Dynamical analysis of COVID-19 and tuberculosis co-infection using mathematical modelling approach

J. O. Akanni; S. Ajao; S. F. Abimbade; Fatmawati; J. O. Akanni; S. Ajao; S. F. Abimbade; Fatmawati

doi:10.3934/mmc.2024018

Mathematical Modelling and Control

2024, Volume 4, Issue 2: 208-229. doi: 10.3934/mmc.2024018

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Research article

Dynamical analysis of COVID-19 and tuberculosis co-infection using mathematical modelling approach

1.
Department of Mathematical and Computing Sciences, Koladaisi University, Ibadan, Oyo State, Nigeria
2.
Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya, Indonesia
3.
Department of Mathematics and Computer Science, Elizade University, Ondo State, Nigeria
4.
Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Oyo State, Nigeria

Received: 26 November 2023 Revised: 04 March 2024 Accepted: 05 March 2024 Published: 23 May 2024

Both tuberculosis (TB) and COVID-19 are infectious diseases with similar clinical manifestations, which mainly affect the lungs. Clinical studies have revealed that the immunosuppressive drugs taken by COVID-19 patients can affect the immunological functions in the body, which can cause the patients to contract active TB via a new infection or reinfection, and the co-infection of the two diseases portends a clinical complexity in the management of the patients. Thus, this paper presents a mathematical model to study the dynamics and control of COVID-19-TB co-infection. The full model of the co-infection is split into two submodels, namely, the TB-only and the COVID-19-only models. The equilibria of the disease-free and endemic situations of the two sub-models are shown to be globally asymptotically stable when their control reproduction numbers $R_{o}^{TV}, R_{o}^{CV} < 1$ and $\tilde {R}_{o}^{TV}, \tilde {R}_{o}^{CV} > 1$ , respectively. However, the disease-free equilibrium of the co-infection model was found to lose its global stability property when the reproduction number $R_{o}^{F} < 1$ , therefore exhibiting a backward bifurcation. Uncertainty and sensitivity analysis of the associated reproduction number of the full model has been performed by using the Latin hypercube sampling/Pearson rank correlation coefficient (LHS/PRCC) method. The rate of transmission of COVID-19 and the proportions of individuals vaccinated with Bacillus Calmette-Guérin (BCG) and against COVID-19 were found to be highly significant in the spread and control of COVID-19-TB co-infection. Furthermore, the simulation results show that decreasing the COVID-19 transmission rate and increasing the proportion of people vaccinated with BCG and against COVID-19 can lower the number of cases of COVID-19-TB co-infection. Therefore, measures to reduce the transmission rate and the provision of adequate resources to increase the proportions of people vaccinated against TB and COVID-19 should be implemented to minimize the cases of co-infection.

Keywords:

Citation: J. O. Akanni, S. Ajao, S. F. Abimbade, Fatmawati. Dynamical analysis of COVID-19 and tuberculosis co-infection using mathematical modelling approach[J]. Mathematical Modelling and Control, 2024, 4(2): 208-229. doi: 10.3934/mmc.2024018

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Abstract

1. Introduction

Time-fractional differential equations arise in the mathematical modeling of a variety of real-world phenomena in many areas of sciences and engineering, such as elasticity, heat transfer, circuits systems, continuum mechanics, fluid mechanics, wave theory, etc. For more details, we refer the reader to ^{[4,6,7,8,14,15,17,24]} and the references therein. Consequently, the study of time-fractional differential equations attracted much attention of many researchers (see e.g. ^{[1,5,9,10,19,22,23]} and the references therein).

Multi-time differential equations arise, for example, in analyzing frequency and amplitude modulation in oscillators, see Narayan and Roychowdhury ^[18]. Some methods for solving Multi-time differential equations can be found in ^[20,21].

The study of blowing-up solutions to time-fractional differential equations was initiated by Kirane and his collaborators, see e.g. ^[3,11,12,13]. In particular, Kirane et al. ^[11] considered the two-times fractional differential equation

$\begin{eqnarray} \left\{\begin{array}{llll} {}^C\!D_{0|t}^\alpha u (t,s)+{}^C\!D_{0|s}^\beta |u|^m (t,s) = |u|^p(t,s),\quad t,s > 0,\\ \\ u(0,s) = u_0(s),\, u(t,0) = u_1(t),\quad t,s > 0, \end{array} \right. \end{eqnarray}$

(1.1)

where $p, m > 1$ , $0 < \alpha, \beta < 1$ , ${}^C\!D_{0|t}^\alpha$ is the Caputo fractional derivative of order $\alpha$ with respect to the first time-variable $t$ , and ${}^C\!D_{0|s}^\beta$ is the Caputo fractional derivative of order $\beta$ with respect to the second time-variable $s$ . Namely, the authors provided sufficient conditions for which any solution to (1.1) blows-up in a finite time. In the same reference, the authors extended their study to the case of systems.

In this paper, we investigate the nonexistence of global solutions to two-times-fractional differential inequalities of the form

$\begin{eqnarray} \left\{\begin{array}{llll} {}^{HC}\!D_{a|t}^\alpha u (t,s)+{}^C\!D_{a|s}^\beta |u|^m (t,s) \geq (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p(t,s),\quad t,s > a,\\ \\ u(a,s) = u_0(s),\, u(t,a) = u_1(t),\quad t,s > a, \end{array} \right. \end{eqnarray}$

(1.2)

where $p > 1$ , $m\geq 1$ , $\gamma, \sigma\in \mathbb{R}$ , $a > 0$ , $0 < \alpha, \beta < 1$ , ${}^{HC}\!D_{a|t}^\alpha$ is the Hadamard-Caputo fractional derivative of order $\alpha$ with respect to the first time-variable $t$ , and ${}^C\!D_{a|s}^\beta$ is the Caputo fractional derivative of order $\beta$ with respect to the second time-variable $s$ . Using the test function method (see e.g. ^[16]) and a judicious choice of a test function, we establish sufficient conditions ensuring the nonexistence of global solutions to (1.2). Our obtained conditions depend on the parameters $\alpha, \beta, p, m, \gamma, \sigma$ , and the initial values.

Our motivation for considering problems of type (1.2) is to study the combination effect of the two fractional derivatives of different nature ${}^{HC}\!D_{a|t}^\alpha$ and ${}^C\!D_{a|s}^\beta$ on the nonexistence of global solutions to (1.2). As far as we know, the study of nonexistence of global solutions for time fractional differential equations (or inequalities) involving both Hadamard-Caputo and Caputo fractional derivatives, was never considered in the literature.

The rest of the paper is organized as follows: In Section 2, we recall some concepts from fractional calculus and provide some useful lemmas. In Section 3, we state our main results and provide some examples. Section 4 is devoted to the proofs of our main results.

2. Some preliminaries

Let $a, T\in \mathbb{R}$ be such that $0 < a < T$ . The left-sided and right-sided Riemann-Liouville fractional integrals of order $\theta > 0$ of a function $\vartheta\in L^1([a, T])$ , are defined respectively by (see ^[10])

$(I_a^\theta\vartheta)(t) = \frac{1}{\Gamma(\theta)} \int_a^t (t-\tau)^{\theta-1} \vartheta(\tau)\,d\tau$

and

$(I_T^\theta\vartheta)(t) = \frac{1}{\Gamma(\theta)} \int_t^T (\tau-t)^{\theta-1} \vartheta(\tau)\,d\tau,$

for almost everywhere $t\in [a, T]$ , where $\Gamma$ is the Gamma function.

