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.
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This paper has been performed under the auspices of the G.N.F.M. of I.N.d.A.M. The Referees’ competence and comments are gratefully acknowledged.
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Appendix
Appendix
1.1 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.
1.2 Proof of Theorem 2
Substituting \((\bar{S},\,\bar{I},\,\bar{B},\,\bar{R})=(N_0,0,0,0)\) in (17), one has that
Hence
and
In view of (56), it follows that
Hence (24) is verified if and only if
i.e., by virtue of (55), if and only if (27) holds.
1.3 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
Hence
and
where \(K_i,\,(i=1,2,3)\) are positive constants given by
and
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
In view of (60)\(_2\) and (61), it follows that (24) is always satisfied.
1.4 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
On choosing
(ii) is obtained. For the endemic equilibrium, since \(b_{21}>0\) and \(b_{13}<0\), on choosing
it follows that \(b_{13}-A_3b_{21}=0\) and
Hence, on taking
(34) is proved.
As concerns (iii), by virtue of (6), (11), (15)\(_1\), the following inequalities hold a.e. in \(\Omega \)
and
Hence, from (32)\(_7\), it turns out that
From (6), (10) and (15)\(_1\) one obtains
and
Hence
On applying the Cauchy–Schwarz, (35) is obtained.
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Capone, F., De Cataldis, V. & De Luca, R. Influence of diffusion on the stability of equilibria in a reaction–diffusion system modeling cholera dynamic. J. Math. Biol. 71, 1107–1131 (2015). https://doi.org/10.1007/s00285-014-0849-9
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DOI: https://doi.org/10.1007/s00285-014-0849-9