Modeling effects of impulsive control strategies on the spread of mosquito borne disease: role of latent period

Abstract

Controlling the mosquito population is a big challenge for humans. In this paper, we have studied the effects of impulsive control strategies on the spread of mosquito-borne diseases considering the latent period. Therefore, we proposed and analyzed a mosquito-borne disease model governed by a system of impulsive delay differential equations. The proposed mosquito-borne disease model also accounts for three different impulsive control strategies, namely vaccination, pesticides, and adulticides. Two thresholds $R_1$ and $R_2$ established for the global attractivity of the disease-free state and the persistence of the endemic state. The non-trivial disease-free solution of the proposed model is globally asymptotically stable if $R_1$ and $R_2$ less than one. It is shown that a unique positive endemic periodic solution exists only when $R_1$ and $R_2$ greater than unity, which makes for the persistence of the disease. Numerical simulation supports the analytical finding and shows the effectiveness of the impulse control strategies.

Appendices

Appendix A: Lemmas

Lemma 1

(see [33]) Let consider the following impulsive system:

$$\begin{aligned} \left\{ \begin{array}{l} p'(t) = a- b p(t), \ \ \ \ \ \ \ t\ne kT\\ p(t^+) = (1- \theta ) p(t), \ \ \ \ t = kT, k \in N^+. \end{array} \right. \end{aligned}$$

(46)

where $a,b >0, 0<\theta <1.$ Then there exist a unique positive periodic solution of (46)

$$\begin{aligned} {{\widetilde{p}}}_e(t)= \frac{a}{b} + ( p^* - \frac{a}{b}) e^{-b(t-kT)}, \ \ \ \ \ \ kT<t\le (k+1)T. \end{aligned}$$

Which is globally asymptotically stable, where

$$\begin{aligned} p^* = \frac{a (1-\theta )(1-e^{-bT})}{b(1-(1-\theta )e^{-bT})}. \end{aligned}$$

Lemma 2

( See [14, 29]) Let us consider the following delay differential equation

$$\begin{aligned} p'(t)= a_1 p(t-\tau )-a_2 p(t), \end{aligned}$$

(47)

where $a_1, a_2, \tau > 0$; $p(t) > 0$ for $-\tau \le t \le 0.$ We have:

• (1) if $a_1 < a_2$, then $\lim _{t\rightarrow \infty } p(t) = 0;$

• (2) if $a_1 > a_2$, then $\lim _{t\rightarrow \infty } p(t) = +\infty .$

Appendix B: Definitions

Definition 1

System (1) is said to be uniformly persistent if there exist positive constants $Q_i \ge q_i, \ \ i= 1,2,3,4.$ ( both are independent of the initial values ), such that every solution $(S_h(t),I_h(t),R_h(t),A_m(t),S_m(t), I_m(t))$ with positive initial conditions (2) of system (1) satisfies:

$$\begin{aligned} \left\{ \begin{array}{l} q_1 \le S_h(t) \le Q_1, \\ q_2\le R_h(t)\le Q_2 ,\\ q \le I_h(t) + I_m(t) \le Q, \\ q_3\le A_m(t)\le Q_3, \\ q_4\le S_m(t)\le Q_4. \end{array} \right. \end{aligned}$$

Definition 2

System (1) is said to be permanent if there exists a compact region $\varOmega _0 \in \varOmega$ such that every solution $(S_h(t),I_h(t),R_h(t),A_m(t),S_m(t), I_m(t))$ with initial condition (2) of system (1) will eventually enter and remain in the region $\varOmega _0.$

Appendix C: Proof

First, If $[I_h(t)+I_m(t)] > q^*$ for all $t> t_1,$ then our aim is fulfilled. Second $I_h(t)+I_m(t)$ oscillates about $q^*$ for all large t; setting $t^* = inf_{t> t_1} (I_h(t)+I_m(t)) \le q^*,$ there are two possible cases for $t^*.$ We hope to show that $(I_h(t)+I_m(t))\ge q$ for all large t. The conclusion is evident in the first case. For the second case, let $t^* >0$ and $\rho >0$ satisfy $I_h(t^*)+I_m(t^*) = I_h(t^*+\rho )+I_m(t^*+\rho )= q^*,$ and $( I_h(t)+I_m(t)) < q^*,$ $S_h(t) > \rho _1$ for $t^*< t < t^*+\rho .$ Therefore, it is certain that there exists a $\psi (0<\psi <\tau )$ such that

$$\begin{aligned} (I_h(t)+I_m(t)) > \frac{q^*}{2} \ \ \ \ for \ \ \ \ t^*< t < t^*+\psi . \end{aligned}$$

In this case, we will discuss three possible cases in terms of the size of $\rho , \psi ,$ and $\tau$.

Case I. If $\rho \le \psi \le \tau$, then $(I_h(t)+I_m(t)) > \frac{q^*}{2} \ \ \ \ for \ \ \ \ t^*< t < t^*+\rho .$

Case II. If $\psi \le \rho \le \tau$, then from equations of system (1), we can deduce $I'_h(t)+I'_m(t) >-\varPi ( I_h(t)+I_m(t))$, where $\varPi = max\{\mu _h + \omega _h+d, \mu _m\}$ for $t \varepsilon [t^*, t^*+\tau$ and $(I_h(t^*)+I_m(t^*)) = q^*$; it is obvious that $I_h(t)+I_m(t) = Q_1$ for $t^*< t < t^*+\psi .$ Case III. If If $\psi \le T \le \rho$, we will consider the following two cases, respectively.

Case IIIa. For $t^*< t < t^*+\tau$, it is easy to obtain $( I_h(t)+I_m(t)) > Q_1$.

Case IIIb. For $t^*+\tau< t < t^*+\rho$, it is easy to obtain $(I_h(t)+I_m(t)) > Q_1.$ Then, proceeding exactly as the proof for the above claim, we see that $( I_h(t)+I_m(t)) \ge q$ for $t^*+\tau< t < t^*+\rho$. Since this kind of interval $[t^*, t^*+\rho ]$ is chosen in an arbitrary way ( we only need $t^*$ to be large), we conclude that of our above discussions, the choices of q are independent of the positive solution, and we have proved that any positive solution of system (1) satisfies $(I_h(t)+I_m(t)) \ge q$ for all large t. The proof is completed.