Notice that, if $\vartheta\in C([a, T])$ , then $I_a^\theta\vartheta, I_T^\theta\vartheta \in C([a, T])$ with

$\begin{equation} (I_a^\theta\vartheta)(a) = (I_T^\theta\vartheta)(T) = 0. \end{equation}$

(2.1)

The Caputo fractional derivative of order $\theta\in (0, 1)$ of a function $\vartheta\in AC([a, \infty))$ , is defined by (see ^[10])

${}^CD_a^\theta \vartheta (t) = (I_a^{1-\theta} \vartheta')(t) = \frac{1}{\Gamma(1-\theta)}\int_a^t (t-\tau)^{-\theta} \vartheta'(\tau)\,d\tau,$

for almost everywhere $t\geq a$ .

Lemma 2.1. [see ^[10]]Let $\kappa > 0$ , $p, q\geq 1$ , and $\frac{1}{p}+\frac{1}{q}\leq 1+\kappa$ ( $p\neq 1$ , $q\neq 1$ , in the case $\frac{1}{p}+\frac{1}{q} = 1+\kappa$ ). Let $\vartheta\in L^p([a, T]$ and $w\in L^q([a, T])$ . Then

$\int_a^T (I_a^\kappa \vartheta)(t) w(t)\,dt = \int_a^T \vartheta(t) (I_T^\kappa w)(t)\,dt.$

The left-sided and right-sided Hadamard fractional integrals of order $\theta > 0$ of a function $\vartheta\in L^1([a, T])$ , are defined respectively by (see ^[10])

$(J_a^\theta\vartheta)(t) = \frac{1}{\Gamma(\theta)} \int_a^t \left(\ln \frac{t}{\tau}\right)^{\theta-1} \vartheta(\tau) \frac{1}{\tau}\,d\tau$

and

$(J_T^\theta\vartheta)(t) = \frac{1}{\Gamma(\theta)} \int_t^T \left(\ln \frac{\tau}{t}\right)^{\theta-1} \vartheta(\tau) \frac{1}{\tau}\,d\tau,$

for almost everywhere $t\in [a, T]$ .

Notice that, if $\vartheta\in C([a, T])$ , then $J_a^\theta\vartheta, J_T^\theta\vartheta \in C([a, T])$ with

$\begin{equation} (J_a^\theta\vartheta)(a) = (J_T^\theta\vartheta)(T) = 0. \end{equation}$

(2.2)

The Hadamard-Caputo fractional derivative of order $\theta\in (0, 1)$ of a function $\vartheta\in AC([a, \infty))$ , is defined by (see ^[2])

${}^{HC}D_a^\theta \vartheta(t) = (J_a^{1-\theta} \delta \vartheta)(t) = \frac{1}{\Gamma(1-\theta)}\int_a^t \left(\ln \frac{t}{\tau}\right)^{-\theta} \delta \vartheta(\tau) \frac{1}{\tau}\,d\tau,$

for almost everywhere $t\geq a$ , where

$\delta \vartheta(t) = t \vartheta'(t).$

We have the following integration by parts rule.

Lemma 2.2. Let $\kappa > 0$ , $p, q\geq 1$ , and $\frac{1}{p}+\frac{1}{q}\leq 1+\kappa$ ( $p\neq 1$ , $q\neq 1$ , in the case $\frac{1}{p}+\frac{1}{q} = 1+\kappa$ ). If $\vartheta \circ \exp \in L^p([\ln a, \ln T])$ and $w \circ \exp \in L^q([\ln a, \ln T])$ , then

$\int_a^T (J_a^\kappa\vartheta)(t) w(t)\frac{1}{t}\,dt = \int_a^T \vartheta(t)(J_T^\kappa w)(t)\frac{1}{t}\,dt.$

Proof. Using the change of variable $x = \ln \tau$ , we obtain

$\begin{eqnarray*} (J_a^\kappa\vartheta)(t)& = &\frac{1}{\Gamma(\kappa)} \int_a^t \left(\ln \frac{t}{\tau}\right)^{\kappa-1} \vartheta(\tau) \frac{1}{\tau}\,d\tau\\ & = & \frac{1}{\Gamma(\kappa)} \int_{\ln a}^{\ln t} \left(\ln t- x\right)^{\kappa-1} (\vartheta\circ \exp) (x)\,dx, \end{eqnarray*}$

that is,

$\begin{equation} (J_a^\kappa\vartheta)(t) = \left(I_{\ln a}^\kappa \vartheta\circ \exp\right) (\ln t). \end{equation}$

(2.3)

Similarly, we have

$\begin{equation} (J_T^\kappa w)(t) = \left(I_{\ln T}^\kappa w\circ \exp\right) (\ln t). \end{equation}$

(2.4)

By (2.3), we obtain

$\int_a^T (J_a^\kappa\vartheta)(t) w(t)\frac{1}{t}\,dt = \int_a^T \left(I_{\ln a}^\kappa \vartheta\circ \exp\right) (\ln t) w(t)\frac{1}{t}\,dt.$

Using the change of variable $x = \ln t$ , we get

$\int_a^T (J_a^\kappa\vartheta)(t) w(t)\frac{1}{t}\,dt = \int_{\ln a}^{\ln T} \left(I_{\ln a}^\kappa \vartheta\circ \exp\right)(x) (w\circ\exp)(x) \,dx.$

Since $\vartheta \circ \exp \in L^p([\ln a, \ln T])$ and $w \circ \exp \in L^q([\ln a, \ln T])$ , by Lemma 2.1, we deduce that

$\int_a^T (J_a^\kappa\vartheta)(t) w(t)\frac{1}{t}\,dt = \int_{\ln a}^{\ln T} (\vartheta\circ \exp)(x) \left(I_{\ln T}^{\kappa} w\circ\exp\right)(x)\,dx.$

Using again the change of variable $x = \ln t$ , there holds

$\int_a^T (J_a^\kappa\vartheta)(t) w(t)\frac{1}{t}\,dt = \int_{a}^{T} \vartheta(t) \left(I_{\ln T}^{\kappa} w\circ\exp\right)(\ln t)\frac{1}{t}\,dt.$

Then, by (2.4), the desired result follows.

By elementary calculations, we obtain the following properties.

