Impact of Awareness Programs on Cholera Dynamics: Two Modeling Approaches

Abstract

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.

Appendix: Global Stability of the Endemic Equilibrium of Model One

Let $E^*=(S^*, I^*, R^*, B^*, M^*)$ denote an endemic equilibrium of model (1). To establish the global stability of $E^*$, we make the following assumptions:

$$\begin{aligned} \left( 1-\dfrac{\beta _1(M)I}{\beta _1(M^*)I^*}\right) \left( 1-\dfrac{M\beta _1(M^*)I^*}{M^*\beta _1(M)I}\right) \ge 0 \end{aligned}$$

(20)

for $0\le I\le N$ and $M^0\le M\le M_{\mathrm{max}}$, and

$$\begin{aligned} \left( 1-\dfrac{\beta _2(M)B/(B+K)}{\beta _2(M^*)B^*/(B^*+K)}\right) \left( 1-\dfrac{\beta _2(M^*)/(B^*+K)}{\beta _2(M)/(B+K)}\right) \ge 0 \end{aligned}$$

(21)

for $M^0\le M\le M_{\mathrm{max}}$ and $0\le B\le B_{\mathrm{max}}$.

Theorem 4.1

Suppose that (i) assumptions (20) and (21) are satisfied; (ii) $\xi (M)\equiv \xi$ is constant; (iii) $\sigma = 0$. If $\mathcal {R}_0>1$, then system (1) has a unique endemic equilibrium $E^*$ that is globally asymptotically stable in $\mathring{\Omega }$.

Proof

For system (1), motivated by Tien and Earn (2010), we consider a Lyapunov function

$$\begin{aligned} \begin{aligned} \mathcal {L}&= c_1 \left( S-S^*-S^*\ln \Big (\dfrac{S}{S^*}\Big )\right) +c_1 \left( I-I^*-I^*\ln \Big (\dfrac{I}{I^*}\Big )\right) \\&\quad + c_2 \left( B-B^*-B^*\ln \Big (\dfrac{B}{B^*}\Big )\right) + c_3 \left( M-M^*-M^*\ln \Big (\dfrac{M}{M^*}\Big )\right) , \end{aligned} \end{aligned}$$

where $c_i>0$ ($i=1,2,3$) are constants to be determined. It is easy to verify that $\mathcal {L}\ge 0$ for all $S, I, B, M>0$, and $\mathcal {L}=0$ iff $(S, I, B, M)=(S^*,I^*,B^*,M^*)$. Differentiating $\mathcal {L}$ along solutions of (1) and applying all equations of (4) except the third one, we obtain

$$\begin{aligned} \begin{aligned} \mathcal {L}'&= c_1 \Big (1-\dfrac{S^*}{S}\Big )S'+c_1 \Big (1-\dfrac{I^*}{I}\Big )I'+c_2 \Big (1-\dfrac{B^*}{B}\Big )B'+c_3 \Big (1-\dfrac{M^*}{M}\Big )M'\\&=c_1 \left[ -\mu \, S \Big (1-\dfrac{S^*}{S}\Big )^2\right. \\&\quad \left. +\, \beta _1(M^*)S^*I^*\left( 2-\dfrac{S^*}{S}-\dfrac{I}{I^*}-\dfrac{\beta _1(M)SII^*}{\beta _1(M^*)S^*I^*I}+\dfrac{\beta _1(M)I}{\beta _1(M^*)I^*}\right) \right] \\&\quad +\, c_1\beta _2(M^*)S^*\dfrac{B^*}{B^*+K} \\&\quad \times \,\left( 2-\dfrac{S^*}{S}-\dfrac{I}{I^*}-\dfrac{\beta _2(M)SB/(B+K)I^*}{\beta _2(M^*)S^*B^*/(B^*+K)I}+\dfrac{\beta _2(M)B/(B+K)}{\beta _2(M^*)B^*/(B^*+K)}\right) \\&\quad +\, c_2\xi I^*\left( \dfrac{I}{I^*}-\dfrac{B}{B^*}-\dfrac{B^* I}{B I^*} +1\right) \\&\quad +\,c_3 \left( -\varLambda \dfrac{M}{M^*}\left( 1-\dfrac{M^*}{M}\right) ^2+\eta I^*\left( \dfrac{I}{I^*}-\dfrac{M}{M^*}-\dfrac{M^* I}{M I^*}+1\right) \right) . \end{aligned} \end{aligned}$$

(22)

Notice that $x-1\ge \ln (x)$ for any $x>0$, and the equality holds iff $x=1.$ Together with (20), we find that

