Optimal control analysis of a COVID-19 and tuberculosis co-dynamics model

Abstract

Tuberculosis and COVID-19 are among the diseases with major global public health concern and great socio-economic impact. Co-infection of these two diseases is inevitable due to their geographical overlap, a potential double blow as their clinical similarities could hamper strategies to mitigate their spread and transmission dynamics. To theoretically investigate the impact of control measures on their long-term dynamics, we formulate and analyze a mathematical model for the co-infection of COVID-19 and tuberculosis. Basic properties of the tuberculosis only and COVID-19 only sub-models are investigated as well as bifurcation analysis (possibility of the co-existence of the disease-free and endemic equilibria). The disease-free and endemic equilibria are globally asymptotically stable. The model is extended into an optimal control system by incorporating five control measures. These are: tuberculosis awareness campaign, prevention against COVID-19 (e.g., face mask, physical distancing), control against co-infection, tuberculosis and COVID-19 treatment. Five strategies which are combinations of the control measures are investigated. Strategy B which focuses on COVID-19 prevention, treatment and control of co-infection yields a better outcome in terms of the number of COVID-19 cases prevented at a lower percentage of the total cost of this strategy.

1. Introduction

Mycobacterium tuberculosis is the causative agent of tuberculosis (TB), one of the top ten leading causes of death due to a single disease worldwide [1]. The transmission route are through cough, sneeze, speak or spit from active pulmonary TB persons, it can also be spread through use of an infected person’s unsterilized eating utensils [2]. The chain of transmission could be broken by isolating and treating infectives [3].

On the other hand, COVID-19 caused by the coronavirus SARS-CoV-2 emerged in December 2019 [4], [5], [6], and spread worldwide like a wildfire [7]. The virus can spread from an infected person’s mouth or nose when they cough, sneeze, speak, sing or breathe. These similarities of COVID-19 spreading pattern and TB call for great attention [8]. While most people who fall sick with COVID-19 may experience mild to moderate symptoms and recover without special treatment, the pandemic has claim millions of lives. COVID-19 prevention interventions (non-therapeutic measures) such as wearing face masks, self-isolation, physical distancing, and the most restrictive lock-downs could affect the transmission dynamic of TB. With the combination of non therapeutic prevention interventions and therapeutic measures, the number of COVID-19 cases and deaths have reduced despite the emergence of new variants of the COVID-19 virus [9].

Geographical overlap of both diseases and their clinical similarities could be a double blow in mitigating their spread because of the potential fatal outcome if they are not properly diagnosed and adequately treated [8]. Co-infection of both diseases is inevitable given the high worldwide prevalence of TB and COVID-19 [10], [11], [12], [13]. Co-interaction of TB and COVID-19 may pose a challenge in mitigating their spread, as TB is a risk factor for COVID-19 both in terms of severity and mortality [14]. In fact, co-infection related mortality is higher (about 12.3%) in the patients with dual infection [15]. Also, COVID-19 patients have a low ability to build an immune response to TB, while in co-infected subjects, TB could impair the ability to mount a SARS-CoV-2 specific immune response [16]. These two diseases (TB and COVID-19) of global public health concern form a deadly duo, with great socio-economic impact worldwide [10]. From the aforementioned reasons, it is important to theoretically investigate the impact of control measures on their long-term dynamics.

We formulate and analyze a mathematical model for the co-infection of COVID-19 and TB transmission. Basic properties of the two sub-models, namely TB only and COVID-19 only) are investigated as well as the possibility of the co-existence of the disease-free and endemic equilibria (bifurcation analysis). Optimal control strategies are incorporated into the model, and conditions for the existence of optimal control and the optimality system for the co-infection model are established using the Pontryagin’s maximum Principle.

The paper is organized as follows. The proposed co-infection model is formulated in Section 2. The model and its sub-models, TB and COVID-19 are rigorously analyzed in Section 3. In order to mitigate the spread of these two diseases and their co-infection, time variant controls are introduced into the full model, and the obtained optimal control problem investigated via the Pontryagin’s Maximum Principle in Section 4. To support the theoretical results, numerical simulations are provided in Section NS, where five scenarios being combinations of various control strategies are investigated. Finally, Section 6 is the conclusion where it is noted that the best scenario in terms of the potential number of COVID-19 cases that could be prevented (at a lower percentage of the total cost) is Strategy B which focuses on COVID-19 prevention, treatment and control of co-infection.

2. The model

The population at time $t$ denoted by $N\left(t\right)$ is divided into sub-populations of susceptible individuals $S\left(t\right)$, individuals exposed to COVID-19 only $E{}_{\text{C}}\left(t\right)$, unreported individuals infected with COVID-19 only $I{}_{\text{CU}}\left(t\right)$, reported individuals infected with COVID-19 only $I{}_{\text{CR}}\left(t\right)$, individuals exposed to tuberculosis only $E{}_{\text{T}}\left(t\right)$, unreported individuals infected with tuberculosis only $I{}_{\text{TU}}\left(t\right)$, reported individuals infected with tuberculosis only $I{}_{\text{TR}}\left(t\right)$, individuals exposed to tuberculosis and COVID-19 $E{}_{\text{CT}}\left(t\right)$, individuals infected with tuberculosis and COVID-19 $I{}_{\text{CT}}\left(t\right)$ and recovered individuals $R\left(t\right)$. It is important to note that all exposed individuals herein are actually asymptomatic and can transmit either of the disease as per their disease status.

The model has the following assumptions:

• i.individuals infected with COVID-19 are susceptible to infection with tuberculosis and vice versa.
• ii.co-infected individuals can transmit either COVID-19 or tuberculosis but not the mixed infections at the same time,
• iii.co-infected individuals can recover either from COVID-19 or tuberculosis but not from the mixed infection at the same time,
• iv.rate of transmissibility for singly infected and co-infected individuals are assumed to be the same.

Individuals are recruited into the population through birth or immigration at the rate $\omega {}_{\text{H}}$. Susceptible humans $S$ acquire COVID-19 following effective contacts with either singly or co-infected individuals with COVID-19 at the rate

$$\lambda {}_{\text{C}}=\frac{\Lambda {}_{\text{C}}(E{}_{\text{C}}+I{}_{\text{CU}})}{N}.$$ (1)

Similarly, the population $S$, is reduced due to infection with tuberculosis at the rate

$$\lambda {}_{\text{T}}=\frac{\Lambda {}_{\text{T}}I{}_{\text{TU}}}{N}.$$ (2)

From the model flow diagram in Fig. 1 and the above description, we derive the following nonlinear system ordinary differential equations for the COVID-19 and tuberculosis co-infection.

$$\left\{\begin{array}{cc}\hfill \frac{dS}{dt}& =\omega {}_{\text{H}}+\omega {}_{\text{R}}R-\lambda {}_{\text{C}}S-\lambda {}_{\text{T}}S-\mu {}_{\text{H}}S,\hfill \\ \hfill \frac{dE{}_{\text{C}}}{dt}& =\lambda {}_{\text{C}}S-\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}-(\eta {}_{\text{R}}+\eta {}_{\text{U}})E{}_{\text{C}}-\mu {}_{\text{H}}E{}_{\text{C}},\hfill \\ \hfill \frac{dI{}_{\text{CR}}}{dt}& =\eta {}_{\text{R}}E{}_{\text{C}}+\alpha {}_{\text{C}}I{}_{\text{CT}}-(\alpha {}_{\text{CR}}+\alpha {}_{\text{C}})I{}_{\text{CR}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CR}})I{}_{\text{CR}},\hfill \\ \hfill \frac{dI{}_{\text{CU}}}{dt}& =\eta {}_{\text{U}}E{}_{\text{C}}-(\gamma {}_{\text{CU}}+\alpha {}_{\text{CU}})I{}_{\text{CU}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}},\hfill \\ \hfill \frac{dE{}_{\text{CT}}}{dt}& =\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}+\beta {}_{\text{T}}\lambda {}_{\text{T}}E{}_{\text{T}}-\gamma {}_{\text{CT}}E{}_{\text{CT}}-\mu {}_{\text{H}}E{}_{\text{CT}},\hfill \\ \hfill \frac{dI{}_{\text{CT}}}{dt}& =\gamma {}_{\text{CT}}E{}_{\text{CT}}+\gamma {}_{\text{TU}}I{}_{\text{TU}}+\gamma {}_{\text{CU}}I{}_{\text{CU}}+\alpha {}_{\text{CR}}I{}_{\text{CR}}+\alpha {}_{\text{TR}}I{}_{\text{TR}}\hfill \\ \hfill & -(\alpha {}_{\text{C}}+\alpha {}_{\text{T}})I{}_{\text{CT}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CT}})I{}_{\text{CT}},\hfill \\ \hfill \frac{dE{}_{\text{T}}}{dt}& =\lambda {}_{\text{T}}S-(\beta {}_{\text{T}}\lambda {}_{\text{T}}+\theta {}_{\text{U}}+\theta {}_{\text{R}})E{}_{\text{T}}-\mu {}_{\text{H}}E{}_{\text{T}},\hfill \\ \hfill \frac{dI{}_{\text{TU}}}{dt}& =\theta {}_{\text{U}}E{}_{\text{T}}-(\gamma {}_{\text{TU}}+\alpha {}_{\text{TU}})I{}_{\text{TU}}-(\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}},\hfill \\ \hfill \frac{dI{}_{\text{TR}}}{dt}& =\theta {}_{\text{R}}E{}_{\text{T}}+\alpha {}_{\text{T}}I{}_{\text{CT}}-(\alpha {}_{\text{TR}}+\alpha {}_{\text{T}})I{}_{\text{TR}}-(\mu {}_{\text{H}}+\varphi {}_{\text{TR}})I{}_{\text{TR}},\hfill \\ \hfill \frac{dR}{dt}& =\alpha {}_{\text{C}}I{}_{\text{CR}}+\alpha {}_{\text{CU}}I{}_{\text{CU}}+\alpha {}_{\text{TU}}I{}_{\text{TU}}+\alpha {}_{\text{T}}I{}_{\text{TR}}-\mu {}_{\text{H}}R-\omega {}_{\text{R}}R,\hfill \end{array}\right)$$ (3)

together with initial conditions

$$\begin{array}{c}S\left(0\right)\ge 0,E{}_{\text{C}}\left(0\right)\ge 0,E{}_{\text{T}}\left(0\right)\ge 0,E{}_{\text{CT}}\left(0\right)\ge 0,I{}_{\text{CR}}\left(0\right)\ge 0,\hfill \\ I{}_{\text{CU}}\left(0\right)\ge 0,I{}_{\text{CT}}\left(0\right)\ge 0,I{}_{\text{TU}}\left(0\right)\ge 0,I{}_{\text{TR}}\left(0\right)\ge 0,R\left(0\right)\ge 0.\hfill \end{array}$$ (4)

Fig. 1.

COVID-19 and tuberculosis co-interaction flow diagram.

3. Model analysis

Two sub-models, namely: Tuberculosis only and COVID-19 only sub-models will first be considered.

3.1. COVID-19 only sub-model

By setting $E{}_{\text{T}}=E{}_{\text{CT}}=I{}_{\text{CT}}=I{}_{\text{TU}}=I{}_{\text{TR}}=0$, we obtain the following COVID-19 only sub-model.

$$\left\{\begin{array}{cc}\hfill \frac{dS}{dt}& =\omega {}_{\text{H}}+\omega {}_{\text{R}}R-\lambda {}_{\text{C}}S-\mu {}_{\text{H}}S,\hfill \\ \hfill \frac{dE{}_{\text{C}}}{dt}& =\lambda {}_{\text{C}}S-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{C}},\hfill \\ \hfill \frac{dI{}_{\text{CR}}}{dt}& =\eta {}_{\text{R}}E{}_{\text{C}}-(\alpha {}_{\text{C}}+\mu {}_{\text{H}}+\varphi {}_{\text{CR}})I{}_{\text{CR}},\hfill \\ \hfill \frac{dI{}_{\text{CU}}}{dt}& =\eta {}_{\text{U}}E{}_{\text{C}}-(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}},\hfill \\ \hfill \frac{dR}{dt}& =\alpha {}_{\text{C}}I{}_{\text{CR}}+\alpha {}_{\text{CU}}I{}_{\text{CU}}-\mu {}_{\text{H}}R-\omega {}_{\text{R}}R,\hfill \end{array}\right)$$ (5)

where $N{}_{\text{C}}=S+E{}_{\text{C}}+I{}_{\text{CR}}+I{}_{\text{CU}}+R$. By adding up all the equations of the system (5), we have

$$\dot{N}{}_{\text{C}}=\omega {}_{\text{H}}-\mu {}_{\text{H}}N{}_{\text{C}}-\varphi {}_{\text{CR}}I{}_{\text{CR}}-\varphi {}_{\text{CU}}I{}_{\text{CU}}⩽\omega {}_{\text{H}}-\mu {}_{\text{H}}N{}_{\text{C}}.$$ (6)

The given initial conditions of the sub-model system (5) ensure that $N\left(0\right)\ge 0$. Thus, the total human population is positive and bounded for all finite time $t>0$. From the theory of differential inequality [31], we have

$$N{}_{\text{C}}\left(t\right)⩽N{}_{\text{C}}\left(0\right)e^{-\mu {}_{\text{H}}t}+\frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}}(1-e^{-\mu {}_{\text{H}}t}).$$ (7)

As $t\to +\infty$, we obtain $0\le N{}_{\text{C}}\left(t\right)\le \frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}}$. The feasible region of the COVID-19 only sub-model (5) is given by

$$\Omega {}_{\text{C}}=\left\{(S,E{}_{\text{C}},I{}_{\text{CR}},I{}_{\text{CU}},R)\in {\mathbb{R}}_{+}^5:N{}_{\text{C}}\left(t\right)\le \frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}}\right\}.$$ (8)

The set ${\Omega }_c$ is positively invariant and attracting [32], and all solutions of the COVID-19 only sub-model (5) starting in ${\Omega }_c$ remain in ${\Omega }_c$ for all $t\ge 0$. Thus, the model (5) is mathematically and epidemiologically well-posed, and it is sufficient to study its dynamics in ${\Omega }_c$ [3], [33].

