{2}y^{4}+2)^{\frac{9}{4}}}{(2\epsilon u^{2}+(\epsilon\mu^{2}y^{4}+2)^{2})^{\frac{3}{4}}(\mu y^{2}(\epsilon \mu^{2}y^{4}+2)-2u)^{\frac{3}{2}}}. \end{equation}$

(3.4)

Consequently, equation is (3.1) Liouvillian integrable.

Example 2. Consider the Liénard (1, 9) equation

$\begin{equation} y_{zz}+(b_{i}y^{i}){y_{z}}+a_{j}y^{j} = 0, \quad i = 0,\ldots 4, \quad j = 0,\ldots,9. \end{equation}$

(3.5)

Here summation over repeated indices is assumed. One can show that this equation is equivalent to (2.1) if it is of the form

$\begin{equation} y_{zz}-9(y+\mu)(y+3\mu)^{3}y_{z}+2y(2y+3\mu)(y+3\mu)^{7} = 0, \end{equation}$

(3.6)

where $\mu$ is an arbitrary constant.

With the help of Corollary 2.1 one can present the first integral of (3.6) in the parametric form as follows:

$\begin{equation} y = \frac{3\sqrt{-2\epsilon}\mu w}{2-\sqrt{-2\epsilon}w}, \quad y_{z} = \frac{7776 \sqrt{2} \epsilon \mu^{5} w w_{\zeta}}{(\sqrt{-2\epsilon}w-2)^{5}(2\sqrt{-\epsilon}w_{\zeta}+(\sqrt{2}\epsilon w +2\sqrt{-\epsilon})w)}, \end{equation}$

(3.7)

where $w$ is given by (2.2). Thus, one can see that (3.5) is completely integrable due to the existence of this parametric autonomous first integral.

Using Corollary 2.2 we find two invariant curves of (3.6):

$\begin{equation} H_{1} = \frac{y^{2}[(2y+3\mu)(y+3\mu)^{4}-2u)]}{(y+3\mu)^{2}[(y+3\mu)^{4}y-u]}, \quad \lambda_{1} = \frac{6\mu(u-y(y+3\mu)^{4})}{y(y+3\mu)}, \end{equation}$

(3.8)

and

$\begin{equation} H_{2} = \frac{y^{2}(y+3\mu)^{2}}{y(y+3\mu)^{4}-u}, \quad \lambda_{2} = \frac{2(2y+3\mu)(u-2y(y+3\mu)^{4})}{y(y+3\mu)}. \end{equation}$

(3.9)

The corresponding Darboux integrating factor is

$\begin{equation} M = \left[y(y+3\mu)^{4}-u\right]^{-\frac{3}{2}}\left[(2y+3\mu)(y+3\mu)^{4}-2u\right]^{-\frac{3}{4}}. \end{equation}$

(3.10)

As a consequence, we see that Eq (3.6) is Liouvillian integrable.

Therefore, we see that equations considered in Examples 1 and 2 are completely integrable from two points of view. First, they possess autonomous parametric first integrals. Second, they have Darboux integrating factors.

4. Conclusions and discussion

In this work we have considered the equivalence problem between family of Eqs (1.1) and its integrable member (2.1), with equivalence transformations given by generalized nonlocal transformations (1.2). We construct the corresponding equivalence criterion in the explicit form, which leads to two new integrable subfamilies of (1.1). We have demonstrated that one can explicitly construct a parametric autonomous first integral for each equation that is equivalent to (2.1) via (1.2). We have also shown that transformations (1.2) preserve autonomous invariant curves for (1.1). As a consequence, we have obtained that equations from the obtained integrable subfamilies posses two autonomous invariant curves, which corresponds to the irreducible polynomial invariant curves of (2.1). This fact demonstrate a connection between nonlocal equivalence approach and Darboux and Liouvillian integrability approach. We have illustrate our results by two examples of integrable equations from (1.1).

Acknowledgments

The author was partially supported by Russian Science Foundation grant 19-71-10003.

Conflict of interest

The author declares no conflict of interest in this paper.

References

[1]	N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, et al., A novel coronavirus from patients with pneumonia in China, N. Engl. J. Med., 382 (2020), 727–733. https://doi.org/10.1056/NEJMoa2001017 doi: 10.1056/NEJMoa2001017
[2]	H. Zhang, F. Du, X. Cao, X. Feng, H. Zhang, Z. Wu, et al., Clinical characteristics of coronavirus disease 2019 (COVID-19) in patients out of Wuhan from China: a case control study, BMC Infect. Dis., 21 (2021), 207. https://doi.org/10.1186/s12879-021-05897-z doi: 10.1186/s12879-021-05897-z
[3]	C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, et al., Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, Lancet, 395 (2020), 497–506. https://doi.org/10.1016/S0140-6736(20)30183-5 doi: 10.1016/S0140-6736(20)30183-5
[4]	M. A. Hassan, A. A. Bala, A. I. Jatau, Low rate of COVID-19 vaccination in Africa: a cause for concern, Ther. Adv. Vaccines Immun., 10 (2022), 1–3. https://doi.org/10.1177/25151355221088159 doi: 10.1177/25151355221088159
[5]	S. O. Minka, F. H. Minka, A tabulated summary of the evidence on humoral and cellular responses to the SARS-CoV-2 Omicron VOC, as well as vaccine efficacy against this variant, Immunol. Lett., 243 (2022), 38–43. https://doi.org/10.1016/j.imlet.2022.02.002 doi: 10.1016/j.imlet.2022.02.002
[6]	W. O. Kermack, A. G. Mckendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc., 115 (1927), 111–124. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[7]	A. Din, Y. Li, F. M. Khan, Z. U. Khan, P. Liu, On Analysis of fractional order mathematical model of Hepatitis B using Atangana Baleanu Caputo (ABC) derivative, Fractals, 30 (2022), 2240017. https://doi.org/10.1142/S0218348X22400175 doi: 10.1142/S0218348X22400175
[8]	B. Boukanjime, M. E. Fatini, A stochastic Hepatitis B epidemic model driven by Levy noise, Physica A, 521 (2019), 796–806. https://doi.org/10.1016/j.physa.2019.01.097 doi: 10.1016/j.physa.2019.01.097
[9]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamical behavior of a stochastic epidemic model for cholera, J. Franklin Inst., 115 (1927), 111–124. https://doi.org/10.1016/j.jfranklin.2018.11.056 doi: 10.1016/j.jfranklin.2018.11.056
[10]	P. Liu, A. Din, Impact of information intervention on stochastic dengue epidemic model, Alex. Eng. J., 60 (2021), 5725–5739. https://doi.org/10.1016/j.aej.2021.03.068 doi: 10.1016/j.aej.2021.03.068
[11]	Z. Wang, G. Rost, S. M. Moghadas, Delay in booster schedule as a control parameter in vaccination dynamics, J. Math. Biol., 79 (2019), 2157–2182. https://doi.org/10.1007/s00285-019-01424-6 doi: 10.1007/s00285-019-01424-6
[12]	R. M. Anderson, B. T. Grenfell, Quantitative investigations of different vaccination policies for the control of congentila rubella syndrome (CRS) in the United Kingdom, Epidemiol. Infect., 96 (1986), 305–333. https://doi.org/10.1017/s0022172400066079 doi: 10.1017/s0022172400066079
[13]	M. E. Alexander, S. M. Moghadas, P. Rohani, A. R. Summers, Modelling the effect of a booster vaccination on disease epidemiology, J. Math. Biol., 52 (2006), 290–306. https://doi.org/10.1007/s00285-005-0356-0 doi: 10.1007/s00285-005-0356-0
[14]	T. Marinov, R. Marinova, Inverse problem for adaptive SIR model: Application to COVID-19 in Latin America, Infect. Dis. Modell., 7 (2022), 234–148. https://doi.org/10.1016/j.idm.2021.12.001 doi: 10.1016/j.idm.2021.12.001
