Influence of diffusion on the stability of equilibria in a reaction–diffusion system modeling cholera dynamic

Abstract

A reaction–diffusion system modeling cholera epidemic in a non-homogeneously mixed population is introduced. The interaction between population and toxigenic Vibrio cholerae concentration in contaminated water has been taken into account. The existence of biologically meaningful equilibria is investigated together with their linear and nonlinear stability. Using the data collected during the Haiti cholera epidemic, a numerical simulation is performed.

Appendix

Proof of Theorem 1

Let us set $\max _{\bar{\Omega }_T}\,B=B({\mathbf x}_1, t_1)$. We have to distinguish two cases.

• (1) If $({\mathbf x}_1,t_1)$ belongs to the interior of $\Omega _T$, then (1)$_3$ and $\Vert I\Vert _\infty \le N_0$ imply that

  $$\begin{aligned} \left[ \frac{\partial B}{\partial t}-eN_0+(\mu _B-\pi _B)B-\gamma _3\Delta B\right] _{({\mathbf x}_1,t_1)}<0. \end{aligned}$$

  (53)

  Since

  $$\begin{aligned} \left[ \frac{\partial B}{\partial t}\right] _{({\mathbf x}_1,t_1)}=0,\qquad \left[ \Delta B\right] _{({\mathbf x}_1,t_1)}<0, \end{aligned}$$

  then (53) can hold if and only if

  $$\begin{aligned} -eN_0+(\mu _B-\pi _B)B({\mathbf x}_1,t_1)<0 \end{aligned}$$

  and hence if and only if $B({\mathbf x}_1,t_1)< \frac{eN_0}{\mu _B-\pi _B}$.

• (2) If $({\mathbf x}_1, t_1)\in \Gamma _T$, in view of the regularity of the domain $\Omega$, since $\Omega$ verifies in any point ${\mathbf x}_0\in \partial \Omega$ the interior ball condition, there exists an open ball $B^*\subset \Omega$ with ${\mathbf x}_0\in \partial B^*$. If $B({\mathbf x}_1, t_1)> \frac{eN_0}{\mu _B-\pi _B}$, on choosing the radius of $B^*$ sufficiently small, it follows that

  $$\begin{aligned} \gamma _3\Delta B- \frac{\partial B}{\partial t}>0,\quad \hbox {in}\,\, B^* \end{aligned}$$

  and by virtue of Hopf’s Lemma (Protter and Weinberger 1967), one obtains that

  $$\begin{aligned} \left( \frac{dB}{d{\mathbf n}}\right) _{({\mathbf x}_1,t_1)}>0. \end{aligned}$$

  Since $\frac{dB}{d{\mathbf n}}=0$ on $\partial \Omega \times {\mathbb R}^+$, (10) follows.

Proof of Theorem 2

Substituting $(\bar{S},\,\bar{I},\,\bar{B},\,\bar{R})=(N_0,0,0,0)$ in (17), one has that

$$\begin{aligned} {\left\{ \begin{array}{ll} b_{11}=-(\mu +\bar{\alpha }\gamma _1),\quad b_{13}=- \frac{\mu _3\beta N_0}{\mu _1K_B},\quad b_{21}=0,\\ b_{22}=-(\sigma +\mu +\bar{\alpha }\gamma _2),\quad b_{23}= \frac{\mu _3\beta N_0}{\mu _2K_B},\\ b_{32}= \frac{\mu _2}{\mu _3}e,\quad b_{33}=-(\mu _B-\pi _B+\bar{\alpha }\gamma _3). \end{array}\right. } \end{aligned}$$

(54)

Hence

$$\begin{aligned} A^{*}&= (\sigma +\mu +\bar{\alpha }\gamma _2)(\mu _B-\pi _B+\bar{\alpha }\gamma _3)- \frac{\beta N_0 e}{K_B}\nonumber \\&= (\sigma + \mu )(\mu _B - \pi _B)\left[ 1 - R_0 + \frac{\bar{\alpha }[\gamma _2(\mu _B - \pi _B) + \gamma _3(\sigma + \mu ) + \bar{\alpha }\gamma _2\gamma _3]}{(\sigma +\mu )(\mu _B-\pi _B)}\right] \nonumber \\&= (\sigma +\mu )(\mu _B-\pi _B)(R^*_0-R_0) \end{aligned}$$

(55)

and

$$\begin{aligned} A^{*}_1=0,\quad A^{*}_2=-b_{11}(b_{11}+\mathtt{I }^{*})<0. \end{aligned}$$

(56)

In view of (56), it follows that

$$\begin{aligned} \max \{A^{*}_1,A^*_2\}=0. \end{aligned}$$

Hence (24) is verified if and only if

$$\begin{aligned} A^{*}>0, \end{aligned}$$

(57)

i.e., by virtue of (55), if and only if (27) holds.