3.1.1. Stability of the disease-free equilibrium

The disease-free equilibrium (DFE) of the COVID-19 only sub-model system (5) is obtained when $E{}_{\text{C}}=I{}_{\text{CR}}=I{}_{\text{CU}}=R=0$. Thus, the DFE of the COVID-19 only sub-model (5) is given by

$${\mathcal{E}}_{C_0}=(S^0,E{}_{\text{C}}^0,I{}_{\text{CR}}^0,I{}_{\text{CU}}^0,R^0)=\left(\frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}},0,0,0,0\right).$$ (9)

The linear stability of ${\mathcal{E}}_{C_0}$ is established using the next generation operator method on system (5) as described in [34]. System (5) can be written as

$$\dot{x}=f\left(x\right)=\mathcal{F}\left(x\right)-\mathcal{V}\left(x\right),$$ (10)

where

$$\mathcal{F}=\left(\begin{array}{c}\hfill \lambda {}_{\text{C}}S\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),$$

and

$$\mathcal{V}=\left(\begin{array}{c}\hfill (\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{C}}\hfill \\ \hfill -\eta {}_{\text{R}}E{}_{\text{C}}+(\alpha {}_{\text{C}}+\mu {}_{\text{H}}+\varphi {}_{\text{CR}})I{}_{\text{CR}}\hfill \\ \hfill -\eta {}_{\text{U}}E{}_{\text{C}}+(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}}\hfill \end{array}\right).$$

are the new infection and transfer terms respectively. Evaluating the Jacobian of $\mathcal{F}$ and $\mathcal{V}$ at the DFE ${\mathcal{E}}_{C_0}$ gives

$$F=\left(\begin{array}{ccc}\hfill \Lambda {}_{\text{C}}\hfill & \hfill 0\hfill & \hfill \Lambda {}_{\text{C}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

and

$$V=\left(\begin{array}{ccc}\hfill \eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\eta {}_{\text{R}}\hfill & \hfill \alpha {}_{\text{C}}+\mu {}_{\text{H}}+\varphi {}_{\text{CR}}\hfill & \hfill 0\hfill \\ \hfill -\eta {}_{\text{U}}\hfill & \hfill 0\hfill & \hfill \alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}}\hfill \end{array}\right).$$

Set

$$A{}_{\text{1}}=\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}},A{}_{\text{2}}=\alpha {}_{\text{C}}+\mu {}_{\text{H}}+\varphi {}_{\text{CR}}\text{and}A{}_{\text{3}}=\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}}.$$ (11)

The inverse of the matrix $V$ is given by

$$V^{-1}=\left(\begin{array}{ccc}\hfill \frac{1}{A{}_{\text{1}}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{\eta {}_{\text{R}}}{A{}_{\text{1}}A{}_{\text{2}}}\hfill & \hfill \frac{1}{A{}_{\text{2}}}\hfill & \hfill 0\hfill \\ \hfill \frac{\eta {}_{\text{U}}}{A{}_{\text{1}}A{}_{\text{3}}}\hfill & \hfill 0\hfill & \hfill \frac{1}{A{}_{\text{3}}}\hfill \end{array}\right).$$

The largest eigenvalue of the next generation matrix $F{V}^{-1}$ denoted by $R_{0_C}$ is given by

$$R_{0_C}=\frac{\Lambda {}_{\text{C}}\eta {}_{\text{U}}}{A{}_{\text{1}}A{}_{\text{3}}}+\frac{\Lambda {}_{\text{C}}}{A{}_{\text{1}}}=\frac{\Lambda {}_{\text{C}}(\eta {}_{\text{U}}+\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})}{(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})}.$$ (12)

The basic reproduction number $R_{0_C}$ is defined as the expected number of secondary cases generated by one infected individual during its entire period of infectiousness in a fully susceptible population [34]. From Theorem 2 of [34], the following result follows.

Lemma 3.1

The disease-free equilibrium ${\mathcal{E}}_{C_0}$ of the COVID-19 only sub-model system (5) is locally asymptotically stable if $R_{0_C}<1$ , and unstable otherwise.

Proof

The stability of ${\mathcal{E}}_{C_0}$ is obtained from the roots of the characteristic polynomial, which states that the equilibrium is stable if the roots of the characteristic polynomial are all negative. For ${\mathcal{E}}_{C_0}$, the Jacobian matrix of the system is obtained as

$$J\left({\mathcal{E}}_{C0}\right)=\left(\begin{array}{ccccc}\hfill -\mu {}_{\text{H}}\hfill & \hfill -\Lambda {}_{\text{C}}\hfill & \hfill 0\hfill & \hfill -\Lambda {}_{\text{C}}\hfill & \hfill \omega {}_{\text{R}}\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \Lambda {}_{\text{C}}-A{}_{\text{1}}\hfill & \hfill 0\hfill & \hfill \Lambda {}_{\text{C}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \eta {}_{\text{R}}\hfill & \hfill -A{}_{\text{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \eta {}_{\text{U}}\hfill & \hfill 0\hfill & \hfill -A{}_{\text{3}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \alpha {}_{\text{C}}\hfill & \hfill \alpha {}_{\text{CU}}\hfill & \hfill -(\mu {}_{\text{H}}+\omega {}_{\text{R}})\hfill \end{array}\right).$$

where $A_i;i=1,2,3$ are given in (11). The characteristic polynomial is given by

$$P\left(\lambda \right)=(\lambda +\mu {}_{\text{H}})(\lambda +\mu {}_{\text{H}}+\omega {}_{\text{R}})(-\lambda -A{}_{\text{2}})((-\lambda +\Lambda {}_{\text{C}}-A{}_{\text{1}})(-\lambda -A{}_{\text{3}})-\Lambda {}_{\text{C}}\eta {}_{\text{U}})=(\lambda +\mu {}_{\text{H}})(\lambda +\mu {}_{\text{H}}+\omega {}_{\text{R}})(-\lambda -A{}_{\text{2}})({\lambda }^2+\lambda (A{}_{\text{1}}-\Lambda {}_{\text{C}}+A{}_{\text{3}})$$$$+A{}_{\text{1}}A{}_{\text{3}}-\Lambda {}_{\text{C}}A{}_{\text{3}}-\Lambda {}_{\text{C}}\eta {}_{\text{U}}).$$ (13)

Using the Routh–Hurwitz criterion for second order polynomials, we have for $\Lambda {}_{\text{C}}<\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}}$ and $R_{0_C}<1$ that $A{}_{\text{1}}-\Lambda {}_{\text{C}}+A{}_{\text{3}}$ and $A{}_{\text{1}}A{}_{\text{3}}-\Lambda {}_{\text{C}}A{}_{\text{3}}-\Lambda {}_{\text{C}}\eta {}_{\text{U}}$ are positive. Hence all eigenvalues are negative which means that the DFE ${\mathcal{E}}_{C_0}$ of the COVID-19 only sub-model system (5) is locally asymptotically stable when $R_{0_C}<1$. ■

Theorem 3.1

The DFE of the COVID-19 only model (5) is globally asymptotically stable for if $R_{0_C}<1$ .

Proof

Consider the following Lyapunov function

$$W=(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})E{}_{\text{C}}+\Lambda {}_{\text{C}}I{}_{\text{CU}}.$$

The time derivative of $W$ computed along the solutions of (5) is given by

$$\begin{array}{ccc}\hfill \dot{W}& \hfill =\hfill & (\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})\dot{E}{}_{\text{C}}+\Lambda {}_{\text{C}}\dot{I}{}_{\text{CU}},\hfill \\ \hfill \\ \hfill & \hfill =\hfill & (\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})\left[\lambda {}_{\text{C}}S-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{C}}\right]+\hfill \\ \hfill \\ \hfill & \hfill \hfill & \Lambda {}_{\text{C}}\left[\eta {}_{\text{U}}E{}_{\text{C}}-(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}}\right],\hfill \\ \hfill \\ \hfill & \hfill =\hfill & (\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})\left[\frac{\Lambda {}_{\text{C}}(E{}_{\text{C}}+I{}_{\text{CU}})}{N}S-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{C}}\right]+\hfill \\ \hfill \\ \hfill & \hfill \hfill & \Lambda {}_{\text{C}}\left[\eta {}_{\text{U}}E{}_{\text{C}}-(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}}\right],\hfill \\ \hfill \\ \hfill & \hfill ⩽\hfill & (\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})\left[\Lambda {}_{\text{C}}(E{}_{\text{C}}+I{}_{\text{CU}})-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{C}}\right]+\hfill \\ \hfill \\ \hfill & \hfill \hfill & \Lambda {}_{\text{C}}\left[\eta {}_{\text{U}}E{}_{\text{C}}-(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}}\right],\hfill \\ \hfill \\ \hfill & \hfill ⩽\hfill & (\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})\Lambda {}_{\text{C}}E{}_{\text{C}}-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})E{}_{\text{C}}\hfill \\ \hfill & \hfill \hfill & +\Lambda {}_{\text{C}}\eta {}_{\text{U}}E{}_{\text{C}},\hfill \\ \hfill \\ \hfill & \hfill ⩽\hfill & \Lambda {}_{\text{C}}(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}}+\eta {}_{\text{U}})E{}_{\text{C}}-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})E{}_{\text{C}},\hfill \\ \hfill \\ \hfill & \hfill ⩽\hfill & (\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})E{}_{\text{C}}(R_{0_C}-1),\hfill \\ \hfill \\ \hfill & \hfill ⩽\hfill & 0,\text{ for }{R}_{0_C}⩽1.\hfill \end{array}$$

Because all model parameters are non-negative, it follows that $\dot{W}⩽0$ for $R_{0_C}⩽1$, with $\dot{W}=0$ if and only if $E{}_{\text{C}}=I{}_{\text{CR}}=I{}_{\text{CU}}=0$. Substituting $(E{}_{\text{C}},I{}_{\text{CR}},I{}_{\text{CU}})=(0,0,0)$ into 3.1 shows that $S\to \frac{\Pi {}_{\text{H}}}{\alpha {}_{\text{H}}}$ as $t\to \infty$. Hence, $W$ is a Lyapunov function on ${\Omega }_c$ and the largest compact invariant set in $\{(S,E{}_{\text{C}},I{}_{\text{CR}},I{}_{\text{CU}},R)\in {\Omega }_c:\dot{W}=0\}$ is ${\mathcal{E}}_c$. Thus, by LaSalle’s invariance principle, every solution of (5), with initial conditions in ${\Omega }_c$ approaches ${\mathcal{E}}_c$, as $t\to \infty$ whenever $R_{0_C}⩽1$. ■

3.1.2. Stability of the endemic equilibrium

We explore the stability of the endemic equilibrium of the COVID-19 only sub-model system (5) given by

$${\mathcal{E}}_{C_1}=(S_{C_1},E{}_{\text{C}}{}_{C_1},I{}_{\text{CR}}{}_{C_1},I{}_{\text{CU}}{}_{C_1},R{}_{C_1}),=\left(\frac{\omega {}_{\text{H}}}{\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}}},\frac{\omega {}_{\text{H}}\lambda {}_{\text{C}}^{\star }}{(\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}})A{}_{\text{1}}},\frac{\omega {}_{\text{H}}\lambda {}_{\text{C}}^{\star }\eta {}_{\text{R}}}{(\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}})A{}_{\text{1}}A{}_{\text{2}}},\frac{\omega {}_{\text{H}}\lambda {}_{\text{C}}^{\star }\eta {}_{\text{U}}}{(\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}})A{}_{\text{1}}A{}_{\text{3}}},\right)$$$$\left(\frac{\omega {}_{\text{H}}\lambda {}_{\text{C}}^{\star }(\alpha {}_{\text{C}}\eta {}_{\text{R}}A{}_{\text{3}}+\alpha {}_{\text{CU}}\eta {}_{\text{U}}A{}_{\text{2}})}{\mu {}_{\text{H}}(\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}})A{}_{\text{1}}A{}_{\text{2}}A{}_{\text{3}}}\right),$$ (14)

where

$$\lambda {}_{\text{C}}^{\star }=\frac{\Lambda {}_{\text{C}}(E{}_{\text{C}}{}_{C_1}+I{}_{\text{CU}}{}_{C_1})}{S_{C_1}+E{}_{\text{C}}{}_{C_1}+I{}_{\text{CR}}{}_{C_1}+I{}_{\text{CU}}{}_{C_1}+R{}_{C_1}}.$$ (15)