[15]	H. Qiu, Q. Wang, Q. Wu, H. Zhou, Does flattening the curve make a difference? An investigation of the COVID-19 pandemic based on an SIR model, Int. Rev. Econ. Finance, 80 (2022), 159–165. https://doi.org/10.1016/j.iref.2022.02.063 doi: 10.1016/j.iref.2022.02.063
[16]	S. Annas, M. I. Pratama, M. Rifandi, W. Sanusi, S. Side, Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia, Chaos Solitons Fractals, 139 (2020), 110072. https://doi.org/10.1016/j.chaos.2020.110072 doi: 10.1016/j.chaos.2020.110072
[17]	S. Paul, A. Mahata, U. Ghosh, B. Roy, Study of SEIR epidemic model and scenario analysis of COVID-19 pandemic, Ecol. Genet. Genom., 19 (2021), 100087. https://doi.org/10.1016/j.egg.2021.100087 doi: 10.1016/j.egg.2021.100087
[18]	D. Efimov, R. Ushirobira, On an interval prediction of COVID-19 development based on a SEIR epidemic model, Annu. Rev. Control, 51 (2021), 477–487. https://doi.org/10.1016/j.arcontrol.2021.01.006 doi: 10.1016/j.arcontrol.2021.01.006
[19]	Z. Chen, L. Feng, H. A. Lay Jr, K. Furati, A. Khaliq, SEIR model with unreported infected population and dynamic parameters for the spread of COVID-19, Math. Comput. Simul., 198 (2022), 31–46. https://doi.org/10.1016/j.matcom.2022.02.025 doi: 10.1016/j.matcom.2022.02.025
[20]	R. Li, Y. Song, H. Wang, G. Jiang, M. Xiao, Reactive diffusion epidemic model on human mobility networks: Analysis and applications to COVID-19 in China, Physica A, 609 (2023), 128337. https://doi.org/10.1016/j.physa.2022.128337 doi: 10.1016/j.physa.2022.128337
[21]	F. A. C. C. Chaluba, M. O. Souza, The SIR epidemic model from a PDE point of view, Math. Comput. Modell., 53 (2011), 1568–1574. https://doi.org/10.1016/j.mcm.2010.05.036 doi: 10.1016/j.mcm.2010.05.036
[22]	L. Bttcher, M. Xia, T. Chou, Why case fatality ratios can be misleading: individual and population based mortality estimates and factors influencing them, Phys. Biol., 17 (2020), 065003. https://doi.org/10.1088/1478-3975/ab9e59 doi: 10.1088/1478-3975/ab9e59
[23]	Q. Wu, X. Fan, H. Hong, Y. Gua, Z. Liu, S. Fang, Comprehensive assessment of side effects in COVID-19 drug pipeline from a network perspective, Food Chem. Toxicol., 145 (2020), e13476. https://doi.org/10.1016/j.fct.2020.111767 doi: 10.1016/j.fct.2020.111767
[24]	U. Tursen, B. Tursen, T. Lotti, Cutaneous side-effects of the potential COVID-19 drugs, Dermatol. Ther., 33 (2020), 31–46. https://doi.org/10.1111/dth.13476 doi: 10.1111/dth.13476
[25]	P. Wintachai, K. Prathom, Stability analysis of SEIR model related to efficiency of vaccines for COVID-19 situation, Heliyon, 7 (2021), e06812. https://doi.org/10.1016/j.heliyon.2021.e06812 doi: 10.1016/j.heliyon.2021.e06812
[26]	P. Jarumaneeroj, P. Dusadeerungsikul, T. Chotivanich, T. Nopsopon, K. Pongpirul, An epidemiology-based model for the operational allocation of COVID-19 vaccines: A case study of Thailand, Comput. Ind. Eng., 167 (2022), 108031. https://doi.org/10.1016/j.cie.2022.108031 doi: 10.1016/j.cie.2022.108031
[27]	L. Böher, J. Nagler, Decisive conditions for strategic vaccination against SARS-CoV-2, Chaos, 31 (2021), 101105. https://doi.org/10.1063/5.0066992 doi: 10.1063/5.0066992
[28]	A. Mahata, S. Paul, S. Mukherjee, B. Roy, Stability analysis and Hopf bifurcation in fractional order SEIRV epidemic model with a time delay in infected individuals, Partial Differ. Equations Appl. Math., 5 (2022), 100282. https://doi.org/10.1016/j.padiff.2022.100282 doi: 10.1016/j.padiff.2022.100282