$$\begin{aligned}&2-\dfrac{S^*}{S}-\dfrac{I}{I^*}-\dfrac{\beta _1(M)SII^*}{\beta _1(M^*)S^*I^*I}+\dfrac{\beta _1(M)I}{\beta _1(M^*)I^*}\nonumber \\&\quad = -\left( 1-\dfrac{\beta _1(M)I}{\beta _1(M^*)I^*}\right) \left( 1-\dfrac{M\beta _1(M^*)I^*}{M^*\beta _1(M)I}\right) +3-\dfrac{S^*}{S}\nonumber \\&\qquad -\dfrac{\beta _1(M)SII^*}{\beta _1(M^*)S^*I^*I}-\dfrac{M\beta _1(M^*)I^*}{M^*\beta _1(M)I} -\dfrac{I}{I^*}+\dfrac{M}{M^*}\nonumber \\&\quad \le -\left( \dfrac{S^*}{S}-1\right) -\left( \dfrac{\beta _1(M)SII^*}{\beta _1(M^*)S^*I^*I}-1\right) -\left( \dfrac{M\beta _1(M^*)I^*}{M^*\beta _1(M)I}-1\right) -\dfrac{I}{I^*}+\dfrac{M}{M^*}\nonumber \\&\quad = -\ln \Big (\dfrac{S^*}{S}\dfrac{\beta _1(M)SII^*}{\beta _1(M^*)S^*I^*I}\dfrac{M\beta _1(M^*)I^*}{M^*\beta _1(M)I}\Big )-\dfrac{I}{I^*}+\dfrac{M}{M^*}\nonumber \\&\quad = \dfrac{M}{M^*}-\ln \Big (\dfrac{M}{M^*}\Big )-\dfrac{I}{I^*}+\ln \Big (\dfrac{I}{I^*}\Big ). \end{aligned}$$

(23)

Likewise, using (21), we obtain

$$\begin{aligned} \begin{aligned}&2-\dfrac{S^*}{S}-\dfrac{I}{I^*}-\dfrac{\beta _2(M)SB/(B+K)I^*}{\beta _2(M^*)S^*B^*/(B^*+K)I}+\dfrac{\beta _2(M)B/(B+K)}{\beta _2(M^*)B^*/(B^*+K)}\\&\quad \le \dfrac{B}{B^*}-\ln \Big (\dfrac{B}{B^*}\Big )-\dfrac{I}{I^*}+\ln \Big (\dfrac{I}{I^*}\Big ). \end{aligned} \end{aligned}$$

(24)

Meanwhile, one can verify that

$$\begin{aligned}&\dfrac{I}{I^*}-\dfrac{B}{B^*}-\dfrac{B^* I}{B I^*}+1 =-\left( \dfrac{B^* I}{B I^*}-1\right) +\dfrac{I}{I^*}-\dfrac{B}{B^*}\nonumber \\&\quad \le -\ln \left( \dfrac{B^* I}{B I^*}\right) +\dfrac{I}{I^*}-\dfrac{B}{B^*} =\dfrac{I}{I^*}-\ln \Big (\dfrac{I}{I^*}\Big )-\dfrac{B}{B^*}+\ln \Big (\dfrac{B}{B^*}\Big ). \end{aligned}$$

(25)

Similarly, we have

$$\begin{aligned} \dfrac{I}{I^*}-\dfrac{M}{M^*}-\dfrac{M^* I}{M I^*}+1\le \dfrac{I}{I^*}-\ln \Big (\dfrac{I}{I^*}\Big )-\dfrac{M}{M^*}+\ln \Big (\dfrac{M}{M^*}\Big ). \end{aligned}$$

(26)

It follows from (23)–(26) that the Eq. (22) yields

$$\begin{aligned} \mathcal {L}'&\le c_1 \beta _1(M^*)S^*I^*\left( \dfrac{M}{M^*}-\ln \Big (\dfrac{M}{M^*}\Big )-\dfrac{I}{I^*}+\ln \Big (\dfrac{I}{I^*}\Big )\right) \nonumber \\&\quad +c_1 \beta _2(M^*)S^*\dfrac{B^*}{B^*+K}\left( \dfrac{B}{B^*}-\ln \Big (\dfrac{B}{B^*}\Big )-\dfrac{I}{I^*}+\ln \Big (\dfrac{I}{I^*}\Big )\right) \nonumber \\&\quad +c_2 \xi I^* \left( \dfrac{I}{I^*}-\ln \Big (\dfrac{I}{I^*}\Big )-\dfrac{B}{B^*}+\ln \Big (\dfrac{B}{B^*}\Big )\right) \nonumber \\&\quad +c_3 \eta I^* \left( \dfrac{I}{I^*}-\ln \Big (\dfrac{I}{I^*}\Big )-\dfrac{M}{M^*}+\ln \Big (\dfrac{M}{M^*}\Big )\right) . \end{aligned}$$

(27)

Take $c_1= \xi \eta I^*$, $c_2 = \eta \beta _2(M^*)S^*B^*/(B^*+K)$ and $c_3=\xi \beta _1(M^*)S^*I^*$. One can verify by direct calculation that the right-hand side of the inequality (27) is zero. This shows $\mathcal {L}'\le 0$ with the chosen positive constants $c_1, c_2$, and $c_3.$ Moreover, if $\mathcal {L}'=0$, then there exists a constant $\hat{k}$ such that

$$\begin{aligned} S=S^*,\,\,\, I= \hat{k} I^*,\,\, \,B= \hat{k} B^*,\,\,\, M= \hat{k} M^*. \end{aligned}$$

(28)

However, by the last equation of (4), $0=\varLambda + \eta \hat{k} I^*-\nu \hat{k}M^*$. This implies that $\hat{k}=1$. Meanwhile, $R=R^*$ which follows from the third equation of (4). Thus, the largest invariant set for which $\mathcal {L}'=0$ contains only the EE. Therefore, by LaSalle’s invariant principle (LaSalle 1976), the EE is globally asymptotically stable in $\mathring{\Omega }$ when $\mathcal {R}_0>1$.

$\square$