Note that

$$E{}_{\text{C}}{}_{C_1}+I{}_{\text{CU}}{}_{C_1}=\frac{\omega {}_{\text{H}}\lambda {}_{\text{C}}^{\star }A{}_{\text{2}}(A{}_{\text{3}}+\eta {}_{\text{U}})}{\mu {}_{\text{H}}(\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}})A{}_{\text{1}}A{}_{\text{2}}A{}_{\text{3}}}.$$ (16)

Also,

$$S_{C_1}+E{}_{\text{C}}{}_{C_1}+I{}_{\text{CR}}{}_{C_1}+I{}_{\text{CU}}{}_{C_1}+R{}_{C_1}=\frac{\lambda {}_{\text{C}}^{\star }\left(A{}_{\text{2}}A{}_{\text{3}}+\eta {}_{\text{R}}A{}_{\text{3}}\left(1+\frac{\alpha {}_{\text{C}}}{\mu {}_{\text{H}}}\right)+\eta {}_{\text{U}}A{}_{\text{2}}\left(1+\frac{\alpha {}_{\text{CU}}}{\mu {}_{\text{H}}}\right)\right)+A{}_{\text{1}}A{}_{\text{2}}A{}_{\text{3}}}{\mu {}_{\text{H}}(\lambda {}_{\text{C}}^{\star }+\mu {}_{\text{H}})A{}_{\text{1}}A{}_{\text{2}}A{}_{\text{3}}}.$$ (17)

Substituting Eqs. (16), (18) into (15), we obtain

$$\lambda {}_{\text{C}}^{\star }\left(A{}_{\text{2}}A{}_{\text{3}}+\eta {}_{\text{R}}A{}_{\text{3}}\left(1+\frac{\alpha {}_{\text{C}}}{\mu {}_{\text{H}}}\right)+\eta {}_{\text{U}}A{}_{\text{2}}\left(1+\frac{\alpha {}_{\text{CU}}}{\mu {}_{\text{H}}}\right)\right)=\frac{A{}_{\text{2}}}{A{}_{\text{1}}A{}_{\text{3}}}\left(R_{0_C}-1\right).$$ (18)

Therefore, $\lambda {}_{\text{C}}^{\star }$ exists if and only if $R_{0_C}>1$. Hence, the following result.

Theorem 3.2

The COVID-19 only model system (5) has a unique endemic equilibrium if $R_{0_C}>1$ .

In the following, we use the center manifold approach to analyze the global stability of the full model. To this end we use the notation: $x_1=S$, $x_2=E{}_{\text{C}}$, $x_3=I{}_{\text{CR}}$, $x_4=I{}_{\text{CU}}$, $x_5=R$ and $N{}_{\text{C}}=x_1+x_2+x_3+x_4+x_5$ to write the model in the form $\dot{x}=f\left(x\right)$ with $x={(x_1,\dots ,x_5)}^T$ and $f=(f_1,\dots ,f_5)$. That is

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1& =f_1=\omega {}_{\text{H}}+\omega {}_{\text{R}}{x}_5-\frac{\Lambda {}_{\text{C}}(x_2+x_4)}{x_1+x_2+x_3+x_4+x_5}{x}_1-\mu {}_{\text{H}}{x}_1,\hfill \\ \hfill {\dot{x}}_2& =f_2=\frac{\Lambda {}_{\text{C}}(x_2+x_4)}{x_1+x_2+x_3+x_4+x_5}{x}_1-(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})x_2,\hfill \\ \hfill {\dot{x}}_3& =f_3=\eta {}_{\text{R}}{x}_2-(\alpha {}_{\text{C}}+\alpha {}_{\text{CR}}+\mu {}_{\text{H}}+\varphi {}_{\text{CR}})x_3,\hfill \\ \hfill {\dot{x}}_4& =f_4=\eta {}_{\text{U}}{x}_2-(\gamma {}_{\text{CU}}+\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})x_4,\hfill \\ \hfill {\dot{x}}_5& =f_5=\alpha {}_{\text{C}}{x}_3+\alpha {}_{\text{CU}}{x}_4-(\mu {}_{\text{H}}+\omega {}_{\text{R}})x_5,\hfill \end{array}\right)$$ (19)

The Jacobian of (19) at the DFE ${\mathcal{E}}_{C_0}$ is given by

$$J\left({\mathcal{E}}_{C_0}\right)=\left(\begin{array}{ccccc}\hfill -\mu {}_{\text{H}}\hfill & \hfill -\Lambda {}_{\text{C}}\hfill & \hfill 0\hfill & \hfill -\Lambda {}_{\text{C}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \Lambda {}_{\text{C}}-A{}_{\text{1}}\hfill & \hfill 0\hfill & \hfill \Lambda {}_{\text{C}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \eta {}_{\text{R}}\hfill & \hfill -A{}_{\text{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \eta {}_{\text{U}}\hfill & \hfill 0\hfill & \hfill -A{}_{\text{3}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \alpha {}_{\text{C}}\hfill & \hfill \alpha {}_{\text{CU}}\hfill & \hfill -\mu {}_{\text{H}}\hfill \end{array}\right),$$

where $A_i;i=1,2,3$ are given in Eq. (11).

We choose $\Lambda {}_{\text{C}}$ as a bifurcation parameter. Therefore, setting $R_{0_C}=1$, we obtain

$$\Lambda {}_{\text{C}}=\Lambda {}_{\text{C}}^{\star }=\frac{(\eta {}_{\text{R}}+\eta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})}{(\eta {}_{\text{U}}+\alpha {}_{\text{CU}}+\mu {}_{\text{H}}+\varphi {}_{\text{CU}})}.$$ (20)

At $\Lambda {}_{\text{C}}=\Lambda {}_{\text{C}}^{\star }$, the Jacobian has a simple zero eigenvalue (since $A{}_{\text{1}}A{}_{\text{3}}-\Lambda {}_{\text{C}}A{}_{\text{3}}-\Lambda {}_{\text{C}}\eta {}_{\text{U}}=0$, see Eq. (13)) and all other eigenvalues have negative real parts. Therefore, the DFE ${\mathcal{E}}_{C_0}$ is a non-hyperbolic equilibrium point. Hence, the center manifold theory [35] can be applied to model system (19) near $\Lambda {}_{\text{C}}=\Lambda {}_{\text{C}}^{\star }$.

The right eigenvector $w={(w_1,w_2,w_3,w_4,w_5)}^T$ associated with the zero eigenvalue of $J\left({\mathcal{E}}_{C_0}\right)$ evaluated at $\Lambda {}_{\text{C}}=\Lambda {}_{\text{C}}^{\star }$ is

$$w_1=\left(\frac{\omega {}_{\text{R}}}{\mu {}_{\text{H}}+\omega {}_{\text{R}}}\left(\frac{\alpha {}_{\text{C}}\eta {}_{\text{R}}}{\mu {}_{\text{H}}{A}_2}+\frac{\alpha {}_{\text{CU}}\eta {}_{\text{U}}}{\mu {}_{\text{H}}{A}_3}\right)-\frac{A_1}{\mu {}_{\text{H}}}\right)w_2,w_3=\frac{\eta {}_{\text{R}}}{A_2}{w}_2,$$$$w_4=\frac{\eta {}_{\text{U}}}{A_3}{w}_2,w_5=\frac{1}{\mu {}_{\text{H}}+\omega {}_{\text{R}}}\left(\frac{\alpha {}_{\text{C}}\eta {}_{\text{R}}}{A_2}+\frac{\alpha {}_{\text{CU}}\eta {}_{\text{U}}}{A_3}\right)w_2,w_2=w_2>0.$$ (21)

Similarly, the left eigenvector $v=(v_1,v_2,v_3,v_4,v_5)$ is given by

$$v_1=0,v_3=0,v_5=0,v_4=\frac{A_1}{\eta {}_{\text{U}}+A_3}{v}_2,v_2=v_2>0.$$ (22)

The left and right eigenvectors satisfy $v.w=1$ that is

$$v_2{w}_2\left(1+\frac{\eta {}_{\text{U}}{A}_1}{(\eta {}_{\text{U}}+A_3)A_3}\right)=1.$$ (23)

For the direction of the bifurcation, we determine the sign of the bifurcation parameters $a$ and $b$. For $a$, one has

$$a=\sum _{k,i,j=1}^5{v}_k{w}_i{w}_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}({\mathcal{E}}_{C_0},\Lambda {}_{\text{C}}^{\star }),=\sum _{i,j=1}^5{v}_2{w}_i{w}_j\frac{{\partial }^2{f}_2}{\partial x_i\partial x_j}({\mathcal{E}}_{C_0},\Lambda {}_{\text{C}}^{\star }).$$ (24)

The partial derivatives are

$$\frac{\partial f_2}{\partial x_1}=\lambda {}_{\text{C}}-\lambda {}_{\text{C}}\frac{x_1}{N{}_{\text{C}}},\frac{\partial f_2}{\partial x_2}=(\Lambda {}_{\text{C}}-\lambda {}_{\text{C}})\frac{x_1}{N{}_{\text{C}}},$$$$\frac{\partial f_2}{\partial x_3}=-\lambda {}_{\text{C}}\frac{x_1}{N{}_{\text{C}}},\frac{\partial f_2}{\partial x_4}=(\Lambda {}_{\text{C}}-\lambda {}_{\text{C}})\frac{x_1}{N{}_{\text{C}}},\frac{\partial f_2}{\partial x_5}=-\lambda {}_{\text{C}}\frac{x_1}{N{}_{\text{C}}}.$$ (25)

Therefore,

$$a=\frac{\mu {}_{\text{H}}\Lambda {}_{\text{C}}^{\star }}{\omega {}_{\text{H}}}{v}_2(w_1{w}_2+w_1{w}_4-w_2^2-w_2{w}_3-w_2{w}_4$$$$-w_2{w}_5-w_4{w}_2-w_4{w}_3-w_4^2-w_4{w}_5).$$ (26)

On the other hand, we have

$$w_1=\left(\frac{\omega {}_{\text{R}}}{\mu {}_{\text{H}}+\omega {}_{\text{R}}}\left(\frac{\alpha {}_{\text{C}}\eta {}_{\text{R}}}{\mu {}_{\text{H}}{A}_2}+\frac{\alpha {}_{\text{CU}}\eta {}_{\text{U}}}{\mu {}_{\text{H}}{A}_3}\right)-\frac{A_1}{\mu {}_{\text{H}}}\right)w_2,⩽\left(\frac{\alpha {}_{\text{C}}\eta {}_{\text{R}}}{\mu {}_{\text{H}}{A}_2}+\frac{\alpha {}_{\text{CU}}\eta {}_{\text{U}}}{\mu {}_{\text{H}}{A}_3}-\frac{A_1}{\mu {}_{\text{H}}}\right)w_2⩽\frac{1}{\mu {}_{\text{H}}}\left[\left(\frac{\alpha {}_{\text{C}}}{A_2}-1\right)\eta {}_{\text{R}}+\left(\frac{\alpha {}_{\text{CU}}}{A_3}-1\right)\eta {}_{\text{U}}-\mu {}_{\text{H}}\right]w_2⩽-\delta w_2.$$ (27)

with

$$\delta =\frac{1}{\mu {}_{\text{H}}}\left[\left(1-\frac{\alpha {}_{\text{C}}}{A_2}\right)\eta {}_{\text{R}}+\left(1-\frac{\alpha {}_{\text{CU}}}{A_3}\right)\eta {}_{\text{U}}+\mu {}_{\text{H}}\right]>0$$ (28)

Taking into account (27) in (29), the bifurcation parameter $a$ satisfies :

$$a⩽-\frac{\mu {}_{\text{H}}\Lambda {}_{\text{C}}^{\star }}{\omega {}_{\text{H}}}{v}_2(\delta w_2^2+\delta w_2{w}_4+w_2^2+w_2{w}_3+w_2{w}_4$$$$+w_2{w}_5+w_4{w}_2+w_4{w}_3+w_4^2+w_4{w}_5),⩽0.$$ (29)

For the bifurcation parameter $b$, we have

$$b=\sum _{k,j=1}^5{v}_k{w}_j\frac{{\partial }^2{f}_k}{\partial x_j\partial \Lambda {}_{\text{C}}}({\mathcal{E}}_{C_0},\Lambda {}_{\text{C}}^{\star }),=\sum _{j=1}^5{v}_2{w}_j\frac{{\partial }^2{f}_2}{\partial x_j\partial \Lambda {}_{\text{C}}}({\mathcal{E}}_{C_0},\Lambda {}_{\text{C}}^{\star }),=v_2{w}_2+v_2{w}_4,=v_2{w}_2\left(1+\frac{\eta {}_{\text{U}}}{A_3}\right)>0.$$ (30)

Since $a<0$ and $b>0$, the COVID-19 only sub-model system (5) does not exhibit the phenomenon of backward bifurcation at $R_{0_C}=1$. Because the direction of the bifurcation is forward (transcritical bifurcation), a stable disease-free equilibrium cannot co-exist with a stable endemic equilibrium. Similar result for the COVID-19 only model was obtained in [36], [37]. Hence, the following result.