Proof of Theorem 3

Substituting $(\bar{S},\,\bar{I},\,\bar{B},\,\bar{R})=(S_2,\,I_2,\,B_2,\, R_2)$ in (17), one has that

$$\begin{aligned} {\left\{ \begin{array}{ll} b_{11}=- \dfrac{\mu (\beta +\mu )R_0+\bar{\alpha }\gamma _1(\beta +\mu R_0)}{\beta +\mu R_0},\\ b_{13}=- \dfrac{\mu _3(\sigma +\mu )(\mu _B-\pi _B)(\beta +\mu )}{\mu _1e(\beta +\mu R_0)},\\ b_{21}= \dfrac{\mu _1\beta \mu (R_0-1)}{\mu _2(\beta +\mu R_0)},\quad b_{22}=-(\sigma +\mu +\bar{\alpha }\gamma _2),\\ b_{23}=- \dfrac{\mu _1}{\mu _2}b_{13},\quad b_{32}= \dfrac{\mu _2}{\mu _3}e,\quad b_{33}=-(\mu _B-\pi _B+\bar{\alpha }\gamma _3).\end{array}\right. } \end{aligned}$$

(58)

Hence

$$\begin{aligned} A^{*}&= (\sigma + \mu + \bar{\alpha }\gamma _2)(\mu _B - \pi _B + \bar{\alpha }\gamma _3) - \frac{(\sigma + \mu )(\mu _B - \pi _B)(\beta + \mu )}{\beta +\mu R_0}\nonumber \\&= \frac{\mu (\sigma + \mu )(\mu _B - \pi _B)(R_0 - 1)}{\beta +\mu R_0} + \bar{\alpha }\left[ \gamma _2(\mu _B - \pi _B) + \gamma _3(\sigma + \mu ) + \bar{\alpha }\gamma _2\gamma _3\right] ,\nonumber \\ \end{aligned}$$

(59)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} A^{*}_1=- \dfrac{\beta \mu (\beta +\mu )(\sigma +\mu )(\mu _B-\pi _B)(R_0-1)}{(\beta +\mu R_0)[\mu R_0(\beta +\mu )+\bar{\alpha }\gamma _1(\beta +\mu R_0)]},\\ A^*_2=- \dfrac{K_1 R^2_0+K_2R_0+K_3}{(\beta +\mu R_0)^2[\sigma +\mu +\mu _B-\pi _B+\bar{\alpha }(\gamma _2+\gamma _3)]}(<\!0), \end{array}\right. } \end{aligned}$$

(60)

where $K_i,\,(i=1,2,3)$ are positive constants given by

$$\begin{aligned} K_1&= \mu ^2(\sigma + \mu + \mu _B - \pi _B + \bar{\alpha }(\gamma _2 + \gamma _3))\left\{ \bar{\alpha }\gamma _1[\sigma + \mu + \mu _B - \pi _B \right. \\&+\, \bar{\alpha }(\gamma _1 + \gamma _2 + \gamma _3)] + (\beta +\mu ) \left[ \beta +\mu +\sigma +\mu +\mu _B-\pi _B\right. \\&+\left. \left. \bar{\alpha }(\gamma _1+\gamma _2+\gamma _3)+\bar{\alpha }\gamma _1\right] \right\} , \end{aligned}$$

$$\begin{aligned} K_2&= \beta \mu \left\{ \left[ \sigma +\mu +\mu _B-\pi _B+\bar{\alpha }(\gamma _2+\gamma _3)\right] \left[ \bar{\alpha }(\beta +\mu )(\gamma _1+\gamma _2+\gamma _3)\right. \right. \\&+\left. \bar{\alpha }\gamma _1(\beta +\mu )+2\bar{\alpha }\gamma _1(\sigma +\mu +\mu _B-\pi _B+\bar{\alpha }(\gamma _1+\gamma _2+\gamma _3))\right] \\&+\!\left. (\beta +\mu )(\mu _B-\pi _B)^2+(\beta +\mu )(\sigma +\mu +\mu _B-\pi _B)[\sigma +\mu +\bar{\alpha }(\gamma _2+\gamma _3)]\right\} \end{aligned}$$

and

$$\begin{aligned} K_3&= \beta ^2\bar{\alpha }\gamma _1[\sigma \!+\! \mu \!+\! \mu _B \!-\! \pi _B \!+\! \bar{\alpha }(\gamma _2 + \gamma _3)] [\sigma + \mu + \mu _B - \pi _B + \bar{\alpha }(\gamma _1 + \gamma _2 + \gamma _3)]\\&+\beta \mu (\mu _B-\pi _B)(\sigma +\mu )(\beta +\mu ). \end{aligned}$$

Since $(S_2,\,I_2,\,B_2,\,R_2)$ exists if and only if $R_0>1$, then, from (59) and (60)$_1$, it turns out that

$$\begin{aligned} A^{*}>0,\quad A^{*}_1<0. \end{aligned}$$

(61)

In view of (60)$_2$ and (61), it follows that (24) is always satisfied.

Proof of Lemma 2

By virtue of Remark 4, the linear stability of the biologically meaningful equilibria guarantees that $A^*>0$. Hence, the proof of (i) can be obtained on following the same procedure by Rionero (2011a, b), Capone et al. (2013, 2014). As concerns (ii), for the disease-free equilibrium, since $b_{21}=0$, one has that

$$\begin{aligned} \Phi ^{*}=b_{13}\langle U_1,U_3 \rangle \le \frac{\mu _3^3\beta ^2N^2_0}{2\mu ^2_1K^2_B|I^*A^*|}\Vert U_1\Vert ^2+ \frac{1}{2}|I^*A^*|(\Vert U_2\Vert ^2+\Vert U_3\Vert ^2). \end{aligned}$$

On choosing

$$\begin{aligned} \frac{\mu ^2_3}{\mu ^2_1}= \frac{|b_{11}I^*A^*|K^2_B}{\beta ^2N^2_0}, \end{aligned}$$

(62)

(ii) is obtained. For the endemic equilibrium, since $b_{21}>0$ and $b_{13}<0$, on choosing

$$\begin{aligned} \frac{\mu ^2_1}{\mu _2\mu _3}=- \frac{(\sigma +\mu )(\mu _B-\pi _B)(\beta +\mu )}{A_3e\beta \mu (R_0-1) }>0, \end{aligned}$$

(63)

it follows that $b_{13}-A_3b_{21}=0$ and

$$\begin{aligned} \Phi ^*\!=\! \frac{\mu _1\beta \mu (R_0-1)A_1}{\mu _2(\beta +\mu R_0)} \langle U_1,U_2 \rangle \!\le \! \frac{\mu _1^2\beta ^2\mu ^2(R_0\!-\!1)^2A^2_1}{2|I^*A^*|\mu _2^2(\beta +\mu R_0)^2}\Vert U_1\Vert ^2+ \frac{1}{2}|I^*A^*|\Vert U_2\Vert ^2. \end{aligned}$$

Hence, on taking

$$\begin{aligned} \frac{\mu ^2_1}{\mu ^2_2}= \frac{|b_{11}I^*A^*|(\beta +\mu R_0)^2}{\beta ^2\mu ^2(R_0-1)^2A^2_1}, \end{aligned}$$

(64)

(34) is proved.

As concerns (iii), by virtue of (6), (11), (15)$_1$, the following inequalities hold a.e. in $\Omega$

$$\begin{aligned} \theta _1\mu _1U_1+\bar{S}=\theta _1X_1+\bar{S}=\theta _1(S-\bar{S})+\bar{S}=\theta _1S+(1-\theta _1)\bar{S}\le N_0 \end{aligned}$$

and

$$\begin{aligned} K_B+\theta _1\mu _3U_3+\bar{B}=K_B+\theta _1B+(1-\theta _1)\bar{B}>K_B. \end{aligned}$$

Hence, from (32)$_7$, it turns out that

$$\begin{aligned} \Phi _2&< \mu _3\left[ c_1 \frac{\mu _3}{\mu _1} \int _{\Omega }|U_1|U^2_3\,d \Omega +c_2 \int _{\Omega }U^2_1|U_3|\,d \Omega \right. \\&\quad +\left. c_3 \frac{\mu _3}{\mu _2} \int _{\Omega }|U_2|U_3^2\,d \Omega +c_4 \frac{\mu _1}{\mu _2} \int _{\Omega }|U_1U_2U_3|\,d \Omega \right. \\&\quad \left. +c_5 \frac{\mu _3}{\mu _2} \int _{\Omega }|U_3|^3\,d \Omega +c_6 \frac{\mu _1}{\mu _2} \int _{\Omega }|U_1|U_3^2\,d \Omega \right] . \end{aligned}$$

From (6), (10) and (15)$_1$ one obtains

$$\begin{aligned} \Vert U_i\Vert _\infty = \frac{1}{\mu _i}\Vert X_i\Vert _\infty \le \frac{2N_0}{\mu _i},\quad i=1,2 \end{aligned}$$

and

$$\begin{aligned} \Vert U_3\Vert _\infty = \frac{1}{\mu _3}\Vert X_3\Vert _\infty = \frac{1}{\mu _3}\Vert B-\bar{B}\Vert _\infty \le \frac{2M}{\mu _3}. \end{aligned}$$

Hence

$$\begin{aligned} \Phi _2&< \mu _3\left[ \frac{2c_1M}{\mu _1} \int _\Omega |U_1||U_3|d\Omega + \frac{2c_2N_0}{\mu _1} \int _\Omega |U_1||U_3|d\Omega + \frac{2c_3M}{\mu _2} \int _\Omega |U_2||U_3|d\Omega \right. \\&+ \left. \frac{2c_4N_0}{\mu _2} \int _\Omega |U_2||U_3|d\Omega + \frac{2c_5M}{\mu _2} \int _\Omega U^2_3d\Omega + \frac{2c_6N_0}{\mu _2} \int _\Omega U^2_3d\Omega \right] . \end{aligned}$$

On applying the Cauchy–Schwarz, (35) is obtained.