[29]	H. Yang, Y. Wang, S. Kundu, Z. Song, Z. Zhang, Dynamics of an SIR epidemic model incorporating time delay and convex incidence rate, Results Phys., 32 (2022), 105025. https://doi.org/10.1016/j.rinp.2021.105025 doi: 10.1016/j.rinp.2021.105025
[30]	A. Khan, R. Ikram, A. Din, U. W. Humphries, A. Akgul, Stochastic COVID-19 SEIQ epidemic model with time-delay, Results Phys., 30 (2021), 104775. https://doi.org/10.1016/j.rinp.2021.104775 doi: 10.1016/j.rinp.2021.104775
[31]	C. Zhu, J. Zhu, Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method, Chaos Solitons Fractals, 143 (2021), 110546. https://doi.org/10.1016/j.chaos.2020.110546 doi: 10.1016/j.chaos.2020.110546
[32]	E. M. Farah, S. Amine, K. Allali, Dynamics of a time-delayed two-strain epidemic model with general incidence rates, Chaos Solitons Fractals, 153 (2021), 111527. https://doi.org/10.1016/j.chaos.2021.111527 doi: 10.1016/j.chaos.2021.111527
[33]	H. Li, X. Liu, R. Yan, C. Liu, Hopf bifurcation analysis of a tumor virotherapy model with two time delays, Physica A, 553 (2020), 124266. https://doi.org/10.1016/j.physa.2020.124266 doi: 10.1016/j.physa.2020.124266
[34]	B. Sun, M. Li, F. Zhang, H. Wang, J. Liu, The characteristics and self-time-delay synchronization of two-time-delay complex Lorenz system, J. Franklin Inst., 356 (2019), 334–350. https://doi.org/10.1016/j.jfranklin.2018.09.031 doi: 10.1016/j.jfranklin.2018.09.031
[35]	M. Akio, S. Ferenc, Time delays and chaos in two competing species revisited, Appl. Math. Comput., 395 (2021), 125862. https://doi.org/10.1016/j.amc.2020.125862 doi: 10.1016/j.amc.2020.125862
[36]	H. Zhou, Z. Wang, D. Yuan, H. Song, Hopf bifurcation of a free boundary problem modeling tumor growth with angiogenesis and two time delays, Chaos Solitons Fractals, 153 (2021), 111578. https://doi.org/10.1016/j.chaos.2021.111578 doi: 10.1016/j.chaos.2021.111578
[37]	A. R. Hakimi, M. Azhdari, T. Binazadeh, Limit cycle oscillator in nonlinear systems with multiple time delays, Chaos Solitons Fractals, 153 (2021), 111454. https://doi.org/10.1016/j.chaos.2021.111454 doi: 10.1016/j.chaos.2021.111454
[38]	A. Adhikary, A. Pal, A six compartments with time-delay model SHIQRD for the COVID-19 pandemic in India: During lockdown and beyond, Alex. Eng. J., 61 (2022), 1403–1412. https://doi.org/10.1016/j.aej.2021.06.027 doi: 10.1016/j.aej.2021.06.027
[39]	A. Mahata, S. Paul, S. Mukherjee, B. Roy, Stability analysis and Hopf bifurcation in fractional order SEIRV epidemic model with a time delay in infected individuals, Partial Differ. Equations Appl. Math., 5 (2022), 100282. https://doi.org/10.1016/j.padiff.2022.100282 doi: 10.1016/j.padiff.2022.100282
[40]	Y. Zhang, J. Jia, Hopf bifurcation of an epidemic model with a nonlinear birth in population and vertical transmission, Appl. Math. Comput., 230 (2014), 164–173. https://doi.org/10.1016/j.amc.2013.12.084 doi: 10.1016/j.amc.2013.12.084
[41]	X. Duan, J. Yin, X. Li, Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery, Chaos Solitons Fractals, 104 (2017), 613–624. https://doi.org/10.1016/j.chaos.2017.09.029 doi: 10.1016/j.chaos.2017.09.029
[42]	Z. Zhang, S. Kundu, J. P. Tripathi, S. Bugalia, Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays, Chaos Solitons Fractals, 131 (2020), 109483. https://doi.org/10.1016/j.chaos.2019.109483 doi: 10.1016/j.chaos.2019.109483
[43]	H. Yang, Y. Wang, S. Kundu, Z. Song, Z. Zhang, Dynamics of an SIR epidemic model incorporating time delay and convex incidence rate, Results Phys., 32 (2022), 105025. https://doi.org/10.1016/j.rinp.2021.105025 doi: 10.1016/j.rinp.2021.105025