Theorem 3.3

The unique endemic equilibrium ${\mathcal{E}}_{C_1}$ of the COVID-19 only sub-model (5) is globally asymptotically stable if $R_{0_C}>1$ .

The above result when $R_{0_C}>1$ is graphically depicted in Fig. 2. The red line in Fig. 2 represents the area of instability of the endemic equilibrium ${\mathcal{E}}_{C_1}$, and the blue line the stability area of the endemic equilibrium ${\mathcal{E}}_{C_1}$. The red dotted line represents the threshold stability switch line $R_{0_C}=1$. When $R_{0_C}>1$, the green line does not cross the dotted line, hence the endemic equilibrium ${\mathcal{E}}_{C_1}$ is globally asymptotically stable.

Fig. 2.

Illustration of the bifurcation by plotting $\Lambda {}_{\text{C}}$ versus $R_{0_C}$.

3.2. Tuberculosis only sub-model

The following tuberculosis only sub-model is obtained from system (3) when $E{}_{\text{C}}=E{}_{\text{CT}}=I{}_{\text{CT}}=I{}_{\text{CU}}=I{}_{\text{CR}}=0$.

$$\left\{\begin{array}{cc}\hfill \frac{dS}{dt}& =\omega {}_{\text{H}}-\lambda {}_{\text{T}}S-\mu {}_{\text{H}}S,\hfill \\ \hfill \frac{dE{}_{\text{T}}}{dt}& =\lambda {}_{\text{T}}S-(\theta {}_{\text{U}}+\theta {}_{\text{R}}+\mu {}_{\text{H}})E{}_{\text{T}},\hfill \\ \hfill \frac{dI{}_{\text{TU}}}{dt}& =\theta {}_{\text{U}}E{}_{\text{T}}-(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}},\hfill \\ \hfill \frac{dI{}_{\text{TR}}}{dt}& =\theta {}_{\text{R}}E{}_{\text{T}}-(\alpha {}_{\text{T}}+\mu {}_{\text{H}}+\varphi {}_{\text{TR}})I{}_{\text{TR}},\hfill \\ \hfill \frac{dR}{dt}& =\alpha {}_{\text{TU}}I{}_{\text{TU}}+\alpha {}_{\text{T}}I{}_{\text{TR}}-\mu {}_{\text{H}}R,\hfill \end{array}\right)$$ (31)

where $N{}_{\text{T}}=S+E{}_{\text{T}}+I{}_{\text{TR}}+I{}_{\text{TU}}+R$.

Arguing as in 3, the feasible region for the tuberculosis only sub-model

$$\Omega {}_{\text{T}}=\left\{(S,E{}_{\text{T}},I{}_{\text{TU}},I{}_{\text{TR}},R)\in {\mathbb{R}}_{+}^5:N{}_{\text{T}}\left(t\right)\le \frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}}\right\}.$$ (32)

is positively invariant and attracting, that is, solution starting in $\Omega {}_{\text{T}}$ will remain in $\Omega {}_{\text{T}}$ for all time $t\ge 0$. Thus, it is sufficient to consider the dynamics of the sub-model system (31) in $\Omega {}_{\text{T}}$.

3.2.1. Stability of the disease-free equilibrium

The DFE of the tuberculosis only sub-model (31) is

$${\mathcal{E}}_{T_0}=(S^0,E{}_{\text{T}}^0,I{}_{\text{TU}}^0,I{}_{\text{TR}}^0,R^0)=\left(\frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}},0,0,0,0\right).$$ (33)

The basic reproductive number $R_{0_T}$ is derived using the next generation operator method [34].

The sub-model system (31) can be written as

$$\dot{x}=f\left(x\right)=\mathcal{F}\left(x\right)-\mathcal{V}\left(x\right),$$ (34)

where

$$\mathcal{F}=\left(\begin{array}{c}\hfill \lambda {}_{\text{T}}S\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),$$

and

$$\mathcal{V}=\left(\begin{array}{c}\hfill (\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{T}}\hfill \\ \hfill -\theta {}_{\text{U}}E{}_{\text{T}}+(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}}\hfill \\ \hfill -\theta {}_{\text{R}}E{}_{\text{T}}+(\alpha {}_{\text{T}}+\mu {}_{\text{H}}+\varphi {}_{\text{TR}})I{}_{\text{TR}}\hfill \end{array}\right)$$

are the new infection and transfer terms respectively. Evaluating the Jacobian of $\mathcal{F}$ and $\mathcal{V}$ at the DFE ${\mathcal{E}}_{T_0}$ gives

$$F=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill \Lambda {}_{\text{T}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

and

$$V=\left(\begin{array}{ccc}\hfill \theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\theta {}_{\text{U}}\hfill & \hfill \alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}}\hfill & \hfill 0\hfill \\ \hfill -\theta {}_{\text{R}}\hfill & \hfill 0\hfill & \hfill \alpha {}_{\text{T}}+\mu {}_{\text{H}}+\varphi {}_{\text{TR}}\hfill \end{array}\right).$$

Set

$$B{}_{\text{1}}=\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}},B{}_{\text{2}}=\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}}\text{and}B{}_{\text{3}}=\alpha {}_{\text{T}}+\mu {}_{\text{H}}+\varphi {}_{\text{TR}}.$$ (35)

The inverse of the matrix $V$ is

$$V^{-1}=\left(\begin{array}{ccc}\hfill \frac{1}{B{}_{\text{1}}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{\theta {}_{\text{U}}}{B{}_{\text{1}}B{}_{\text{2}}}\hfill & \hfill \frac{1}{B{}_{\text{2}}}\hfill & \hfill 0\hfill \\ \hfill \frac{\theta {}_{\text{R}}}{B{}_{\text{1}}B{}_{\text{3}}}\hfill & \hfill 0\hfill & \hfill \frac{1}{B{}_{\text{3}}}\hfill \end{array}\right).$$

Therefore, the basic reproduction number $R_{0_T}$ which is the largest eigenvalue or spectral radius of the next generation matrix $F{V}^{-1}$ is given by

$$R_{0_T}=\frac{\Lambda {}_{\text{T}}\theta {}_{\text{U}}}{B{}_{\text{1}}B{}_{\text{2}}}=\frac{\Lambda {}_{\text{T}}\theta {}_{\text{U}}}{(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})}.$$ (36)

The basic reproduction number $R_{0_T}$ represents the average number of cases directly generated by one infectious TB case in a population which is assumed totally susceptible [38].

Thus, using Theorem 2 of [34], we establish the following result.

Theorem 3.4

The DFE of the tuberculosis only sub-model (31) is locally asymptotically stable if $R_{0_T}<1$ , and unstable otherwise.

Proof

For ${\mathcal{E}}_{T_0}$, the Jacobian matrix of the system is obtained as

$$J\left({\mathcal{E}}_{T_0}\right)=\left(\begin{array}{ccccc}\hfill -\mu {}_{\text{H}}\hfill & \hfill 0\hfill & \hfill -\Lambda {}_{\text{T}}\hfill & \hfill 0\hfill & \hfill \omega {}_{\text{R}}\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill -B{}_{\text{1}}\hfill & \hfill \Lambda {}_{\text{T}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \theta {}_{\text{U}}\hfill & \hfill -B{}_{\text{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \theta {}_{\text{R}}\hfill & \hfill 0\hfill & \hfill -B{}_{\text{3}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \alpha {}_{\text{TU}}\hfill & \hfill \alpha {}_{\text{T}}\hfill & \hfill -\mu {}_{\text{H}}-\omega {}_{\text{R}}\hfill \end{array}\right),$$

where $B_i;i=1,2,3$ are given in Eq. (35). The characteristic polynomial is given by

$$P\left(\lambda \right)=(\lambda +\mu {}_{\text{H}})(\lambda +\mu {}_{\text{H}}+\omega {}_{\text{R}})(-\lambda -B{}_{\text{3}})((-\lambda -B{}_{\text{1}})(-\lambda -B{}_{\text{2}})-\Lambda {}_{\text{T}}\theta {}_{\text{U}}),=(\lambda +\mu {}_{\text{H}})(\lambda +\mu {}_{\text{H}}+\omega {}_{\text{R}})(-\lambda -B{}_{\text{3}})({\lambda }^2+\lambda (B{}_{\text{1}}+B{}_{\text{2}})+B{}_{\text{1}}B{}_{\text{2}}-\Lambda {}_{\text{T}}\theta {}_{\text{U}}).$$ (37)

The eigenvalues of the characteristic polynomial (37) are $-\mu {}_{\text{H}},-\mu {}_{\text{H}}-\omega {}_{\text{R}},-B{}_{\text{3}}$,

$$-(B{}_{\text{1}}+B{}_{\text{1}})-\sqrt{{(B{}_{\text{1}}-B{}_{\text{1}})}^2+4\Lambda {}_{\text{T}}\theta {}_{\text{U}}}\text{and}-(B{}_{\text{1}}+B{}_{\text{1}})+\sqrt{{(B{}_{\text{1}}-B{}_{\text{1}})}^2+4\Lambda {}_{\text{T}}\theta {}_{\text{U}}}.$$

The first four eigenvalues are negative, and since $R_{0_T}<1$ the last one is also negative. Hence, the DFE ${\mathcal{E}}_{T_0}$ of the tuberculosis only sub-model system (31) is locally asymptotically stable when $R_{0_T}<1$. ■

Theorem 3.5

The DFE of the tuberculosis only sub-model (31) is globally asymptotically stable if $R_{0_T}<1$ , and unstable otherwise.

Proof

Consider the following Lyapunov function

$$W=(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})E{}_{\text{T}}+\Lambda {}_{\text{T}}I{}_{\text{TU}}.$$

The time derivative of $W$ computed along the solutions of (31) is given by

$$\begin{array}{ccc}\dot{W}\hfill & =\hfill & (\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})\dot{E}{}_{\text{T}}+\Lambda {}_{\text{T}}\dot{I}{}_{\text{TU}},\hfill \\ \hfill \\ \hfill & =\hfill & (\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})\left[\lambda {}_{\text{T}}S-(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{C}}\right]\hfill \\ \hfill & \hfill & +\Lambda {}_{\text{T}}\left[\theta {}_{\text{U}}E{}_{\text{T}}-(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}}\right],\hfill \\ \hfill \\ \hfill & =\hfill & (\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})\left[\frac{\Lambda {}_{\text{T}}I{}_{\text{CU}}}{N}S-(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{T}}\right]\hfill \\ \hfill & \hfill & +\Lambda {}_{\text{T}}\left[\theta {}_{\text{U}}E{}_{\text{T}}-(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}}\right],\hfill \\ \hfill \\ \hfill & ⩽\hfill & (\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})\left[\Lambda {}_{\text{T}}I{}_{\text{TU}}-(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{T}}\right]\hfill \\ \hfill & \hfill & +\Lambda {}_{\text{T}}\left[\theta {}_{\text{U}}E{}_{\text{T}}-(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}}\right],\hfill \\ \hfill \\ \hfill & ⩽\hfill & -(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{T}}+\Lambda {}_{\text{T}}\theta {}_{\text{U}}E{}_{\text{T}},\hfill \\ \hfill \\ \hfill & ⩽\hfill & (\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})E{}_{\text{T}}(R_{0_T}-1),\hfill \\ \hfill \\ \hfill & ⩽\hfill & 0,\text{ for }{R}_{0_T}⩽1.\hfill \end{array}$$

Because all model parameters are non-negative, it follows that $\dot{W}$, for $R_{0_T}⩽1$ with $\dot{W}=0$ if and only if $E{}_{\text{T}}=I{}_{\text{TR}}=I{}_{\text{TU}}=0$. Substituting $(E{}_{\text{T}},I{}_{\text{TR}},I{}_{\text{TU}})=(0,0,0)$ into (31) shows that $S\to \frac{\omega {}_{\text{H}}}{\alpha {}_{\text{H}}}$ as $t\to \infty$. Hence, $W$ is a Lyapunov function on ${\Omega }_T$, and the largest compact invariant set in $\{(S,E{}_{\text{T}},I{}_{\text{TR}},I{}_{\text{TU}},R)\in {\Omega }_T:\dot{W}=0\}$ is ${\mathcal{E}}_T$. Thus, by LaSalle’s invariance principle, every solution of (31), with initial conditions in ${\Omega }_T$ approaches ${\mathcal{E}}_T$, as $t\to \infty$ whenever $R_{0_T}⩽1$. ■

3.2.2. Stability of the endemic equilibrium

We study the stability of the endemic equilibrium of the tuberculosis only sub-model system (31). From (5), this equilibrium denoted by ${\mathcal{E}}_{T_1}$ is given by

$$S_{T_1}=\frac{\omega {}_{\text{H}}(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}}{(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})},$$$$E{}_{\text{T}}{}_{T_1}=\frac{\omega {}_{\text{H}}{\lambda }^{\star }{}_{\text{T}}(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{2}}B{}_{\text{3}}}{(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})},$$$$I{}_{\text{TU}}{}_{T_1}=\frac{\omega {}_{\text{H}}\lambda {}_{\text{T}}^{\star }\theta {}_{\text{U}}(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{3}}}{(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})},$$$$I{}_{\text{TR}}{}_{T_1}=\frac{\omega {}_{\text{H}}\lambda {}_{\text{T}}^{\star }\theta {}_{\text{R}}(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{2}}}{(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})},$$$$R{}_{T_1}=\frac{\omega {}_{\text{H}}\lambda {}_{\text{T}}^{\star }(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})}{(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})},$$ (38)

where

$$\lambda {}_{\text{T}}^{\star }=\frac{\Lambda {}_{\text{T}}I{}_{\text{TU}}{}_{T_1}}{S_{T_1}+E{}_{\text{T}}{}_{T_1}+I{}_{\text{TU}}{}_{T_1}+I{}_{\text{TR}}{}_{T_1}+R{}_{T_1}}=\frac{\Lambda {}_{\text{T}}I{}_{\text{TU}}{}_{T_1}}{N{}_{T_1}},=\frac{\Lambda {}_{\text{T}}\omega {}_{\text{H}}\lambda {}_{\text{T}}^{\star }\theta {}_{\text{U}}(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{3}}}{N{}_{T_1}(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})}$$ (39)

Note that $N{}_{T_1}$ given in Box I.

Box I.

$$N{}_{T_1}=\frac{\omega {}_{\text{H}}\lambda {}_{\text{T}}^{\star }\left((\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{2}}B{}_{\text{3}}+\theta {}_{\text{R}}B{}_{\text{2}}\left(\omega {}_{\text{R}}+\mu {}_{\text{H}}+\alpha {}_{\text{T}}\right)+\theta {}_{\text{U}}B{}_{\text{3}}\left(\omega {}_{\text{R}}+\mu {}_{\text{H}}+\alpha {}_{\text{TU}}\right)\right)+\omega {}_{\text{H}}(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}}{(\lambda {}_{\text{T}}^{\star }+\mu {}_{\text{H}})(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}-\lambda {}_{\text{T}}^{\star }\omega {}_{\text{R}}(\alpha {}_{\text{T}}\theta {}_{\text{R}}B{}_{\text{2}}+\alpha {}_{\text{TU}}\theta {}_{\text{U}}B{}_{\text{3}})}$$ (40)

Substituting (40) into (39), we obtain

$$\lambda {}_{\text{T}}^{\star }\left((\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{2}}B{}_{\text{3}}+\theta {}_{\text{R}}B{}_{\text{2}}\left(\omega {}_{\text{R}}+\mu {}_{\text{H}}+\alpha {}_{\text{T}}\right)+\theta {}_{\text{U}}B{}_{\text{3}}\left(\omega {}_{\text{R}}+\mu {}_{\text{H}}+\alpha {}_{\text{TU}}\right)\right)=(\omega {}_{\text{R}}+\mu {}_{\text{H}})B{}_{\text{1}}B{}_{\text{2}}B{}_{\text{3}}\left(R_{0_T}-1\right).$$ (41)

Therefore, $\lambda {}_{\text{T}}^{\star }$ exist if and only if $R_{0_T}>1$. Hence, we have established the following result.

Theorem 3.6

The tuberculosis only sub-model system (31) has one unique endemic equilibrium if $R_{0_T}>1$ .

We again use the center manifold approach to analyze the global stability of the tuberculosis only sub-model (31). To this end, we use the notation $x_1=S$, $x_2=E{}_{\text{C}}$, $x_3=I{}_{\text{CT}}$, $x_4=I{}_{\text{CR}}$, $x_5=R$ and $N{}_{\text{T}}=x_1+x_2+x_3+x_4+x_5$ to write the model in the form $\dot{x}=f\left(x\right)$, with $x={(x_1,\dots ,x_5)}^T$ and $f=(f_1,\dots ,f_5)$ as follows

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1& =f_1=\omega {}_{\text{H}}+\omega {}_{\text{R}}{x}_5-\frac{\Lambda {}_{\text{T}}{x}_3}{x_1+x_2+x_3+x_4+x_5}{x}_1-\mu {}_{\text{H}}{x}_1,\hfill \\ \hfill {\dot{x}}_2& =f_2=\frac{\Lambda {}_{\text{T}}{x}_3}{x_1+x_2+x_3+x_4+x_5}{x}_1-(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})x_2,\hfill \\ \hfill {\dot{x}}_3& =f_3=\theta {}_{\text{U}}{x}_2-(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})x_3,\hfill \\ \hfill {\dot{x}}_4& =f_4=\theta {}_{\text{R}}{x}_2-(\alpha {}_{\text{T}}+\mu {}_{\text{H}}+\varphi {}_{\text{TR}})x_4,\hfill \\ \hfill {\dot{x}}_5& =f_5=\alpha {}_{\text{TU}}{x}_3+\alpha {}_{\text{T}}{x}_4-(\mu {}_{\text{H}}+\omega {}_{\text{R}})x_5,\hfill \end{array}\right)$$ (42)

The Jacobian of (42) at the DFE ${\mathcal{E}}_{T_0}$ is

$$J\left({\mathcal{E}}_{T_0}\right)=\left(\begin{array}{ccccc}\hfill -\mu {}_{\text{H}}\hfill & \hfill 0\hfill & \hfill -\Lambda {}_{\text{T}}\hfill & \hfill 0\hfill & \hfill \omega {}_{\text{R}}\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill -B{}_{\text{1}}\hfill & \hfill \Lambda {}_{\text{T}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \theta {}_{\text{U}}\hfill & \hfill -B{}_{\text{2}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \theta {}_{\text{R}}\hfill & \hfill 0\hfill & \hfill -B{}_{\text{3}}\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \alpha {}_{\text{TU}}\hfill & \hfill \alpha {}_{\text{T}}\hfill & \hfill -\mu {}_{\text{H}}-\omega {}_{\text{R}}\hfill \end{array}\right),$$

where $B_i;i=1,2,3$ are given in Eq. (35). We choose $\Lambda {}_{\text{T}}$ as a bifurcation parameter. Therefore, setting $R_{0_T}=1$, we obtain

$$\Lambda {}_{\text{T}}=\Lambda {}_{\text{T}}^{\star }=\frac{B_1{B}_2}{\theta {}_{\text{U}}}=\frac{(\theta {}_{\text{R}}+\theta {}_{\text{U}}+\mu {}_{\text{H}})(\alpha {}_{\text{TU}}+\mu {}_{\text{H}}+\varphi {}_{\text{TU}})}{\theta {}_{\text{U}}}.$$ (43)

At $\Lambda {}_{\text{T}}=\Lambda {}_{\text{T}}^{\star }$ the Jacobian has a simple zero eigenvalue and all other eigenvalues have negative real parts. Therefore, the disease free equilibrium point ${\mathcal{E}}_{T_0}$ is a non-hyperbolic equilibrium point. Hence, the center manifold theory can be applied to model system (42) near $\Lambda {}_{\text{T}}=\Lambda {}_{\text{T}}^{\star }$.

The right eigenvector $w={(w_1,w_2,w_3,w_4,w_5)}^T$ associated with the zero eigenvalue of $J\left({\mathcal{E}}_{T_0}\right)$ evaluated at $\Lambda {}_{\text{T}}=\Lambda {}_{\text{T}}^{\star }$ is

$$w_1=\frac{\omega {}_{\text{R}}}{\mu {}_{\text{H}}+\omega {}_{\text{R}}}\left(\frac{\alpha {}_{\text{T}}\theta {}_{\text{R}}}{\mu {}_{\text{H}}{B}_3}+\frac{\alpha {}_{\text{TU}}\eta {}_{\text{U}}}{\mu {}_{\text{H}}{B}_2}\right)w_2-\frac{B_1}{\mu {}_{\text{H}}}{w}_2,w_3=\frac{\theta {}_{\text{U}}}{B_2}{w}_2,w_4=\frac{\theta {}_{\text{R}}}{B_3}{w}_2,$$$$w_5=\frac{1}{\mu {}_{\text{H}}+\omega {}_{\text{R}}}\left(\frac{\alpha {}_{\text{T}}\theta {}_{\text{R}}}{B_3}+\frac{\alpha {}_{\text{TU}}\eta {}_{\text{U}}}{B_2}\right)w_2,w_2=w_2>0.$$ (44)

Similarly, the left eigenvector $v=(v_1,v_2,v_3,v_4,v_5)$ is given by

$$v_1=0,v_4=0,v_5=0,v_3=\frac{B_1}{\theta {}_{\text{U}}}{v}_2,v_2=v_2>0.$$ (45)

The left and right eigenvectors satisfy $v.w=1$, that is,

$$v_2{w}_2\left(1+\frac{B_1}{B_2}\right)=1.$$ (46)

For the direction of the bifurcation, we determine the sign of the bifurcation parameters $a$ et $b$.

$$a=\sum _{k,i,j=1}^5{v}_k{w}_i{w}_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}({\mathcal{E}}_{T_0},\Lambda {}_{\text{T}}^{\star })=\sum _{i,j=1}^5{v}_2{w}_i{w}_j\frac{{\partial }^2{f}_2}{\partial x_i\partial x_j}({\mathcal{E}}_{T_0},\Lambda {}_{\text{T}}^{\star }).$$ (47)

The partial derivatives are

$$\frac{\partial f_2}{\partial x_1}=\lambda {}_{\text{T}}-\lambda {}_{\text{T}}\frac{x_1}{N{}_{\text{T}}},\frac{\partial f_2}{\partial x_2}=-\lambda {}_{\text{T}}\frac{x_1}{N{}_{\text{T}}},$$$$\frac{\partial f_2}{\partial x_3}=(\Lambda {}_{\text{T}}-\lambda {}_{\text{T}})\frac{x_1}{N{}_{\text{T}}},\frac{\partial f_2}{\partial x_4}=-\lambda {}_{\text{T}}\frac{x_1}{N{}_{\text{T}}},\frac{\partial f_2}{\partial x_5}=-\lambda {}_{\text{T}}\frac{x_1}{N{}_{\text{T}}}.$$ (48)

Therefore,

$$a=\frac{\mu {}_{\text{H}}\Lambda {}_{\text{T}}^{\star }}{\omega {}_{\text{H}}}{v}_2(w_1{w}_3+w_3{w}_2-w_3^3-w_3{w}_4-w_3{w}_5),=\frac{\mu {}_{\text{H}}\Lambda {}_{\text{T}}^{\star }}{\omega {}_{\text{H}}}\frac{\theta {}_{\text{U}}}{B_2}{v}_2{w}_2^2\left(-\frac{B_1}{\mu {}_{\text{H}}}-\frac{\theta {}_{\text{U}}}{B_2}-\frac{\theta {}_{\text{R}}}{B_3}-\frac{\alpha {}_{\text{T}}\theta {}_{\text{R}}}{\mu {}_{\text{H}}{B}_3}-\frac{\alpha {}_{\text{TU}}\eta {}_{\text{U}}}{\mu {}_{\text{H}}{B}_2}-1\right),=-\frac{\mu {}_{\text{H}}{B}_1}{\omega {}_{\text{H}}}{v}_2{w}_2^2\left(\frac{B_1}{\mu {}_{\text{H}}}+\frac{\theta {}_{\text{U}}}{B_2}+\frac{\theta {}_{\text{R}}}{B_3}+\frac{\alpha {}_{\text{T}}\theta {}_{\text{R}}}{\mu {}_{\text{H}}{B}_3}+\frac{\alpha {}_{\text{TU}}\eta {}_{\text{U}}}{\mu {}_{\text{H}}{B}_2}+1\right)<0.$$ (49)

For the bifurcation parameter $b$, we have

$$b=\sum _{k,j=1}^5{v}_k{w}_j\frac{{\partial }^2{f}_k}{\partial x_j\partial \Lambda {}_{\text{T}}}({\mathcal{E}}_{T_0},\Lambda {}_{\text{T}}^{\star }),=\sum _{j=1}^5{v}_2{w}_j\frac{{\partial }^2{f}_2}{\partial x_j\partial \Lambda {}_{\text{T}}}({\mathcal{E}}_{T_0},\Lambda {}_{\text{T}}^{\star }),=v_2{w}_3,=\frac{\theta {}_{\text{U}}}{B_2}{v}_2{w}_2>0.$$ (50)

Since $a<0$ and $b>0$, our proposed tuberculosis only sub-model system (31) does not exhibit the phenomenon of backward bifurcation at $R_{0_T}=1$. Hence, the following result.

Theorem 3.7

The unique endemic equilibrium ${\mathcal{E}}_{T_1}$ of the tuberculosis only sub-model (31) is globally asymptotically stable if $R_{0_T}>1$ .

The above result when $R_{0_T}>1$ is graphically depicted in Fig. 3. The red line in Fig. 2 represents the stability area of the DFE ${\mathcal{E}}_{T_0}$, and the blue line is the instability area of the DFE. The red dotted line represents the threshold stability switch line $R_{0_T}=1$. When $R_{0_T}>1$, the green line does not cross the dotted line, hence the endemic equilibrium ${\mathcal{E}}_{T_1}$ is globally asymptotically stable.

Fig. 3.

Illustration of the bifurcation by plotting $\Lambda {}_{\text{T}}$ versus $R_{0_T}$.

3.3. Tuberculosis-COVID-19 model

The feasible region of the full model system (3) is

$$\Omega {}_{\text{CT}}=\Omega {}_{\text{C}}\times \Omega {}_{\text{T}},$$ (51)

where $\Omega {}_{\text{C}}$ and $\Omega {}_{\text{T}}$ are defined in (8), (32), respectively.

The DFE of the COVID-19 and tuberculosis co-infection model is given by

$${\mathcal{E}}_0=(S^0,E{}_{\text{C}}^0,I{}_{\text{CR}}^0,I{}_{\text{CU}}^0,E{}_{\text{CT}}^0,I{}_{\text{CT}}^0,E{}_{\text{T}}^0,I{}_{\text{TU}}^0,I{}_{\text{TR}}^0,R^0)=\left(\frac{\omega {}_{\text{H}}}{\mu {}_{\text{H}}},0,0,0,0,0,0,0,0,0\right).$$ (52)

From the basic reproduction number of the COVID-19 only and tuberculosis only sub-models, the basic reproduction number of the full system is given as

$$R_{0_{CT}}=max(R_{0_C},R_{0_T}),$$ (53)

where $R_{0_C}$ and $R_{0_T}$ are respectively defined in (12), (36).

Using Theorem 2 of [34],

Theorem 3.8

The DFE of the tuberculosis–COVID-19 model (31) is locally asymptotically stable if $R_{0_{CT}}<1$ , and unstable otherwise.

Since the COVID-19 only and tuberculosis only sub-models do not undergo the phenomenon of backward bifurcation, the full tuberculosis–COVID-19 model will not undergo backward bifurcation.

4. Optimal control model

This section is devoted to investigating optimal interventions for mitigating the spread of COVID-19 and tuberculosis and their co-infection. We incorporate the following five controls into the full model (3)

• $u_1$: Public awareness campaign control of tuberculosis,

• $u_2$: Prevention control against COVID-19 (e.g., face mask, physical distancing),

• $u_3$: Control against co-infection with a second disease,

• $u_4$: Tuberculosis treatment control, and

• $u_5$: COVID-19 treatment control.

The model system (3) now reads

$$\left\{\begin{array}{cc}\hfill \frac{dS}{dt}& =\omega {}_{\text{H}}+\omega {}_{\text{CU}}I{}_{\text{CU}}+\omega {}_{\text{CR}}I{}_{\text{CR}}+\omega {}_{\text{TU}}I{}_{\text{TU}}+\omega {}_{\text{TR}}I{}_{\text{TR}}\hfill \\ \hfill & +\omega {}_{\text{CT}}I{}_{\text{CT}}-(1-u_2)\lambda {}_{\text{C}}S-(1-u_1)\lambda {}_{\text{T}}S-\mu {}_{\text{H}}S,\hfill \\ \hfill \frac{dE{}_{\text{C}}}{dt}& =(1-u_2)\lambda {}_{\text{C}}S-(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}-(\eta {}_{\text{R}}+\eta {}_{\text{U}})E{}_{\text{C}}-\mu {}_{\text{H}}E{}_{\text{C}},\hfill \\ \hfill \frac{dI{}_{\text{CR}}}{dt}& =\eta {}_{\text{R}}E{}_{\text{C}}+(1+u_5)\alpha {}_{\text{C}}I{}_{\text{CT}}-(\omega {}_{\text{CR}}+\alpha {}_{\text{CR}}+(1+u_5)\alpha {}_{\text{C}})I{}_{\text{CR}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CR}})I{}_{\text{CR}},\hfill \\ \hfill \frac{dI{}_{\text{CU}}}{dt}& =\eta {}_{\text{U}}E{}_{\text{C}}-(\omega {}_{\text{CU}}+\gamma {}_{\text{CU}}+\alpha {}_{\text{CU}})I{}_{\text{CU}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}},\hfill \\ \hfill \frac{dE{}_{\text{CT}}}{dt}& =(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}+(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}E{}_{\text{T}}-\gamma {}_{\text{CT}}E{}_{\text{CT}}-\mu {}_{\text{H}}E{}_{\text{CT}},\hfill \\ \hfill \frac{dI{}_{\text{CT}}}{dt}& =\gamma {}_{\text{CT}}E{}_{\text{CT}}+\gamma {}_{\text{TU}}I{}_{\text{TU}}+\gamma {}_{\text{CU}}I{}_{\text{CU}}+\alpha {}_{\text{CR}}I{}_{\text{CR}}+\alpha {}_{\text{TR}}I{}_{\text{TR}}\hfill \\ \hfill & -(\omega {}_{\text{CT}}+(1+u_5)\alpha {}_{\text{C}}+(1+u_4)\alpha {}_{\text{T}})I{}_{\text{CT}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CT}})I{}_{\text{CT}},\hfill \\ \hfill \frac{dE{}_{\text{T}}}{dt}& =(1-u_1)\lambda {}_{\text{T}}S-(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}E{}_{\text{T}}-(\theta {}_{\text{U}}+\theta {}_{\text{R}})E{}_{\text{T}}-\mu {}_{\text{H}}E{}_{\text{T}},\hfill \\ \hfill \frac{dI{}_{\text{TU}}}{dt}& =\theta {}_{\text{U}}E{}_{\text{T}}-(\omega {}_{\text{TU}}+\gamma {}_{\text{TU}}+\alpha {}_{\text{TU}})I{}_{\text{TU}}-(\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}},\hfill \\ \hfill \frac{dI{}_{\text{TR}}}{dt}& =\theta {}_{\text{R}}E{}_{\text{T}}+(1+u_4)\alpha {}_{\text{T}}I{}_{\text{CT}}-(\omega {}_{\text{TR}}+\alpha {}_{\text{TR}}+(1+u_4)\alpha {}_{\text{T}})I{}_{\text{TR}}-(\mu {}_{\text{H}}+\varphi {}_{\text{TR}})I{}_{\text{TR}},\hfill \\ \hfill \frac{dR}{dt}& =(1+u_5)\alpha {}_{\text{C}}I{}_{\text{CR}}+\alpha {}_{\text{CU}}I{}_{\text{CU}}+\alpha {}_{\text{TU}}I{}_{\text{TU}}+(1+u_4)\alpha {}_{\text{T}}I{}_{\text{TR}}-\mu {}_{\text{H}}R,\hfill \end{array}\right)$$ (54)

with initial conditions

$$\begin{array}{c}S\left(0\right)\ge 0,E{}_{\text{C}}\left(0\right)\ge 0,E{}_{\text{T}}\left(0\right)\ge 0,E{}_{\text{CT}}\left(0\right)\ge 0,I{}_{\text{CR}}\left(0\right)\ge 0,\hfill \\ I{}_{\text{CU}}\left(0\right)\ge 0,I{}_{\text{CT}}\left(0\right)\ge 0,I{}_{\text{TU}}\left(0\right)\ge 0,I{}_{\text{TR}}\left(0\right)\ge 0,R\left(0\right)\ge 0.\hfill \end{array}$$ (55)

In what follows, because the positive balancing cost factors transfer the integral into monetary quantity over a finite period of time, we choose a quadratic control function, see [36] and the references therein. Thus, consider the following quadratic objective functional which measures the cost of the control. This cost includes strategies and treatment for mitigating at the population level the spread of COVID-19 and tuberculosis, as well as their co-infection. Thus, the nonlinear objective function is

$$J\left(u{}_{\text{1}},u{}_{\text{2}},u{}_{\text{3}},u{}_{\text{4}},u{}_{\text{5}}\right)={\int }_0^T[c_{1}E{}_{\text{C}}\left(t\right)+c_{2}E{}_{\text{CT}}\left(t\right)+c_{3}E{}_{\text{T}}\left(t\right)+c_{4}I{}_{\text{CU}}\left(t\right)+c_{5}I{}_{\text{TU}}\left(t\right)+c_{6}I{}_{\text{CR}}\left(t\right)+c_{7}I{}_{\text{CT}}\left(t\right)+c_{8}I{}_{\text{TR}}\left(t\right)+\frac{w_1}{2}u{}_{\text{1}}^2$$$$+\frac{w_2}{2}u{}_{\text{2}}^2+\frac{w_3}{2}u{}_{\text{3}}^2+\frac{w_4}{2}u{}_{\text{4}}^2+\frac{w_5}{2}u{}_{\text{5}}^2]dt,$$ (56)

where $T$ is the final time, $c_i,i=1,\dots ,8$ are positive weight constants, and $w_i,i=1,\dots ,5$ are weight constants for the strategies and treatments against proliferation of the COVID-19 and tuberculosis. The linear and quadratic form of the controls in (54) and in the objective function allow for the Hamiltonian associated to the optimal control problem to be maximized. Therefore, we seek to find, using the maximum principle of Pontryagin [39], an optimal control $(u{}_{\text{1}}^{\ast },u{}_{\text{2}}^{\ast },u{}_{\text{3}}^{\ast },u{}_{\text{4}}^{\ast },u{}_{\text{5}}^{\ast })\in U$ satisfying (54), such that

$$J\left(u{}_{\text{1}}^{\ast },u{}_{\text{2}}^{\ast },u{}_{\text{3}}^{\ast },u{}_{\text{4}}^{\ast },u{}_{\text{5}}^{\ast }\right)=min\left\{J\left(u{}_{\text{1}},u{}_{\text{2}},u{}_{\text{3}},u{}_{\text{4}},u{}_{\text{5}}\right)|(u{}_{\text{1}},u{}_{\text{2}},u{}_{\text{3}},u{}_{\text{4}},u{}_{\text{5}})\in U\right\}.$$ (57)

The associated pseudo-Hamiltonian is

$$\mathbb{H}=c_{1}E{}_{\text{C}}\left(t\right)+c_{2}E{}_{\text{CT}}\left(t\right)+c_{3}E{}_{\text{T}}\left(t\right)+c_{4}I{}_{\text{CU}}\left(t\right)$$$$+c_{5}I{}_{\text{TU}}\left(t\right)+c_{6}I{}_{\text{CR}}\left(t\right)+c_{7}I{}_{\text{CT}}\left(t\right)+c_{8}I{}_{\text{TR}}\left(t\right)+\frac{w_1}{2}u{}_{\text{1}}^2+\frac{w_2}{2}u{}_{\text{2}}^2+\frac{w_3}{2}u{}_{\text{3}}^2+\frac{w_4}{2}u{}_{\text{4}}^2+\frac{w_5}{2}u{}_{\text{5}}^2+{\xi }_1(\omega {}_{\text{H}}+\omega {}_{\text{CU}}I{}_{\text{CU}}+\omega {}_{\text{CR}}I{}_{\text{CR}}+\omega {}_{\text{TU}}I{}_{\text{TU}}+\omega {}_{\text{TR}}I{}_{\text{TR}}+\omega {}_{\text{CT}}I{}_{\text{CT}}$$$$-(1-u_2)\lambda {}_{\text{C}}S-(1-u_1)\lambda {}_{\text{T}}S-\mu {}_{\text{H}}S)+{\xi }_2\left((1-u_2)\lambda {}_{\text{C}}S-(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}-(\eta {}_{\text{R}}+\eta {}_{\text{U}})E{}_{\text{C}}-\mu {}_{\text{H}}E{}_{\text{C}}\right)+{\xi }_3\left(\eta {}_{\text{R}}E{}_{\text{C}}+(1+u_5)\alpha {}_{\text{C}}I{}_{\text{CT}}-(\omega {}_{\text{CR}}+\alpha {}_{\text{CR}}+(1+u_5)\alpha {}_{\text{C}})I{}_{\text{CR}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CR}})I{}_{\text{CR}}\right)+{\xi }_4\left(\eta {}_{\text{U}}E{}_{\text{C}}-(\omega {}_{\text{CU}}+\gamma {}_{\text{CU}}+\alpha {}_{\text{CU}})I{}_{\text{CU}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CU}})I{}_{\text{CU}}\right)+{\xi }_5\left((1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}+(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}E{}_{\text{T}}-\gamma {}_{\text{CT}}E{}_{\text{CT}}-\mu {}_{\text{H}}E{}_{\text{CT}}\right)+{\xi }_6(\gamma {}_{\text{CT}}E{}_{\text{CT}}+\gamma {}_{\text{TU}}I{}_{\text{TU}}+\gamma {}_{\text{CU}}I{}_{\text{CU}}+\alpha {}_{\text{CR}}I{}_{\text{CR}}+\alpha {}_{\text{TR}}I{}_{\text{TR}}$$$$-(\omega {}_{\text{CT}}+(1+u_5)\alpha {}_{\text{C}}+(1+u_4)\alpha {}_{\text{T}})I{}_{\text{CT}}-(\mu {}_{\text{H}}+\varphi {}_{\text{CT}})I{}_{\text{CT}})+{\xi }_7\left((1-u_1)\lambda {}_{\text{T}}S-(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}E{}_{\text{T}}-(\theta {}_{\text{U}}+\theta {}_{\text{R}})E{}_{\text{T}}-\mu {}_{\text{H}}E{}_{\text{T}}\right)+{\xi }_8\left(\theta {}_{\text{U}}E{}_{\text{T}}-(\omega {}_{\text{TU}}+\gamma {}_{\text{TU}}+\alpha {}_{\text{TU}})I{}_{\text{TU}}-(\mu {}_{\text{H}}+\varphi {}_{\text{TU}})I{}_{\text{TU}}\right)+{\xi }_9\left(\theta {}_{\text{R}}E{}_{\text{T}}+(1+u_4)\alpha {}_{\text{T}}I{}_{\text{CT}}-(\omega {}_{\text{TR}}+\alpha {}_{\text{TR}}+(1+u_4)\alpha {}_{\text{T}})I{}_{\text{TR}}-(\mu {}_{\text{H}}+\varphi {}_{\text{TR}})I{}_{\text{TR}}\right)+{\xi }_{10}\left((1+u_5)\alpha {}_{\text{C}}I{}_{\text{CR}}+\alpha {}_{\text{CU}}I{}_{\text{CU}}+\alpha {}_{\text{TU}}I{}_{\text{TU}}+(1+u_4)\alpha {}_{\text{T}}I{}_{\text{TR}}-\mu {}_{\text{H}}R\right),$$ (58)

where ${\xi }_i,i=1,\dots ,10$ are the adjoint variables satisfying

$$\begin{array}{ccc}{\xi }_1^{\prime }\hfill & =-\frac{\partial \mathbb{H}}{\partial S}\hfill & {\xi }_2^{\prime }=-\frac{\partial \mathbb{H}}{\partial E{}_{\text{C}}},\hfill \\ {\xi }_3^{\prime }\hfill & =-\frac{\partial \mathbb{H}}{\partial I{}_{\text{CR}}}\hfill & {\xi }_4^{\prime }=-\frac{\partial \mathbb{H}}{\partial I{}_{\text{CU}}},\hfill \\ {\xi }_5^{\prime }\hfill & =-\frac{\partial \mathbb{H}}{\partial E{}_{\text{CT}}}\hfill & {\xi }_6^{\prime }=-\frac{\partial \mathbb{H}}{\partial I{}_{\text{CT}}},\hfill \\ {\xi }_7^{\prime }\hfill & =-\frac{\partial \mathbb{H}}{\partial E{}_{\text{T}}}\hfill & {\xi }_8^{\prime }=-\frac{\partial \mathbb{H}}{\partial I{}_{\text{TU}}},\hfill \\ {\xi }_9^{\prime }\hfill & =-\frac{\partial \mathbb{H}}{\partial I{}_{\text{TR}}}\hfill & {\xi }_{10}^{\prime }=-\frac{\partial \mathbb{H}}{\partial R}.\hfill \end{array}$$ (59)

Writing (61) in details gives

$$\left\{\begin{array}{cc}\hfill {\dot{\xi }}_1& =(1-u_2)\lambda {}_{\text{C}}({\xi }_1-{\xi }_2)\left(1-\frac{S}{N}\right)+(1-u_1)\lambda {}_{\text{T}}({\xi }_1-{\xi }_7)\left(1-\frac{S}{N}\right)\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+\mu {}_{\text{H}}{\xi }_1,\hfill \\ \hfill {\dot{\xi }}_2& =(1-u_2)(\Lambda {}_{\text{C}}-\lambda {}_{\text{C}})({\xi }_1-{\xi }_2)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}(\Lambda {}_{\text{C}}-\lambda {}_{\text{C}})({\xi }_2-{\xi }_5)\frac{E{}_{\text{C}}}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_2-{\xi }_5)+(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}\hfill \\ \hfill & +({\xi }_2-{\xi }_3)\eta {}_{\text{R}}+({\xi }_2-{\xi }_4)\eta {}_{\text{U}}+\mu {}_{\text{H}}{\xi }_2-c_1,\hfill \\ \hfill {\dot{\xi }}_3& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}+(\mu {}_{\text{H}}+\varphi {}_{\text{CR}}){\xi }_3\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+\alpha {}_{\text{CR}}({\xi }_3-{\xi }_6)\hfill \\ \hfill & +\omega {}_{\text{CR}}({\xi }_3-{\xi }_1)+(1+u_5)\alpha {}_{\text{C}}({\xi }_3-{\xi }_{10})-c_6,\hfill \\ \hfill {\dot{\xi }}_4& =(1-u_2)(\Lambda {}_{\text{C}}-\lambda {}_{\text{C}})({\xi }_1-{\xi }_2)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}(\Lambda {}_{\text{C}}-\lambda {}_{\text{C}})({\xi }_2-{\xi }_5)\frac{E{}_{\text{C}}}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+\alpha {}_{\text{CU}}({\xi }_4-{\xi }_{10})+\omega {}_{\text{CU}}({\xi }_4-{\xi }_1)\hfill \\ \hfill & +\gamma {}_{\text{CU}}({\xi }_4-{\xi }_6)+(\mu {}_{\text{H}}+\varphi {}_{\text{CU}}){\xi }_4-c_4,\hfill \\ \hfill {\dot{\xi }}_5& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+\gamma {}_{\text{CT}}({\xi }_5-{\xi }_6)+\mu {}_{\text{H}}{\xi }_5-c_2,\hfill \\ \hfill {\dot{\xi }}_6& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}+(\mu {}_{\text{H}}+\varphi {}_{\text{CT}}){\xi }_6\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+(1+u_5)\alpha {}_{\text{C}}({\xi }_6-{\xi }_3)\hfill \\ \hfill & +(1+u_4)\alpha {}_{\text{T}}({\xi }_6-{\xi }_9)+\omega {}_{\text{CT}}({\xi }_6-{\xi }_1)-c_7,\hfill \\ \hfill {\dot{\xi }}_7& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_7-{\xi }_5\right)\hfill \\ \hfill & +\theta {}_{\text{R}}({\xi }_7-{\xi }_9)+\theta {}_{\text{U}}({\xi }_7-{\xi }_8)+\mu {}_{\text{H}}{\xi }_7-c_3,\hfill \\ \hfill {\dot{\xi }}_8& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)(\Lambda {}_{\text{T}}-\lambda {}_{\text{T}})({\xi }_1-{\xi }_7)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}+(\mu {}_{\text{H}}+\varphi {}_{\text{TU}}){\xi }_8\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}(\Lambda {}_{\text{T}}-\lambda {}_{\text{T}})\left({\xi }_7-{\xi }_5\right)\frac{E{}_{\text{T}}}{N}+\alpha {}_{\text{TU}}({\xi }_8-{\xi }_{10})\hfill \\ \hfill & +\omega {}_{\text{TU}}({\xi }_8-{\xi }_1)+\gamma {}_{\text{TU}}({\xi }_8-{\xi }_6)-c_5,\hfill \\ \hfill {\dot{\xi }}_9& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}+(\mu {}_{\text{H}}+\varphi {}_{\text{TR}}){\xi }_9\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+\alpha {}_{\text{TR}}({\xi }_9-{\xi }_6)\hfill \\ \hfill & +\omega {}_{\text{TR}}({\xi }_9-{\xi }_1)+(1+u_4)\alpha {}_{\text{T}}({\xi }_9-{\xi }_{10})-c_8,\hfill \\ \hfill {\dot{\xi }}_{10}& =(1-u_2)\lambda {}_{\text{C}}({\xi }_2-{\xi }_1)\frac{S}{N}+(1-u_1)\lambda {}_{\text{T}}({\xi }_7-{\xi }_1)\frac{S}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{C}}\lambda {}_{\text{C}}({\xi }_5-{\xi }_2)\frac{E{}_{\text{C}}}{N}\hfill \\ \hfill & +(1-u_3)\beta {}_{\text{T}}\lambda {}_{\text{T}}\left({\xi }_5-{\xi }_7\right)\frac{E{}_{\text{T}}}{N}+\mu {}_{\text{H}}{\xi }_{10},\hfill \end{array}\right)$$ (60)

with the final conditions ${\xi }_i\left(T\right),i=1,\dots ,10$.

The necessary and sufficient optimality conditions are

$$\begin{array}{c}\frac{\partial \mathbb{H}}{\partial u{}_{\text{1}}^{\ast }}=0,\frac{\partial \mathbb{H}}{\partial u{}_{\text{2}}^{\ast }}=0,\frac{\partial \mathbb{H}}{\partial u{}_{\text{3}}^{\ast }}=0,\frac{\partial \mathbb{H}}{\partial u{}_{\text{4}}^{\ast }}=0\text{and}\frac{\partial \mathbb{H}}{\partial u{}_{\text{5}}^{\ast }}=0,\hfill \end{array}$$ (61)

which in turns give the optimal controls

$$u{}_{\text{1}}^{\ast }=max\left\{0,min\left(1,\frac{({\xi }_7-{\xi }_1)\lambda {}_{\text{T}}S}{w_1}\right)\right\},$$$$u{}_{\text{2}}^{\ast }=max\left\{0,min\left(1,\frac{({\xi }_2-{\xi }_1)\lambda {}_{\text{C}}S}{w_2}\right)\right\},$$$$u{}_{\text{3}}^{\ast }=max\left\{0,min\left(1,\frac{({\xi }_5-{\xi }_2)\beta {}_{\text{C}}\lambda {}_{\text{C}}E{}_{\text{C}}+({\xi }_5-{\xi }_7)\beta {}_{\text{T}}\lambda {}_{\text{T}}E{}_{\text{T}}}{w_3}\right)\right\},$$$$u{}_{\text{4}}^{\ast }=max\left\{0,min\left(1,\frac{({\xi }_6-{\xi }_9)\alpha {}_{\text{T}}I{}_{\text{CT}}+({\xi }_9-{\xi }_{10})\alpha {}_{\text{T}}I{}_{\text{TR}}}{w_3}\right)\right\},$$$$u{}_{\text{5}}^{\ast }=max\left\{0,min\left(1,\frac{({\xi }_6-{\xi }_3)\alpha {}_{\text{C}}I{}_{\text{CT}}+({\xi }_3-{\xi }_{10})\alpha {}_{\text{C}}I{}_{\text{CR}}}{w_3}\right)\right\}.$$ (62)

5. Numerical simulations

To support the analytical results, the optimal control model system 4 is simulated using the model parameter values in Table 1. The positive weight constant $w_1=w_2=w_3=w_4=w_5=2$.

Table 1.

Description of the variables and parameters.

Parameter	Interpretation	Value	Reference
$\omega {}_{\text{H}}$	Recruitment rate	$\frac{10000}{59\times 365}$	[17], [18]
$\omega {}_{\text{R}}$	Loss of immunity after recovery	0.1	Assumed
$\Lambda {}_{\text{C}}$	Effective contact rate transmission of COVID-19	0.6	[19]
$\Lambda {}_{\text{T}}$	Effective contact rate transmission of tuberculosis	1. 3	[20]
$\beta {}_{\text{C}}$	Modification parameter accounting for susceptibility of COVID-19-infected
	Individuals to tuberculosis	1	Assumed
$\beta {}_{\text{T}}$	Modification parameter accounting for susceptibility of tuberculosis-infected
	Individuals to COVID-19	1	Assumed
$\eta {}_{\text{U}}$	Progression rate from asymptomatic to unreported symptomatic COVID-19	0. 785	[21]
$\eta {}_{\text{R}}$	Progression rate from asymptomatic to reported symptomatic COVID-19	0. 2	[22]
$\theta {}_{\text{U}}$	Progression rate from exposed to unreported infectious tuberculosis class	0.7	[23]
$\theta {}_{\text{R}}$	Progression rate from exposed to reported infectious tuberculosis class	0.166	[23]
$\gamma {}_{\text{CT}}$	Fraction of individuals moving to the co-infection class	0.0333	Assumed
$\alpha {}_{\text{CR}}$	Tuberculosis infection rate of reported individuals already infected with COVID-19	0.0028	Assumed
$\gamma {}_{\text{CU}}$	Tuberculosis infection rate of unreported individuals already infected with COVID-19	0.0044	Assumed
$\alpha {}_{\text{TR}}$	COVID-19 infection rate of reported individuals already infected with tuberculosis	0.13	Assumed
$\gamma {}_{\text{TU}}$	COVID-19 infection rate of unreported individuals already infected with tuberculosis	0.0333	Assumed
$\alpha {}_{\text{CU}}$	Recovery rate of unreported COVID-19 infected individuals	0.142	[21]
$\alpha {}_{\text{C}}$	Recovery rate of reported COVID-19 infected individuals	0.68	[24]
$\alpha {}_{\text{TU}}$	Recovery rate of unreported tuberculosis infected individuals	0.175	[25]
$\alpha {}_{\text{T}}$	Recovery rate of reported tuberculosis infected individuals	0.35	[25]
$\varphi {}_{\text{CU}}$	Death rate of unreported COVID-19 infected individuals	0.0065	[26], [27]
$\varphi {}_{\text{CR}}$	Death rate of reported COVID-19 infected individuals	0 .0018	[28]
$\varphi {}_{\text{TU}}$	Death rate of unreported tuberculosis infected individuals	0. 004	[29]
$\varphi {}_{\text{TR}}$	Death rate of reported tuberculosis infected individuals	0.000179	[30]
$\mu {}_{\text{H}}$	Natural death rate of the population	$\frac{1}{59\times 365}$	[17], [18]

To investigate the impact of various control strategies to mitigate the spread of both diseases, the following five scenarios are considered.