Lemma 2.3. For sufficiently large $\lambda$ , let

$\begin{equation} \phi_1(t) = \left(\ln \frac{T}{a}\right)^{-\lambda} \left(\ln \frac{T}{t}\right)^{\lambda},\quad a\leq t\leq T. \end{equation}$

(2.5)

Let $\kappa\in (0, 1)$ . Then

$\begin{eqnarray} (J_{T}^{\kappa}\phi_1)(t)& = & \frac{\Gamma(\lambda+1)}{\Gamma(\kappa+\lambda+1)} \left(\ln \frac{T}{a}\right)^{-\lambda} \left(\ln \frac{T}{t}\right)^{\kappa+\lambda}, \end{eqnarray}$

(2.6)

$\begin{eqnarray} (J_{T}^{\kappa}\phi_1)'(t)& = &-\frac{\Gamma(\lambda+1)}{\Gamma(\kappa+\lambda)} \left(\ln \frac{T}{a}\right)^{-\lambda} \left(\ln \frac{T}{t}\right)^{\kappa+\lambda-1} \frac{1}{t}. \end{eqnarray}$

(2.7)

Lemma 2.4. For sufficiently large $\lambda$ , let

$\begin{equation} \phi_2(s) = (T-a)^{-\lambda} (T-s)^{\lambda},\quad a\leq s\leq T. \end{equation}$

(2.8)

Let $\kappa\in (0, 1)$ . Then

$\begin{eqnarray} (I_{T}^{\kappa}\phi_2)(s)& = & \frac{\Gamma(\lambda+1)}{\Gamma(\kappa+\lambda+1)} (T-a)^{-\lambda} (T-s)^{\kappa+\lambda}, \end{eqnarray}$

(2.9)

$\begin{eqnarray} (I_{T}^{\kappa}\phi_2)'(s)& = &-\frac{\Gamma(\lambda+1)}{\Gamma(\kappa+\lambda)} (T-a)^{-\lambda} (T-s)^{\kappa+\lambda-1}. \end{eqnarray}$

(2.10)

3. Main results

First, let us define global solutions to (1.2). To do this, we need to introduce the functional space

$X_a: = \left\{u\in C([a,\infty)\times [a,\infty)):\, u(\cdot,s)\in AC([a,\infty)),\, |u|^m(t,\cdot)\in AC([a,\infty))\right\}.$

We say that $u$ is a global solution to (1.2), if $u\in X_a$ and $u$ satisfies the fractional differential inequality

${}^{HC}\!D_{a|t}^\alpha u (t,s)+{}^C\!D_{a|s}^\beta |u|^m (t,s) \geq (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p(t,s)$

for almost everywhere $t, s\geq a$ , as well as the initial conditions

$u(a,s) = u_0(s),\, u(t,a) = u_1(t),\quad t,s > a.$

Now, we state our main results.

Theorem 3.1. Let $u_0\in L^1([a, \infty))$ , $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$ , and $u_1\not\equiv 0$ .Let

$\begin{equation} 0 < \beta < \frac{1}{m}\leq 1,\quad \gamma > \max\left\{\frac{m-1}{1-m\beta}, m(\sigma+1)-1\right\}\beta. \end{equation}$

(3.1)

$\begin{equation} m\max\left\{\gamma+1,\sigma+1\right\} < p < 1+\frac{\gamma}{\beta}, \end{equation}$

(3.2)

then, for all $\alpha\in (0, 1)$ , (1.2) admits no global solution.

Remark 3.1. Notice that by (3.1), the set of exponents $p$ satisfying (3.2) is nonempty.

Theorem 3.2. Let $u_0\in L^1([a, \infty))$ , $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$ , and $u_1\not\equiv 0$ . Let

$\begin{equation} 0 < \beta < \frac{1}{m}\leq 1,\quad 1-\frac{1}{m} < \alpha < 1,\quad \sigma > \frac{(m-1)(1-\alpha)}{1-m\beta}-\alpha. \end{equation}$

(3.3)

$\begin{equation} \beta \max\left\{ \frac{m-1}{1-m\beta},m(\sigma+1)-1\right\} < \gamma < \frac{(\sigma+\alpha)\beta}{1-\alpha} \end{equation}$

(3.4)

and

$\begin{equation} p = 1+\frac{\gamma}{\beta}, \end{equation}$

(3.5)

then (1.2) admits no global solution.

Remark 3.2. Notice that by (3.3), the set of real numbers $\gamma$ satisfying (3.4) is nonempty.

We illustrate our obtained results by the following examples.

Example 3.1. Consider the fractional differential inequality

$\begin{eqnarray} \left\{\begin{array}{llll} {}^{HC}\!D_{a|t}^\alpha u (t,s)+{}^C\!D_{a|s}^{\frac{1}{4}} u^2 (t,s) \geq (s-a) \left(\ln \frac{t}{a}\right)^{-1} |u|^p(t,s),\quad t,s > a,\\ \\ u(a,s) = (1+s^2)^{-1},\, u(t,a) = \exp(-t),\quad t,s > a, \end{array} \right. \end{eqnarray}$

(3.6)

where $a > 0$ and $0 < \alpha < 1$ . Observe that (3.6) is a special case of (1.2) with

$\beta = \frac{1}{4},\,\, m = 2,\,\, \sigma = -1,\,\,\gamma = 1,\,\, u_0(s) = (1+s^2)^{-1},\,\, u_1(t) = \exp(-t).$

Moreover, we have

$0 < \beta = \frac{1}{4} < \frac{1}{2} = \frac{1}{m} < 1, \quad \max\left\{\frac{m-1}{1-m\beta}, m(\sigma+1)-1\right\}\beta = \frac{\max\{2,0\}}{4} = \frac{1}{2} < \gamma = 1,$

and $u_0\in L^1([a, \infty))$ , $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$ . Hence, condition (3.1) is satisfied. Then, by Theorem 3.1, we deduce that, if

$m\max\left\{\gamma+1,\sigma+1\right\} < p < 1+\frac{\gamma}{\beta},$

that is,

$4 < p < 5,$

then (3.6) admits no global solution.

Example 3.2. Consider the fractional differential inequality

$\begin{eqnarray} \left\{\begin{array}{llll} {}^{HC}\!D_{a|t}^{\frac{3}{4}} u (t,s)+{}^C\!D_{a|s}^{\frac{1}{2}} |u| (t,s) \geq (s-a)^{\frac{1}{4}} \left(\ln \frac{t}{a}\right)^{-\frac{1}{2}} |u|^{\frac{3}{2}}(t,s),\quad t,s > a,\\ \\ u(a,s) = (1+s^2)^{-1},\, u(t,a) = \exp(-t),\quad t,s > a, \end{array} \right. \end{eqnarray}$

(3.7)

where $a > 0$ . Then (3.7) is a special case of (1.2) with

$\alpha = \frac{3}{4},\,\, \beta = \frac{1}{2},\,\, m = 1,\,\, \sigma = -\frac{1}{2},\,\, \gamma = \frac{1}{4},\,\, p = \frac{3}{2},\,\, u_0(s) = (1+s^2)^{-1},\,\, u_1(t) = \exp(-t).$

On the other hand, we have

$0 < \beta = \frac{1}{2} < 1 = \frac{1}{m},\quad 1-\frac{1}{m} = 0 < \alpha = \frac{3}{4} < 1,\quad \sigma = -\frac{1}{2} > -\frac{3}{4} = \frac{(m-1)(1-\alpha)}{1-m\beta}-\alpha,$

which shows that condition (3.3) is satisfied. Moreover, we have

$\beta \max\left\{\frac{m-1}{1-m\beta},m(\sigma+1)-1\right\} = -\frac{1}{4} < \gamma = \frac{1}{4} < \frac{1}{2} = \frac{(\sigma+\alpha)\beta}{1-\alpha},\quad p = \frac{3}{2} = 1+\frac{\gamma}{\beta},$

which shows that conditions (3.4) and (3.5) are satisfied. Then, by Theorem 3.2, we deduce that (3.7) admits no global solution.

4. Proofs of the main results

In this section, $C$ denotes a positive constant independent on $T$ , whose value may change from line to line.

