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February  2022, 15(2): 441-456. doi: 10.3934/dcdss.2021116

## 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 Published  February 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (2) : 441-456. doi: 10.3934/dcdss.2021116
##### References:
 [1] CHOLERA EPIDEMIOLOGY AND RESPONSE FACTSHEET NIGERIA, at https://www.unicef.org/cholera/files/UNICEF-Factsheet-Nigeria-EN-FINAL.pdf, (Accessed April 2020). [2] City Population, 2020. Accessed from: https://citypopulation.de/php/nigeria-admin.php?adm1id=NGA002. [3] M. M. Dalhat, N. A. Isa, P. Nguku, S. G. Nasir, K. Urban, M. Abdulaziz, R. S. Dankoli, P. Nsubuga and G. Poggensee, Descriptive characterization of the 2010 cholera outbreak in Nigeria, BMC Public Health, 14 (2014). doi: 10.1186/1471-2458-14-1167. [4] C. C. Dan-Nwafor, U. Ogbonna and P. Onyiah et al, A cholera outbreak in a rural north central Nigerian community: An unmatched case-control study, BMC Public Health, 19 (2019). doi: 10.1186/s12889-018-6299-3. [5] W. Feller, An Introduction to Probability Theory and its Applications, Volume II. John Wiley and Sons, Inc. New York, 1966. [6] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [7] D. M. Hartley, J. G. Morris Jr and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLOS Medicine, 3 (2005). doi: 10.1371/journal.pmed.0030007. [8] K. Hattaf, M. Mahrouf, J. Adnani and N. Yousfi, Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Physica A: Statistical Mechanics and its Applications, 490 (2018), 591-600.  doi: 10.1016/j.physa.2017.08.043. [9] D. J. Higham, Stochastic ordinary differential equations in applied and computational mathematics, IMA J. Appl. Math., 76 (2011), 449-474.  doi: 10.1093/imamat/hxr016. [10] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115. [11] T. O. Lawoyin, N. A. Ogunbodede, E. A. A. Olumide and M. O. Onadeko, Outbreak of cholera in Ibadan, Nigeria, European J. Epidemiology, 15 (1999). [12] G. Kolaye et al., Mathematical assessment of the role of environmental factors on the dynamical transmission of cholera, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 203-222.  doi: 10.1016/j.cnsns.2018.06.023. [13] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamical behavior of a stochastic epidemic model for cholera, J. Franklin Inst., 356 (2019), 7486-7514.  doi: 10.1016/j.jfranklin.2018.11.056. [14] Y. M. Marwa, I. S. Mbalawata, S. Mwalili and W. M. Charles, Stochastic dynamics of cholera epidemic model: Formulation, analysis and numerical simulation, J. Appl. Math. Physics, 7 (2019), 1097-1125.  doi: 10.4236/jamp.2019.75074. [15] V. A. Meszaros, M. Miller-Dickson, F. Baffour-Awuah Jr, S. Almagro-Moreno and C.B. Ogbunugafor, Direct transmission via households informs models of disease and intervention dynamics in cholera, PLoS ONE, 15 (2020). doi: 10.1371/journal.pone.0229837. [16] National Monthly Update for Cholera in Nigeria: NCDC Situation, Report (October, 2019), 01 November 2019, https://reliefweb.int/report/nigeria/national-monthly-update-cholera-nigeria-ncdc-situation-report-october-2019 [17] , Nigeria life expectancy, (2020), 1950-2020, Accessed from: https://www.macrotrends.net/countries/NGA/nigeria/life-expectancy. [18] Nigeria - Regional Cholera Platform., https://plateformecholera.info/index.php/country-monitoring/nigeria, (Accessed January 2020). [19] Nigeria Demographics., https://www.worldometers.info/demographics/nigeria-demographics/#life-exp, (Accessed January 2020). [20] M. Pascual, M. Bouma, A. King and E. L. Ionides, Inapparent infections and cholera dynamics, Nature, 454 (2008). [21] R. Piarroux and B. Faucher, Cholera epidemics in 2010: Respective roles of environment, strain changes, and human-driven dissemination, Clin Microbiol Infect, 18 (2012), 231-238.  doi: 10.1111/j.1469-0691.2012.03763.x. [22] T. Shelton, E. K. Groves and S. Adrian, A Model of the transmission of cholera in a population with contaminated water, CODEE Journal, 12 (2019), Available at: https://scholarship.claremont.edu/codee/vol12/iss1/5. doi: 10.5642/codee.201912.01.05. [23] J. Wang and C. Modnak, Modeling cholera dynamics with controls, Can. Appl. Math. