A dynamical model for HIV-typhoid co-infection with typhoid vaccine

Abstract

Individuals living with HIV/AIDS are significantly at higher risk of infection with Salmonella Typhi. A deterministic nonlinear mathematical model that describes the co-infection dynamics of HIV and typhoid incorporating typhoid vaccine and treatment has been developed. The basic reproduction number for the co-infection model is computed. The co-infection model exhibits four steady states, namely, the disease-free, HIV alone endemic, typhoid alone endemic, and the co-infection endemic states. The local stability of the disease-free state has been investigated. The co-infection endemic state, if it exists, is found to be locally stable. The minimum vaccination rate that eliminates the typhoid infection is determined. Sensitivity analysis has been performed to ascertain model parameters that have a strong impact on the disease dynamics. It is demonstrated that the co-infection basic reproduction number can be reduced to below unity by simultaneous preventive measures, thereby eliminating both diseases. Numerical simulation of the co-infection model is carried out to examine the effects of parameters on disease spread. The study suggests that the two diseases need to be managed simultaneously for effective control of the co-infection.

Appendices

Proof of Theorem 1

Proof

The right hand side of the co-infection model system (2.2) is continuous and satisfies locally Lipschitz condition on the space of continuous functions. Thus, its solution $(S(t), I_H(t), I_T(t), I_{HT}(t), I_A(t), I_{AT}(t), R(t), B(t))$ exists and is unique over [0, T), where $0<T\le \infty$.

Let us first establish that $S(t)>0$, $\forall t\in [0,T)$. Suppose this statement is false. Then $\exists t_1\in [0,T)$ such that $S(t_1)=0$, $\frac{dS(t_1)}{dt}\le 0$ and $S(t)>0$, $\forall t\in [0,t_1)$. Then there must be $I_H(t)>0$, $I_T(t)>0$ and $R(t)>0$, $\forall t\in [0,t_1)$. Suppose not. Then $\exists t_2\in [0,t_1)$ such that

$$\begin{aligned}&(i)~~I_H(t_2)=0,~ \frac{dI_H(t_2)}{dt}\le 0 ~~and~~ I_H(t)>0, \forall t\in [0,t_2) \nonumber \\&(ii)~~I_T(t_2)=0, ~\frac{dI_T(t_2)}{dt}\le 0 ~~and~~ I_T(t)>0, \forall t\in [0,t_2) \nonumber \\&(iii)~~R(t_2)=0, ~\frac{dR(t_2)}{dt}\le 0 ~~and~~ R(t)>0, \forall t\in [0,t_2). \end{aligned}$$

(A.1)

Then there must have $I_{HT}(t)>0$, $I_A(t)>0$ and $B(t)>0$, $\forall t\in [0,t_2)$. Suppose this is not true. Then $\exists t_3\in [0,t_2)$ such that

$$\begin{aligned}&(i)~~I_{HT}(t_3)=0,~ \frac{dI_{HT}(t_3)}{dt}\le 0~~ and~~I_{HT}(t)>0, \forall t\in [0,t_3)\nonumber \\&(ii)~~I_A(t_3)=0,~ \frac{dI_A(t_3)}{dt}\le 0~~ and~~ I_A( t)>0, \forall t\in [0,t_3) \nonumber \\&(iii)~~B(t_3)=0,~ \frac{dB(t_3)}{dt}\le 0 ~~and~~ B(t)>0, \forall t\in [0,t_3). \end{aligned}$$

(A.2)

Our claim is $I_{AT}(t)>0$, $\forall t\in [0,t_3)$. If this is not true, $\exists t_4\in [0,t_3)$ such that $I_{AT}(t_4)=0$, $\frac{dI_{AT}(t_4)}{dt}\le 0$ and $I_{AT}( t)>0$, $\forall t\in [0,t_4)$. From the sixth equation of system (2.2)

$$\begin{aligned} \frac{dI_{AT}(t_4)}{dt}=\phi I_{HT}(t_4)+\epsilon \lambda _T(t_4)I_A(t_4)>0 \end{aligned}$$

which contradicts $\frac{dI_{AT}(t_4)}{dt}\le 0$. Therefore, $I_{AT}(t)>0$, $\forall t\in [0,t_3)$.

So the fourth, fifth, and eighth equations of system (2.2) give

$$\begin{aligned} \frac{dI_{HT}(t_3)}{dt}= & {} \delta _2I_{AT}(t_3)>0,~~\frac{dI_A(t_3)}{dt}=\phi I_H(t_3)+\tau _3I_{AT}(t_3)>0,\\ \frac{dB(t_3)}{dt}= & {} \alpha _1I_T(t_3)+\alpha _3I_{AT}(t_3)>0 \end{aligned}$$

which contradicts $\frac{dI_{HT}(t_3)}{dt}\le 0$, $\frac{dI_A(t_3)}{dt}\le 0$ and $\frac{dB(t_3)}{dt}\le 0$, respectively (See Eq. (A.2)). Hence, $I_{HT}(t)>0$, $I_A(t)>0$ and $B(t)>0$, $\forall t\in [0,t_2)$. Similarly, $I_{AT}>0$, $\forall t\in [0,t_2)$.

Now from the second, third, and seventh equations of system (2.2)

$$\begin{aligned} \frac{dI_H(t_2)}{dt}= & {} \lambda _H(t_2)S(t_2)+(\tau _2+(1-p)\xi )I_{HT}(t_2)+\delta _1I_A(t_2)>0,\\ \frac{dI_T(t_2)}{dt}= & {} \lambda _T(t_2)S(t_2)>0,~~\frac{dR(t_2)}{dt}=\theta S(t_2)>0 \end{aligned}$$

which is a contradiction to $\frac{dI_H(t_2)}{dt}\le 0$, $\frac{dI_T(t_2)}{dt}\le 0$ and $\frac{dR(t_2)}{dt}\le 0$, respectively (See Eq. (A.1)). Hence, $I_H(t)>0$, $I_T(t)>0$ and $R(t)>0$, $\forall t\in [0,t_1)$. This implies that $I_{HT}>0$, $I_A(t)>0$, $I_{AT}>0$ and $B(t)>0$, $\forall t\in [0,t_1)$. It follows from the first equation of system (2.2)

$$\begin{aligned} \frac{dS(t_1)}{dt}=\varLambda +\omega R(t_1)>0 \end{aligned}$$

which contradicts $\frac{dS(t_1)}{dt}\le 0$. Hence, $S(t)>0$, $\forall t\in [0,T)$.

The above step-by-step discussion follows that $I_H(t)>0$, $I_T(t)>0$, $I_{HT}(t)>0$, $I_A(t)>0$, $I_{AT}(t)>0$, $R(t)>0$ and $B(t)>0$ for all $t\in [0,T)$. Thus, the theorem is proved. $\square$

Proof of Theorem 2

Proof

The dynamics of the total human population is

$$\begin{aligned} \frac{dN}{dt}= & {} \varLambda +(1-p)\xi (I_H+I_{HT})-\mu N-d_T(I_T+I_{AT})-d_{HT} I_{HT}-d_A(I_A+I_{AT}).\\&\Rightarrow ~~~\frac{dN}{dt}\le \varLambda -(\mu -(1-p)\xi )N. \end{aligned}$$

Accordingly,

$$\begin{aligned} 0\le N(t)\le \frac{\varLambda }{\mu -(1-p)\xi }+\left( N(0)-\frac{\varLambda }{\mu -(1-p)\xi }\right) e^{-(\mu -(1-p)\xi )t}. \end{aligned}$$

Thus,

$$\begin{aligned} 0\le N(t)\le \frac{\varLambda }{\mu -(1-p)\xi }~~as ~~t\rightarrow \infty . \end{aligned}$$

Thus, the total human population N(t) is bounded. Hence each of its coordinates $(S(t), I_H(t), I_T(t), I_{HT}(t), I_A(t), I_{AT}(t), R(t))$ is bounded.

Further, the dynamics of Salmonella Typhi bacteria in the environment is given by

$$\begin{aligned} \frac{dB}{dt}= & {} \alpha _1 I_T+\alpha _2 I_{HT}+\alpha _3 I_{AT}-\bar{\mu }B.\\&\Rightarrow ~~~\frac{dB}{dt}+\bar{\mu }B \le (\alpha _1+\alpha _2+\alpha _3)\frac{\varLambda }{\mu -(1-p)\xi }. \end{aligned}$$

This gives

$$\begin{aligned} 0\le B(t)\le \frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }(\mu -(1-p)\xi )}+\left( B(0)-\frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }(\mu -(1-p)\xi )}\right) e^{-\bar{\mu }t}. \end{aligned}$$

Thus,

$$\begin{aligned} 0\le B(t)\le \frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }(\mu -(1-p)\xi )}~~as ~~t\rightarrow \infty . \end{aligned}$$

Hence, B(t) is also bounded.

Therefore, every solution of the model system (2.2) with initial conditions in $\mathbb {R}_+^{8}$ will enter and remains in region

$$\begin{aligned} \varOmega= & {} \left\{ (S,I_H,I_T,I_{HT},I_A,I_{AT},R,B)\in \mathbb {R}_+^8:0\le N(t)\le \frac{\varLambda }{Q};0\le B(t)\le \frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }Q}\right\} ; \\ Q= & {} \mu -(1-p)\xi>0 \quad and \quad \bar{\mu }=\mu _b-r>0. \end{aligned}$$

$\square$