Proof of Theorem 3.1. Suppose that $u\in X_a$ is a global solution to (1.2). For sufficiently large $T$ and $\lambda$ , let

$\varphi(t,s) = \phi_1(t)\phi_2(s),\quad a\leq t,s\leq T,$

where $\phi_1$ and $\phi_2$ are defined respectively by (2.5) and (2.8). Multiplying the inequality in (1.2) by $\frac{1}{t}\varphi$ and integrating over $\Omega_T: = (a, T)\times (a, T)$ , we obtain

$\begin{equation} \begin{aligned} &\int_{\Omega_T} (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p \varphi(t,s)\frac{1}{t}\,dt\,ds\\ & \leq \int_{\Omega_T}{}^{HC}\!D_{a|t}^\alpha u \,\varphi(t,s)\frac{1}{t}\,dt\,ds + \int_{\Omega_T} {}^C\!D_{a|s}^\beta |u|^m\,\varphi(t,s)\frac{1}{t}\,dt\,ds. \end{aligned} \end{equation}$

(4.1)

On the other hand, using Lemma 2.2, integrating by parts, using the initial conditions, and taking in consideration (2.2), we obtain

$\begin{aligned} &\int_a^T {}^{HC}\!D_{a|t}^\alpha u \,\varphi(t,s)\frac{1}{t}\,dt\\ & = \int_a^T \left(J_{a|t}^{1-\alpha} t \frac{\partial u}{\partial t}\right)(t,s)\varphi(t,s)\frac{1}{t}\,dt\\ & = \int_a^T \frac{\partial u}{\partial t}(t,s) \left(J_{T|t}^{1-\alpha}\varphi\right)(t,s) \,dt\\ & = \left[u(t,s) \left(J_{T|t}^{1-\alpha}\varphi\right)(t,s)\right]_{t = a}^T-\int_a^T u(t,s) \frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}(t,s)\,dt\\ & = -u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)-\int_a^T u(t,s) \frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}(t,s)\,dt. \end{aligned}$

Integrating over $(a, T)$ , we get

$\begin{equation} \begin{aligned} &\int_{\Omega_T}{}^{HC}\!D_{a|t}^\alpha u \,\varphi(t,s)\frac{1}{t}\,dt\,ds\\ & = -\int_a^T u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)\,ds-\int_{\Omega_T} u(t,s) \frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}(t,s)\,dt\,ds. \end{aligned} \end{equation}$

(4.2)

Similarly, using Lemma 2.1, integrating by parts, using the initial conditions, and taking in consideration (2.1), we obtain

$\begin{aligned} & \int_a^T {}^C\!D_{a|s}^\beta |u|^m\,\varphi(t,s)\,ds\\ & = \int_a^T \left(I_{a|s}^{1-\beta} \frac{\partial |u|^m}{\partial s}(t,s)\right)\,\varphi(t,s)\,ds\\ & = \int_a^T \frac{\partial |u|^m}{\partial s}(t,s) \left(I_{T|s}^{1-\beta}\varphi\right)(t,s)\,ds\\ & = \left[|u|^m(t,s)\left(I_{T|s}^{1-\beta}\varphi\right)(t,s)\right]_{s = a}^T-\int_a^T |u|^m(t,s) \frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}(t,s)\,ds\\ & = -|u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)-\int_a^T |u|^m(t,s) \frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}(t,s)\,ds. \end{aligned}$

Integrating over $(a, T)$ , there holds

$\begin{equation} \begin{aligned} &\int_{\Omega_T} {}^C\!D_{a|s}^\beta |u|^m\,\varphi(t,s)\frac{1}{t}\,dt\,ds\\ & = -\int_a^T |u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)\frac{1}{t}\,dt-\int_{\Omega_T}|u|^m(t,s) \frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}(t,s)\frac{1}{t}\,dt\,\,ds. \end{aligned} \end{equation}$

(4.3)

It follows from (4.1)–(4.3) that

$\begin{equation} \begin{aligned} &\int_{\Omega_T} (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p \varphi(t,s)\frac{1}{t}\,dt\,ds\\ &+\int_a^T u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)\,ds +\int_a^T |u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)\frac{1}{t}\,dt\\ &\leq \int_{\Omega_T} |u| \left|\frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}\right|\,dt\,ds+\int_{\Omega_T}|u|^m \left|\frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}\right|\frac{1}{t}\,dt\,\,ds. \end{aligned} \end{equation}$

(4.4)

On the other hand, by Young's inequality, we have

$\begin{equation} \begin{aligned} &\int_{\Omega_T} |u| \left|\frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}\right|\,dt\,ds\\ &\leq \frac{1}{2} \int_{\Omega_T} (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p \varphi(t,s)\frac{1}{t}\,dt\,ds\\ &+C \int_{\Omega_T} t^{\frac{1}{p-1}} (s-a)^{\frac{-\gamma}{p-1}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \varphi^{\frac{-1}{p-1}}(t,s) \left|\frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}\right|^{\frac{p}{p-1}}\,dt\,ds. \end{aligned} \end{equation}$

(4.5)

Similarly, since $p > m$ , we have

$\begin{equation} \begin{aligned} &\int_{\Omega_T}|u|^m \left|\frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}\right|\frac{1}{t}\,dt\,\,ds\\ &\leq \frac{1}{2} \int_{\Omega_T} (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p \varphi(t,s)\frac{1}{t}\,dt\,ds\\ &+C \int_{\Omega_T} \frac{1}{t} (s-a)^{\frac{-\gamma m}{p-m}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}} \varphi^{\frac{-m}{p-m}}(t,s) \left|\frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}\right|^{\frac{p}{p-m}}\,dt\,\,ds. \end{aligned} \end{equation}$

(4.6)

Hence, combining (4.4)–(4.6), we deduce that

$\begin{equation} \int_a^T u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)\,ds +\int_a^T |u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)\frac{1}{t}\,dt\leq C (K_1+K_2), \end{equation}$

(4.7)

where

$K_1 = \int_{\Omega_T} t^{\frac{1}{p-1}} (s-a)^{\frac{-\gamma}{p-1}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \varphi^{\frac{-1}{p-1}}(t,s) \left|\frac{\partial \left(J_{T|t}^{1-\alpha}\varphi\right)}{\partial t}\right|^{\frac{p}{p-1}}\,dt\,ds$

and

$K_2 = \int_{\Omega_T} \frac{1}{t} (s-a)^{\frac{-\gamma m}{p-m}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}} \varphi^{\frac{-m}{p-m}}(t,s) \left|\frac{\partial \left(I_{T|s}^{1-\beta}\varphi\right)}{\partial s}\right|^{\frac{p}{p-m}}\,dt\,\,ds.$

By the definition of the function $\varphi$ , we have

$\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s) = \phi_2(s) \left(J_{T|t}^{1-\alpha}\phi_1\right)(a).$

Thus, using (2.6), we obtain

$\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s) = C \phi_2(s) \left(\ln \frac{T}{a}\right)^{1-\alpha}.$

Integrating over $(a, T)$ , we get

$\begin{equation} \int_a^T u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)\,ds = C \left(\ln \frac{T}{a}\right)^{1-\alpha} \int_a^T u_0(s)(T-a)^{-\lambda} (T-s)^{\lambda}\,ds. \end{equation}$

(4.8)

Similarly, by the definition of the function $\varphi$ , we have

$\left(I_{T|s}^{1-\beta}\varphi\right)(t,a) = \phi_1(t) \left(I_{T|s}^{1-\beta}\phi_2\right)(a).$

Thus, using (2.9), we obtain

$\left(I_{T|s}^{1-\beta}\varphi\right)(t,a) = C \phi_1(t)(T-a)^{1-\beta}.$

Integrating over $(a, T)$ , we get

$\begin{equation} \int_a^T |u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)\frac{1}{t}\,dt = C (T-a)^{1-\beta} \int_a^T |u_1(t)|^m \left(\ln \frac{T}{a}\right)^{-\lambda} \left(\ln \frac{T}{t}\right)^{\lambda}\frac{1}{t}\,dt. \end{equation}$

(4.9)

Combining (4.8) with (4.9), there holds

$\begin{aligned} &\int_a^T u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)\,ds+ \int_a^T |u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)\frac{1}{t}\,dt\\ & = C \left(\ln \frac{T}{a}\right)^{1-\alpha} \int_a^T u_0(s)(T-a)^{-\lambda} (T-s)^{\lambda}\,ds\\ &+C (T-a)^{1-\beta} \int_a^T |u_1(t)|^m \left(\ln \frac{T}{a}\right)^{-\lambda} \left(\ln \frac{T}{t}\right)^{\lambda}\frac{1}{t}\,dt. \end{aligned}$

Since $u_0\in L^1([a, \infty))$ , $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$ , and $u_1\not\equiv 0$ , by the dominated convergence theorem, we deduce that for sufficiently large $T$ ,

$\begin{equation} \begin{aligned} &\int_a^T u_0(s)\left(J_{T|t}^{1-\alpha}\varphi\right)(a,s)\,ds+ \int_a^T |u_1(t)|^m \left(I_{T|s}^{1-\beta}\varphi\right)(t,a)\frac{1}{t}\,dt\\ &\geq C (T-a)^{1-\beta} \int_a^\infty |u_1(t)|^m \frac{1}{t}\,dt. \end{aligned} \end{equation}$

(4.10)

Now, we shall estimate the terms $K_i$ , $i = 1, 2$ . By the definition of the function $\varphi$ , the term $K_1$ can be written as

$\begin{equation} K_1 = \left(\int_a^T (s-a)^{\frac{-\gamma}{p-1}} \phi_2(s)\,ds\right) \left(\int_a^T t^{\frac{1}{p-1}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \phi_1^{\frac{-1}{p-1}}(t) \left|\left(J_{T|t}^{1-\alpha}\phi_1\right)'(t)\right|^{\frac{p}{p-1}}\,dt\right). \end{equation}$

(4.11)

Next, by (2.8), we obtain

$\begin{eqnarray*} \int_a^T (s-a)^{\frac{-\gamma}{p-1}} \phi_2(s)\,ds& = & (T-a)^{-\lambda} \int_a^T (s-a)^{\frac{-\gamma}{p-1}} (T-s)^\lambda\,ds\\ &\leq & \int_a^T (s-a)^{\frac{-\gamma}{p-1}}\,ds. \end{eqnarray*}$

On the other hand, by (3.1) and (3.2), it is clear that $\gamma < p-1$ . Thus, we deduce that

$\begin{equation} \int_a^T (s-a)^{\frac{-\gamma}{p-1}} \phi_2(s)\,ds\leq C (T-a)^{1-\frac{\gamma}{p-1}}. \end{equation}$

(4.12)

By (2.5) and (2.7), we have

$\begin{aligned} & \int_a^T t^{\frac{1}{p-1}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \phi_1^{\frac{-1}{p-1}}(t) \left|\left(J_{T|t}^{1-\alpha}\phi_1\right)'(t)\right|^{\frac{p}{p-1}}\,dt\\ & = \left(\ln \frac{T}{a}\right)^{-\lambda} \int_a^T \left(\ln \frac{T}{t}\right)^{\lambda-\frac{\alpha p}{p-1}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \frac{1}{t}\,dt\\ &\leq \left(\ln \frac{T}{a}\right)^{-\frac{\alpha p}{p-1}} \int_a^T \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \frac{1}{t}\,dt. \end{aligned}$

Notice that by (3.1) and (3.2), we have $\sigma < p-1$ . Thus, we get

$\begin{equation} \begin{aligned} & \int_a^T t^{\frac{1}{p-1}} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma}{p-1}} \phi_1^{\frac{-1}{p-1}}(t) \left|\left(J_{T|t}^{1-\alpha}\phi_1\right)'(t)\right|^{\frac{p}{p-1}}\,dt\\ &\leq C \left(\ln \frac{T}{a}\right)^{1-\frac{\alpha p+\sigma}{p-1}}. \end{aligned} \end{equation}$

(4.13)

Hence, it follows from (4.11)–(4.13) that

$\begin{equation} K_1\leq C (T-a)^{1-\frac{\gamma}{p-1}} \left(\ln \frac{T}{a}\right)^{1-\frac{\alpha p+\sigma}{p-1}}. \end{equation}$

(4.14)

Similarly, we can write the term $K_2$ as

$\begin{equation} K_2 = \left(\int_a^T \frac{1}{t} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}}\phi_1(t)\,dt\right) \left(\int_a^T (s-a)^{\frac{-\gamma m}{p-m}} \phi_2^{\frac{-m}{p-m}}(s) \left|\left(I_{T|s}^{1-\beta}\phi_2\right)'(s)\right|^{\frac{p}{p-m}}\,ds\right). \end{equation}$

(4.15)

By (2.5), we have

$\begin{eqnarray*} \int_a^T \frac{1}{t} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}}\phi_1(t)\,dt& = & \left(\ln \frac{T}{a}\right)^{-\lambda} \int_a^T \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}} \left(\ln \frac{T}{t}\right)^{\lambda} \frac{1}{t}\,dt\\ &\leq & \int_a^T \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}}\frac{1}{t}\,dt. \end{eqnarray*}$

Notice that by (3.2), we have $\sigma m < p-m$ . Thus, we get

$\begin{equation} \int_a^T \frac{1}{t} \left(\ln \frac{t}{a}\right)^{\frac{-\sigma m}{p-m}}\phi_1(t)\,dt\leq C \left(\ln \frac{T}{a}\right)^{1-\frac{\sigma m}{p-m}}. \end{equation}$

(4.16)

On the other hand, by (2.8) and (2.10), we have

$\begin{aligned} &\int_a^T (s-a)^{\frac{-\gamma m}{p-m}} \phi_2^{\frac{-m}{p-m}}(s) \left|\left(I_{T|s}^{1-\beta}\phi_2\right)'(s)\right|^{\frac{p}{p-m}}\,ds\\ & = (T-a)^{-\lambda} \int_a^T (T-s)^{\lambda-\frac{\beta p}{p-m}} (s-a)^{\frac{-\gamma m}{p-m}}\,ds \\ &\leq (T-a)^{-\frac{\beta p}{p-m}} \int_a^T(s-a)^{\frac{-\gamma m}{p-m}}\,ds. \end{aligned}$

Notice that by (3.2), we have $p > m(\gamma+1)$ . Therefore, we obtain

$\begin{equation} \int_a^T (s-a)^{\frac{-\gamma m}{p-m}} \phi_2^{\frac{-m}{p-m}}(s) \left|\left(I_{T|s}^{1-\beta}\phi_2\right)'(s)\right|^{\frac{p}{p-m}}\,ds\leq C (T-a)^{1- \frac{\gamma m+\beta p}{p-m}}. \end{equation}$

(4.17)

Combining (4.16) with (4.17), there holds

$\begin{equation} K_2\leq C \left(\ln \frac{T}{a}\right)^{1-\frac{\sigma m}{p-m}}(T-a)^{1- \frac{\gamma m+\beta p}{p-m}}. \end{equation}$

(4.18)

Hence, it follows from (4.14) and (4.18) that

$\begin{equation} K_1+K_2\leq C\left[\left(\ln \frac{T}{a}\right)^{1-\frac{\alpha p+\sigma}{p-1}}(T-a)^{1-\frac{\gamma}{p-1}} +\left(\ln \frac{T}{a}\right)^{1-\frac{\sigma m}{p-m}}(T-a)^{1- \frac{\gamma m+\beta p}{p-m}}\right]. \end{equation}$

(4.19)

Thus, by (4.7), (4.10), and (4.19), we deduce that

$\begin{equation} \begin{aligned} &\int_a^\infty |u_1(t)|^m \frac{1}{t}\,dt\\ &\leq C\left[\left(\ln \frac{T}{a}\right)^{1-\frac{\alpha p+\sigma}{p-1}}(T-a)^{\beta-\frac{\gamma}{p-1}} +\left(\ln \frac{T}{a}\right)^{1-\frac{\sigma m}{p-m}}(T-a)^{\beta- \frac{\gamma m+\beta p}{p-m}}\right]. \end{aligned} \end{equation}$

(4.20)

Notice that by (3.1) and (3.2), we have

$\beta-\frac{\gamma}{p-1} < 0,\quad \beta- \frac{\gamma m+\beta p}{p-m} < 0.$

Hence, passing to the limit as $T\to \infty$ in (4.20), we obtain a contradiction with $u_1\not\equiv 0$ . Consequently, (1.2) admits no global solution. The proof is completed.

Proof of Theorem 3.2. Suppose that $u\in X_a$ is a global solution to (1.2). Notice that in the proof of Theorem 3.1, to obtain (4.20), we used that

$p > m\geq 1,\quad p > \sigma+1,\quad p > m(\sigma+1),\quad p > m(\gamma+1).$

On the other hand, by (3.3)–(3.5), it can be easily seen that the above conditions are satisfied. Thus, (4.20) holds. Hence, taking $p = 1+\frac{\gamma}{\beta}$ in (4.20), we obtain

$\begin{equation} \int_a^\infty |u_1(t)|^m \frac{1}{t}\,dt \leq C\left[\left(\ln \frac{T}{a}\right)^{1-\frac{\alpha p+\sigma}{p-1}}+\left(\ln \frac{T}{a}\right)^{1-\frac{\sigma m}{p-m}}(T-a)^{\beta- \frac{\gamma m+\beta p}{p-m}}\right]. \end{equation}$

(4.21)

On the other hand, by (3.3)–(3.5), we have

$1-\frac{\alpha p+\sigma}{p-1} < 0,\quad \beta- \frac{\gamma m+\beta p}{p-m} < 0.$

Hence, passing to the limit as $T\to \infty$ in (4.21), we obtain a contradiction with $u_1\not\equiv 0$ . This shows that (1.2) admits no global solution. The proof is completed.

5. Conclusions

The two-times fractional differential inequality (1.2) is investigated. Namely, using the test function method and a judicious choice of a test function, sufficient conditions ensuring the nonexistence of global solutions to (1.2) are obtained. Two cases are discussed separately: $1 < p < 1+\frac{\gamma}{\beta}$ (see Theorem 3.1) and $p = 1+\frac{\gamma}{\beta}$ (see Theorem 3.2). In the first case, no assumption is imposed on the fractional order $\alpha\in (0, 1)$ of the Hadamard-Caputo fractional derivative, while in the second case, it is supposed that $\alpha > 1-\frac{1}{m}$ . About the initial conditions, in both cases, it is assumed that $u_0\in L^1([a, \infty))$ , $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$ , and $u_1\not\equiv 0$ .

Finally, it would be interesting to extend this study to two-times fractional evolution equations. For instance, the tow-times fractional semi-linear heat equation

$\begin{eqnarray*} \left\{\begin{array}{llll} {}^{HC}\!D_{a|t}^\alpha u (t,s,x)+{}^C\!D_{a|s}^\beta |u|^m (t,s,x) \geq (s-a)^\gamma \left(\ln \frac{t}{a}\right)^\sigma |u|^p(t,s,x),\quad t,s > a,\, x\in \mathbb{R}^N,\\ \\ u(a,s,x) = u_0(s,x),\, u(t,a,x) = u_1(t,x),\quad t,s > a, \, x\in \mathbb{R}^N, \end{array} \right. \end{eqnarray*}$