• 1.Strategy A: COVID-19 prevention and treatment ($u_2\ne 0,u_5\ne 0$);
• 2.Strategy B: COVID-19 prevention, treatment and control of co-infection ($u_2\ne 0,u_3\ne 0,u_5\ne 0$);
• 3.Strategy C: Tuberculosis prevention and treatment ($u_1\ne 0,u_4\ne 0$));
• 4.Strategy D: Tuberculosis prevention, treatment and control of co-infection ($u_1\ne 0,u_3\ne 0,u_4\ne 0$); and
• 5.Strategy E: COVID-19 prevention with both TB and COVID-19 treatment ($u_2\ne 0,u_4\ne 0,u_5\ne 0$).

For all the five strategies, the reproduction number calculated using model parameter values in Table 1 is $R_{0_{CT}}=max(R_{0_C},R_{0_T})=5.8038>1$.

5.1. Strategy A: COVID-19 prevention and treatment ($u_2\ne 0,u_5\ne 0$)

Simulations of the optimal control system 4 when the strategy that prevents COVID-19 infection ($u_2\ne 0$) and the treatment of COVID-19 ($u_5\ne 0$) are implemented. The results of this strategy are shown in Figs. 4, 5, 6, Fig. 7, Fig. 8, respectively. When this intervention strategy is implemented, the number of symptomatic individuals not reported $I_{cu}$ drastically decreases, mainly due to the high disease (COVID-19-induced death rate reported in this class). Strategy A could reduce by 1,820 the number of new cases of reported COVID-19 (Fig. 5), and by $600$ the number of new co-infections (Fig. 8). The control profiles depicted in Fig. 9 show that treatment is at optimal from the onset of the implementation and remain so throughout, while COVID-19 prevention drops at around 180 days for few days before picking up again. This drop likely corresponds to the relaxation of the COVID-19 prevention measures at the end of the first wave, while the sharp increase corresponds to the beginning of the second COVID-19 wave. For this strategy, we choose the positive weight constants $c_1=1.134,c_2=1,c_3=1,c_4=2,c_5=1,c_6=2,c_7=1,c_8=1$. Percentage estimation of the cost components of this Strategy A is as follows: unreported COVID-19 symptomatic 20% of the total cost of this strategy, reported symptomatic COVID-19 individuals 20%, co-infected 10%. This strategy does not reduce the number of people infected with tuberculosis as shown in Fig. 6, Fig. 7.

Fig. 4.

Individuals $I_{cu}$ with strategy A.

Fig. 5.

Individuals $I_{cr}$ with strategy A .

Fig. 6.

Individuals $I_{tr}$ with strategy A .

Fig. 7.

Individuals $I_{tu}$ with strategy A .

Fig. 8.

Co-infected individuals $I_{ct}$ with strategy A.

Fig. 9.

Control profile for strategy A.

Because the optimal control Figs. 4, 5, 6, Fig. 7, Fig. 8 are similar in the remaining strategies, we will only discuss the results without displaying the figures for the sake of avoiding redundancy of graphs.

5.2. Strategy B: COVID-19 prevention, treatment and control of co-infection ($u_2\ne 0,u_3\ne 0,u_5\ne 0$)

Simulations of the optimal control system 4 when the strategy that prevents COVID-19 infection ($u_2\ne 0$), the treatment of COVID-19 ($u_5\ne 0$) and control against co-infection ($u_3\ne 0$) are implemented. When this intervention strategy is implemented, it could reduce by 1,830 the number of new cases of reported COVID-19 $I_{tr}$, and prevent about $615$ the number of new co-infections. The number of symptomatic COVID-19 individuals not reported $I_{cu}$ decreases drastically due to the lack of treatment in this class. The control profiles in Fig. 10 show that prevention and treatment are optimal throughout the simulation period, while the control against co-infection is optimal from day 5 and will remain so up to 5 days before the duration of the simulation period. For this strategy, we choose the positive weight constants $c_1=1,c_2=1,c_3=1,c_4=1,c_5=1,c_6=1,c_7=1,c_8=1$. Percentage estimation of the cost components of Strategy B is as follows: both reported and unreported COVID-19 symptomatic as well as co-infected individuals 12.5% each. As in Strategy A, this strategy also does not reduce the number of people infected with tuberculosis.

Fig. 10.

Control profile for strategy B.

5.3. Strategy C: Tuberculosis prevention and treatment ($u_1\ne 0,u_4\ne 0$))

Optimal control simulations for system 4 for Strategy C when tuberculosis prevention and treatment $u_1\ne 0$ and $u_4\ne 0$ are implemented. This Strategy C could reduce 2,515 new cases of reported tuberculosis $I_{tr}$, and $90$ new cases of co-infection. The control profiles in Fig. 11 shows that prevention is optimal throughout the simulation period, while treatment is optimal from day 22 through the remainder of the simulation period. For this strategy, we choose the positive weight constants $c_1=1,c_2=1,c_3=1,c_4=1,c_5=10,c_6=1,c_7=1,c_8=1$. Percentage estimation of the cost components Strategy C is as follows: unreported tuberculosis infected individuals 60% of the total cost of this strategy, reported tuberculosis infected and co-infected individuals 6% each. This strategy does not reduce the number of people infected with COVID-19.

Fig. 11.

Control profile for strategy C.

5.4. Strategy D: Tuberculosis prevention, treatment and control of co-infection ($u_1\ne 0,u_3\ne 0,u_4\ne 0$)

Optimal control simulations for system 4 when the strategy for tuberculosis prevention ($u_1\ne 0$), the treatment of tuberculosis ($u_4\ne 0$), and control against co-infection ($u_3\ne 0$) are implemented. This strategy could reduce 2,510 new cases of reported tuberculosis $I_{tr}$, and by $80$ the number of new cases of co-infection. This strategy does not reduce the number of people infected with tuberculosis. The control profiles in Fig. 12 show that control against co-infection which starts approximately from day 10 is optimal throughout the simulation period. The prevention against tuberculosis is optimal from day 105, then decreases between days 154 and 169 (which likely coincides with the peak of COVID-19 first wave), then increases again from day 170. Tuberculosis treatment is optimal for the first 5 days of the simulations, then decreases between days 5 and 65 when we observe an increase that reaches the optimal on day 58. For this Strategy D, we choose the positive weight constants $c_1=5,c_2=10,c_3=5,c_4=1,c_5=14,c_6=1,c_7=4,c_8=20$. Percentage estimation of the cost components of this strategy is as follows: unreported tuberculosis infected individuals 25% of the total cost of this strategy, reported tuberculosis infected individuals 33.33%, and co-infection 6.66%.

Fig. 12.

Control profile for strategy D.

5.5. Strategy E: COVID-19 prevention with both TB and COVID-19 treatment ($u_2\ne 0,u_4\ne 0,u_5\ne 0$)

Optimal control simulations for system 4 for COVID-19 prevention ($u_2\ne 0$), the treatment of tuberculosis and COVID-19 (respectively $u_4\ne 0$ and $u_5\ne 0$) are implemented. This strategy could potentially reduce by 1,520 the number of new cases of reported COVID-19 $I_{cr}$, $300$ new co-infection, and $800$ new cases of infected and reported tuberculosis $I_{tr}$. The control profiles in Fig. 13 show that prevention control of COVID-19 is optimal throughout the simulation period. The COVID-19 treatment is optimal for the first 7 days, then decreases drastically for 5 days and then remains optimal throughout the remainder of the simulation period. Treatment for tuberculosis begins around day 43 and remains at its optimum until the end of the simulation. For this Strategy E, we choose the positive weight constants $c_1=7,c_2=6,c_3=3.5,c_4=3.3,c_5=1,c_6=10,c_7=1,c_8=1$. Percentage estimation of the cost components of this strategy is as follows: The cost of mitigating the number of unreported COVID-19 infected individuals is 18% of the total cost, reported individuals COVID-19 infected 30%, while co-infection and reported individuals infected with tuberculosis is 3% each. This strategy does not reduce the number of people infected and unreported with tuberculosis.

Fig. 13.

Control profile for strategy E.

The outcome of all these strategies are summarized in Table 2. Recall that the model variables in Table 2 are defined as follows:

• .$I{}_{\text{CU}}\left(t\right)$ unreported individuals infected with COVID-19 only,
• .$I{}_{\text{CR}}\left(t\right)$ reported individuals infected with COVID-19 only,
• .$I{}_{\text{TU}}\left(t\right)$ unreported individuals infected with tuberculosis only,
• .$I{}_{\text{TR}}\left(t\right)$ reported individuals infected with tuberculosis only, and
• .$I{}_{\text{CT}}\left(t\right)$ co-infected individuals.

Table 2.

Summary of the optimal control strategies A - E.

Infections averted	A	B	C	D	E
COVID-19	1,820	1,830	–	–	–
TB	–	–	2,515	2,510	300
Co-infection	600	615	90	80	800
% cost
$I{}_{\text{CU}}\left(t\right)$	20%	12.5%	–	–	18%
$I{}_{\text{CR}}\left(t\right)$	20%	12.5%	6%	6.6%	30%
$I{}_{\text{CT}}\left(t\right)$	10%	12.5%	–	–	3%
$I{}_{\text{TU}}\left(t\right)$	–	–	60%	25%	–
$I{}_{\text{TR}}\left(t\right)$	–	–	6%	33.33%	3%

6. Conclusion

We formulated and analyzed a deterministic compartmental model for the transmission dynamics of tuberculosis and COVID-19. Theoretical results show that for both the tuberculosis (31) and COVID-19 only (5) sub-models, the DFE of each sub-model is globally asymptotically stable when the associated basic reproduction numbers $R_{0_T}$ and $R_{0_C}$ are less than unity, unstable otherwise. From the bifurcation analysis (using the central manifold theory), co-existence of both the DFE and the endemic equilibrium is not possible, and consequently, the endemic equilibria of the sub-models are also globally asymptotically stable whenever $R_{0_C}>1$ and $R_{0_T}>1$. That is, the bifurcation parameters $a<0$ and $b>0$, and the DFE of the two sub-models exchanges their stability with the endemic equilibrium at the threshold $R_{0_C}=1$ (for the COVID-19 only sub-model (5)) and $R_{0_T}=1$ (for the tuberculosis only sub-model (31)).

The basic model is then extended to included five control measures. The appropriate conditions for the existence of optimal control and the optimality system for the full model are established using Pontryagin’s maximum principle. To support the analytical results, numerical simulations of the model with optimal control are carried out using model parameters from the literature (Table 1).

Five strategies which are a combination of these control measures are investigated. Strategies A and B focus on COVID-19 mitigation, and from Table 2, Strategy B will prevent more COVID-19 and co-infections than Strategy A at a lowest total cost percentage (respectively 2,445; 38% vs 2,420; 50%). Similarly, Strategies C and D focus on TB mitigation. Strategy C will prevent 15 more infections than Strategy D, but at the expense of 7% higher percentage of the total cost of the intervention (2,605; 72% vs 2.590; 65%). Strategy E focuses on both COVID-19 and tuberculosis, and will prevent the least number of infections, 1,110 at 54% of the total cost. Because Strategies C and D focus on tuberculosis mitigation, the results suggest that during the course of the COVID-19 pandemic, Strategy B is a better option compared to Strategies A and E, while Strategies C, D and E will also come at a higher cost. As the COVID-19 pandemic is still ongoing, the best strategy of interest to health policy and decision-makers to mitigate its spread is Strategy B which focuses on COVID-19 prevention, treatment and control of co-infection. This strategy yields a better outcome in terms of the number of COVID-19 cases prevented at a lower percentage of the total cost.

The proposed model can be extended in several ways by (1) incorporating vaccination against COVID-19 and tuberculosis (2) inflow of infective immigrants (3) exogenous TB re-infection and COVID-19 re-infection after recovery (as several variants have recently emerged), (4) Generally, representations of real-life situations will inherit the loss of information, and sensitivity analysis is warranted. Also, investigating the impact of reducing the transmission rate and speeding up the time to detect infected individual [40].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.