# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021116
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## A stochastic population model of cholera disease

 1 Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, 7535, South Africa 2 SA MRC Bioinformatics Unit, South African National Bioinformatics Institute, University of the Western Cape, Bellville, 7535, South Africa 3 Department of Mathematics, , Vaal University of Technology, , Vanderbijlpark, South Africa

*Corresponding author: Peter J. Witbooi

Received  October 2020 Revised  September 2021 Early access November 2021

Fund Project: The co-author Ibrahim H.I. Ahmed is funded through the South African Research Chairs Initiative of the Department of Science and Innovation and the South African National Research Foundation UID:64751

A cholera population model with stochastic transmission and stochasticity on the environmental reservoir of the cholera bacteria is presented. It is shown that solutions are well-behaved. In comparison with the underlying deterministic model, the stochastic perturbation is shown to enhance stability of the disease-free equilibrium. The main extinction theorem is formulated in terms of an invariant which is a modification of the basic reproduction number of the underlying deterministic model. As an application, the model is calibrated as for a certain province of Nigeria. In particular, a recent outbreak (2019) in Nigeria is analysed and featured through simulations. Simulations include making forward projections in the form of confidence intervals. Also, the extinction theorem is illustrated through simulations.

Citation: Peter J. Witbooi, Grant E. Muller, Marshall B. Ongansie, Ibrahim H. I. Ahmed, Kazeem O. Okosun. A stochastic population model of cholera disease. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021116
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##### References:
The classes $I(t)$ and $B(t)$ in the deterministic model as calculated from the data over the outbreak period $[t_0, t_2]$
Evolution of the $I(t)$ as calculated the stochastic model from the data over the post-outbreak period $[t_1, t_2]$. The date $t_1$ which is 13 weeks later than $t_0$, corresponds to $t = 0$
Evolution of the $I(t)$ as calculated for the stochastic model from the data over the post-outbreak period $[t_2, t_2+30]$
Evolution of the model over the post-outbreak period $[t_2, t_2+78]$. We show the deterministic $I(t)$, and for the stochastic model the mean, tenth and ninetieth percentiles
Numerical values of model parameters for Adamawa State, Nigeria, 2019.
 Param. Description Numerical value Reference/comment $P$ Population size when disease-free 4 890 000 [18], [19] $\mu$ mortality rate, excluding death directly due to cholera $0.0003548$ per week [2] $\Lambda$ rate of inflow 1.735 per week $\Lambda=\mu P$ $\epsilon$ rate of human deaths due to cholera $0.0149$ per week [16] $\alpha$ Transfer rate from I-class to R-class (recovery rate) $1$ per week [22] $\omega$ Removal rate of pathogen from the environment $0.33\times 7$ per week [7] [12] [15] $\beta_c$ a contact rate $4.095\times 10^{-7}$ per week Fitted $\beta_h$ a contact rate $1.024\times 10^{-7}$per week per unit of $B$ Fitted $K$ A threshold value of $B$ 1 Remark 5.1 $\sigma$ rate of increase of the levels of the pathogen 1.155 per week Fitted.
 Param. Description Numerical value Reference/comment $P$ Population size when disease-free 4 890 000 [18], [19] $\mu$ mortality rate, excluding death directly due to cholera $0.0003548$ per week [2] $\Lambda$ rate of inflow 1.735 per week $\Lambda=\mu P$ $\epsilon$ rate of human deaths due to cholera $0.0149$ per week [16] $\alpha$ Transfer rate from I-class to R-class (recovery rate) $1$ per week [22] $\omega$ Removal rate of pathogen from the environment $0.33\times 7$ per week [7] [12] [15] $\beta_c$ a contact rate $4.095\times 10^{-7}$ per week Fitted $\beta_h$ a contact rate $1.024\times 10^{-7}$per week per unit of $B$ Fitted $K$ A threshold value of $B$ 1 Remark 5.1 $\sigma$ rate of increase of the levels of the pathogen 1.155 per week Fitted.
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