deserves to be studied.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-02.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	World Health Organization, Coronavirus disease (COVID-19), 2023. Available from: https://www.who.int/news-room/fact-sheets/detail/coronavirus-disease-(covid-19).
[2]	COVID-19, 2020. Available from: https://www.cdc.gov/coronavirus/2019-ncov/your-health/about-covid-19.html.
[3]	Tuberculosis (TB)-basic TB facts, 2019. Available from: https://www.cdc.gov/tb/topic/basics/default.html.
[4]	World Health Organization, Tuberculosis and COVID- 19, 2020. Available from: https://www.who.int/teams/global-tuberculosis-programme/covid-19.
[5]	H. Yang, S. Lu, COVID-19 and tuberculosis, J. Transl. Int. Med., 8 (2020), 59–65. https://doi.org/10.2478/jtim-2020-0010
[6]	G. T. Mousquer, A. Peres, M. Fiegenbaum, Pathology of TB/COVID-19 co-infection: the phantom menace, Tuberculosis, 126 (2021), 102020. https://doi.org/10.1016/j.tube.2020.102020 doi: 10.1016/j.tube.2020.102020
[7]	B. Diao, C. Wang, Y. Tan, X. Chen, Y. Liu, L. Ning, et al., Reduction and functional exhaustion of T cells in patients with coronavirus disease 2019 (COVID-19), Front. Immunol., 11 (2020), 827. https://doi.org/10.3389/fimmu.2020.00827 doi: 10.3389/fimmu.2020.00827
[8]	M. Khayat, H. Fan, Y. Vali, COVID-19 promoting the development of active tuberculosis in a patient with latent tuberculosis infection: a case report, Respir. Med. Case Rep., 32 (2021), 101344. https://doi.org/10.1016/j.rmcr.2021.101344 doi: 10.1016/j.rmcr.2021.101344
[9]	M. Tadolini, L. R. Codecasa, J. M. García-García, F. X. Blanc, S. Borisov, J. W. Alffenaar, et al., Active tuberculosis, sequelae and COVID-19 co-infection: first cohort of 49 cases, Eur. Respir. J., 56 (2020), 2001398. https://doi.org/10.1183/13993003.01398-2020 doi: 10.1183/13993003.01398-2020
[10]	A. Abdoli, S. Falahi, A. Kenarkoohi, COVID-19-associated opportunistic infections: a snapshot on the current reports, Clin. Exp. Med., 22 (2022), 327–346. https://doi.org/10.1007/s10238-021-00751-7 doi: 10.1007/s10238-021-00751-7
[11]	M. Kretzschmar, Disease modeling for public health: added value, challenges, and institutional constraints, J. Public Health Policy, 41 (2020), 39–51. https://doi.org/10.1057/s41271-019-00206-0 doi: 10.1057/s41271-019-00206-0
[12]	Y. Wu, M. Huang, X. Wang, Y. Li, L. Jiang, Y. Yuan, The prevention and control of tuberculosis: an analysis based on a tuberculosis dynamic model derived from the cases of Americans, BMC Publ. Health, 20 (2020), 1173. https://doi.org/10.1186/s12889-020-09260-w doi: 10.1186/s12889-020-09260-w
[13]	L. N. Nkamba, T. T. Manga, F. Agouanet, M. L. Mann Manyombe, Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis, J. Biol. Dyn., 13 (2019), 26–42. https://doi.org/10.1080/17513758.2018.1563218 doi: 10.1080/17513758.2018.1563218
[14]	S. Liu, Y. Bi, Y. Liu, Modeling and dynamic analysis of tuberculosis in mainland China from 1998 to 2017: the effect of DOTS strategy and further control, Theor. Biol. Med. Modell., 2020 (2020), 17. https://doi.org/10.1186/s12976-020-00124-9 doi: 10.1186/s12976-020-00124-9
[15]	K. C. Chong, C. C. Leung, W. W. Yew, B. C. Y. Zee, G. C. H. Tam, M. H. Wang, et al., Mathematical modelling of the impact of treating latent tuberculosis infection in the elderly in a city with intermediate tuberculosis burden. Sci. Rep., 9 (2019), 4869. https://doi.org/10.1038/s41598-019-41256-4 doi: 10.1038/s41598-019-41256-4
[16]	T. A. Perkins, G. Espa $\tilde{n}$ a, Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions, Bull. Math. Biol., 82 (2020), 118. https://doi.org/10.1007/s11538-020-00795-y doi: 10.1007/s11538-020-00795-y
[17]	S. I. Oke, M. I. Ekum, O. J. Akintande, M. O. Adeniyi, T. A. Adekiya, O. J. Achadu, et al., Optimal control of the coronavirus pandemic with both pharmaceutical and non-pharmaceutical interventions, Int. J. Dyn. Control, 11 (2023), 2295–2319. https://doi.org/10.1007/s40435-022-01112-2 doi: 10.1007/s40435-022-01112-2
[18]	M. Zamir, F. Nadeem, M. A. Alqudah, T. Abdeljawad, Future implications of COVID-19 through mathematical modeling, Results Phys., 33 (2022), 105097. https://doi.org/10.1016/j.rinp.2021.105097 doi: 10.1016/j.rinp.2021.105097
[19]	L. Masandawa, S. S. Mirau, I. S. Mbalawata, Mathematical modeling of COVID-19 transmission dynamics between healthcare workers and community, Results Phys., 29 (2021), 104731. https://doi.org/10.1016/j.rinp.2021.104731 doi: 10.1016/j.rinp.2021.104731
[20]	A. O. Atede, A. Omame, S. C. Inyama, A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data, Bull. Biomath., 1 (2023), 78–110. https://doi.org/10.59292/bulletinbiomath.2023005 doi: 10.59292/bulletinbiomath.2023005
[21]	B. Yang, Z. Yu, Y. Cai, The impact of vaccination on the spread of COVID-19: studying by a mathematical model, Phys. A, 590 (2022), 12671. https://doi.org/10.1016/j.physa.2021.126717 doi: 10.1016/j.physa.2021.126717
[22]	A. Kouidere, O. Balatif, M. Rachik, Cost-effectiveness of a mathematical modeling with optimal control approach of spread of COVID-19 pandemic: a case study in Peru, Chaos Solitons Fract., 10 (2023), 100090. https://doi.org/10.1016/j.csfx.2022.100090 doi: 10.1016/j.csfx.2022.100090
[23]	C. N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C. R. MacIntyre, M. H. Bonds, et al., Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus, Math. Biosci., 325 (2020), 108364. https://doi.org/10.1016/j.mbs.2020.108364 doi: 10.1016/j.mbs.2020.108364
[24]	O. Sharomi, C. N. Podder, A. B. Gumel, B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng., 5 (2008), 145–174. https://doi.org/10.3934/mbe.2008.5.145 doi: 10.3934/mbe.2008.5.145
[25]	Z. Mukandavire, A. B. Gumel, W. Garira, J. M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Math. Biosci. Eng., 6 (2009), 333–362. https://doi.org/10.3934/mbe.2009.6.333 doi: 10.3934/mbe.2009.6.333
[26]	H. T. Alemneh, A co-infection model of dengue and leptospirosis diseases, Adv. Differ. Equations, 2020 (2020), 664. https://doi.org/10.1186/s13662-020-03126-6 doi: 10.1186/s13662-020-03126-6
[27]	I. M. Hezam, A. Foul, A. Alrasheedi, A dynamic optimal control model for COVID-19 and cholera co-infection in Yemen, Adv. Differ. Equations, 2021 (2021), 108. htttps://doi.org/10.1186/s13662-021-03271-6 doi: 10.1186/s13662-021-03271-6
[28]	V. Guseva, N. Doktorova, O. Krivorotko, O. Otpushchennikova, L. Parolina, I. Vasilyeva, et al., Building a seir-model for predicting the HIV/tuberculosis coinfection epidemic for russian territories with low TB burden, Int. J. Infect. Dis., 134 (2023), S4-S5. https://doi.org/10.1016/j.ijid.2023.05.028 doi: 10.1016/j.ijid.2023.05.028
[29]	A. Ahmad, M. Farman, A. Akgül, N. Bukhari, S. Imtiaz, Mathematical analysis and numerical simulation of co-infection of TB-HIV, Arab J. Basic Appl. Sci., 27 (2020), 431–441. https://doi.org/10.1080/25765299.2020.1840771 doi: 10.1080/25765299.2020.1840771