[44]	Y. Ding, L. Zheng, Mathematical modeling and dynamics analysis of delayed nonlinear VOC emission system, Nonlinear Dyn., 109 (2022), 3157–3167. https://doi.org/10.1007/s11071-022-07532-1 doi: 10.1007/s11071-022-07532-1
[45]	Y. Ding, L. Zheng, J. Guo, Stability analysis of nonlinear glue flow system with delay, Math. Methods Appl. Sci., 45 (2022), 6861–6877. https://doi.org/10.1002/mma.8211 doi: 10.1002/mma.8211
[46]	Y. Song, Y. Peng, T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differ. Equations, 300 (2021), 597–624. https://doi.org/10.1016/j.jde.2021.08.010 doi: 10.1016/j.jde.2021.08.010
[47]	Y. Song, H. Jiang, Y. Yuan, Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model, J. Appl. Anal. Comput., 9 (2019), 1132–1164. https://doi.org/10.1016/j.cnsns.2015.10.002 doi: 10.1016/j.cnsns.2015.10.002
[48]	I. Alam, A. Radovanovic, R. Incitti, A. A. Kamau, M. Alarawi, E. I. Azhar, et al., CovMT: an interactive SARS-CoV-2 mutation tracker, with a focus on critical variants, Lancet Infect. Dis., 21 (2021), 602. https://doi.org/10.1016/s1473-3099(21)00078-5 doi: 10.1016/s1473-3099(21)00078-5
[49]	T. Phan, Genetic diversity and evolution of SARS-CoV-2, Infect. Genet. Evol., 81 (2020), 104260. https://doi.org/10.1016/j.meegid.2020.104260 doi: 10.1016/j.meegid.2020.104260
[50]	M. Pachetti, B. Marini, F. Giudici, F. Benedetti, S. Angeletti, M. Ciccozzi, et al., Impact of lockdown on COVID-19 case fatality rate and viral mutations spread in 7 countries in europe and north america, J. Transl. Med., 18 (2020), 1–7. https://doi.org/10.1186/s12967-020-02501-x doi: 10.1186/s12967-020-02501-x
[51]	S. Y. Tartof, J. M. Slezak, H. Fischer, V. Hong, B. K. Ackerson, O. N. Ranasinghe, et al., Effectiveness of mRNA BNT162b2 COVID-19 vaccine up to 6 months in a large integrated health system in the USA: a retrospective cohort study, Lancet, 398 (2021), 1407–1416. https://doi.org/10.1016/S0140-6736(21)02183-8 doi: 10.1016/S0140-6736(21)02183-8
[52]	J. L. Bayart, J. Douxfils, C. Gillot, C. David, F. Mullier, M. Elsen, et al., Waning of IgG, total and neutralizing antibodies 6 months post-vaccination with BNT162b2 in healthcare workers, Vaccines, 9 (2021), 1092. https://doi.org/10.3390/vaccines9101092 doi: 10.3390/vaccines9101092
[53]	S. Liu, X. Yang, Y. Bia, Y. Li, Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity, Discrete Contin. Dyn. Syst., 24 (2019), 1469–1483. https://doi.org/10.3934/dcdsb.2018216 doi: 10.3934/dcdsb.2018216
[54]	C. T. Ng, T. C. E. Heng, D. Tsadikovich, E. Levner, A. Elalouf, S. Hovav, A multi-criterion approach to optimal vaccination planning: Method and solution, Comput. Ind. Eng., 126 (2018), 637–649. https://doi.org/10.1016/j.cie.2018.10.018 doi: 10.1016/j.cie.2018.10.018
[55]	M. Xia, L. Böher, T. Chou, Controlling epidemics through optimal allocation of test kits and vaccine doses across networks, IEEE Trans. Network Sci. Eng., 9 (2022), 1422–1436. https://doi.org/10.1109/TNSE.2022.3144624 doi: 10.1109/TNSE.2022.3144624
[56]	Z. Lv, J. Zeng, Y. Ding, X. Liu, Stability analysis of time-delayed SAIR model for duration of vaccine in the context of temporary immunity for COVID-19 situation, Electron. Res. Arch., 31 (2023), 1004–1030. https://doi.org/10.3934/era.2023050 doi: 10.3934/era.2023050

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