[30]	A. Omame, A. D. Okuonghae, U. E. Nwafor, B. U. Odionyenma, A co-infection model for HPV and syphilis with optimal control and cost-effectiveness analysis, Int. J. Biomath., 14 (2021), 2150050. https://doi.org/10.1142/S1793524521500509 doi: 10.1142/S1793524521500509
[31]	A. Omame, M. Abbas, C. P. Onyenegecha, A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana-Baleanu derivative, Chaos Solitons Fract., 153 (2021), 111486. https://doi.org/10.1016/j.chaos.2021.111486 doi: 10.1016/j.chaos.2021.111486
[32]	A. Omame, M. Abbas, C. P. Onyenegecha, A fractional order model for the co-interaction of COVID-19 and Hepatitis B virus, Results Phys., 37 (2022), 105498. https://doi.org/10.1016/j.rinp.2022.105498 doi: 10.1016/j.rinp.2022.105498
[33]	H. Rwezaura, M. L. Diagne, A. Omame, A. L. de Espindola, J. M. Tchuenche, Mathematical modeling and optimal control of SARS-CoV-2 and tuberculosis co-infection: a case study of Indonesia, Model. Earth Syst. Environ., 8 (2022), 5493–5520. https://doi.org/10.1007/s40808-022-01430-6 doi: 10.1007/s40808-022-01430-6
[34]	R. I. Gweryina, C. E. Madubueze, V. P. Bajiya, F. E. Esla, Modeling and analysis of tuberculosis and pneumonia co-infection dynamics with cost-effective strategies, Results Control Optim., 10 (2023), 100210. https://doi.org/10.1016/j.rico.2023.100210 doi: 10.1016/j.rico.2023.100210
[35]	K. G. Mekonen, L. L. Obsu, Mathematical modeling and analysis for the co-infection of COVID-19 and tubercu-losis, Heliyon, 8 (2022), e11195. https://doi.org/10.1016/j.heliyon.2022.e11195 doi: 10.1016/j.heliyon.2022.e11195
[36]	K. G. Mekonen, S. F. Balcha, L. L. Obsu, A. Hassen, Mathematical modeling and analysis of TB and COVID-19 coinfection, J. Appl. Math., 2022 (2022), 2449710. https://doi.org/10.1155/2022/2449710 doi: 10.1155/2022/2449710
[37]	A. Selvam, S. Sabarinathan, B. V. S. Kumar, H. Byeon, K. Guedri, S. M. Eldin, et al., Ulam-Hyers stability of tuberculosis and COVID-19 co-infection model under Atangana-Baleanu fractal-fractional operator, Sci. Rep., 13 (2023), 9012. https://doi.org/10.1038/s41598-023-35624-4 doi: 10.1038/s41598-023-35624-4
[38]	S. R. Bandekar, M. Ghosh, A co-infection model on TB- COVID-19 with optimal control and sensitivity analysis, Math. Comput. Simul., 200 (2022), 1–31. https://doi.org/10.1016/j.matcom.2022.04.001 doi: 10.1016/j.matcom.2022.04.001
[39]	F. Inayaturohmat, N. Anggriani, A. K. Supriatna, A mathematical model of tuberculosis and COVID-19 coinfection with the effect of isolation and treatment, Front. Appl. Math. Stat., 8 (2022), 958081. htttps://doi.org/10.3389/fams.2022.958081 doi: 10.3389/fams.2022.958081
[40]	Z. S. Kifle, L. L. Obsu, Co-dynamics of COVID-19 and TB with COVID-19 vaccination and exogenous reinfection for TB: an optimal control application, Infect. Dis. Modell., 8 (2023), 574–602. https://doi.org/10.1016/j.idm.2023.05.005 doi: 10.1016/j.idm.2023.05.005
[41]	S. W. Teklu, Y. F. Abebaw, B. B. Terefe, D. K. Mamo, HIV/AIDS and TB co-infection deterministic model bifurcation and optimal control analysis, Inf. Med. Unlocked, 41 (2023), 101328. https://doi.org/10.1016/j.imu.2023.101328 doi: 10.1016/j.imu.2023.101328
[42]	B. S. Kotola, S. W. Teklu, Y. F. Abebaw, Bifurcation and optimal control analysis of HIV/AIDS and COVID-19 co-infection model with numerical simulation, PLoS One, 18 (2023), e0284759. https://doi.org/10.1371/journal.pone.0284759 doi: 10.1371/journal.pone.0284759
[43]	S. W. Teklu, Mathematical analysis of the transmission dynamics of COVID-19 infection in the presence of intervention strategies, J. Biol. Dyn., 16 (2022), 640–664. https://doi.org/10.1080/17513758.2022.2111469 doi: 10.1080/17513758.2022.2111469
[44]	S. W. Teklu, B. B. Terefe, D. K. Mamo, Y. F. Abebaw, Optimal control strategies on HIV/AIDS and pneumonia co-infection with mathematical modelling approach, J. Biol. Dyn., 18 (2024), 2288873. https://doi.org/10.1080/17513758.2023.2288873 doi: 10.1080/17513758.2023.2288873
[45]	D. Okuonghae, S. Omosigho, Analysis of a mathematical model for tuberculosis: what could be done to increase case detection, J. Theor. Biol., 269 (2011), 31–45. https://doi.org/10.1016/j.jtbi.2010.09.044 doi: 10.1016/j.jtbi.2010.09.044
[46]	S. M. Garba, J. M. S. Lubuma, B. Tsanou, Modeling the transmission dynamics of the COVID-19 pandemic in South Africa, Math. Biosci., 328 (2020), 108441. https://doi.org/10.1016/j.mbs.2020.108441 doi: 10.1016/j.mbs.2020.108441
[47]	S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. Int. Med., 172 (2020), 9. https://doi.org/10.7326/M20-0504 doi: 10.7326/M20-0504
[48]	A. Alemu, Z. W. Bitew, G. Seid, G. Diriba, E. Gashu, N. Berhe, et al., Tuberculosis in individuals who recovered from COVID-19: a systematic review of case reports, PLOS One, 17 (2022), 0277807. https://doi.org/10.1371/journal.pone.0277807 doi: 10.1371/journal.pone.0277807
[49]	D. Okuonghae, A. Omame, Analysis of a mathematical model for COVID-19 population dynamics in Lagos, Nigeria, Chaos Solitons Fract., 139 (2020), 110032. https://doi.org/10.1016/j.chaos.2020.110032 doi: 10.1016/j.chaos.2020.110032
[50]	C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of R0 and its role in global stability, Inst. Math. Appl., 125 (2002), 229.
[51]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6 doi: 10.1016/s0025-5564(02)00108-6
[52]	J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, 1976.
[53]	J. O. Akanni, A non-linear optimal control model for illicit drug use and terrorism dynamics in developing countries with time-dependent control variables, Decis. Anal. J., 8 (2023), 100281. https://doi.org/10.1016/j.dajour.2023.100281 doi: 10.1016/j.dajour.2023.100281
[54]	C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361
[55]	A. Abidemi, J. O. Akanni, O. D. Makinde, A non-linear mathematical model for analysing the impact of COVID-19 disease on higher education in developing countries, Healthcare Anal., 3 (2023), 100193. https://doi.org/10.1016/j.health.2023.100193 doi: 10.1016/j.health.2023.100193
[56]	S. M. Blower, H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example, Int. Stat. Rev., 62 (1994), 229–243. https://doi.org/10.2307/1403510 doi: 10.2307/1403510
[57]	M. A. Sanchez, S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: tuberculosis as an example, Amer. J. Epidemiol., 145 (1997), 1127–1137. https://doi.org/10.1093/oxfordjournals.aje.a009076 doi: 10.1093/oxfordjournals.aje.a009076
[58]	J. Wu, R. Dhingra, M. Gambhir, J. V. Remais, Sensitivity analysis of infectious disease models: methods, advances and their application, J. R. Soc. Interface, 10 (2013), 1018. https://doi.org/10.1098/rsif.2012.1018 doi: 10.1098/rsif.2012.1018
[59]	S. Olaniyi, J. O. Akanni, O. A. Adepoju, Optimal control and cost-effectiveness analysis of an illicit drug use population dynamics, J. Appl. Nonlinear Dyn., 12 (2023), 133–146. https://doi.org/10.5890/JAND.2023.03.010 doi: 10.5890/JAND.2023.03.010

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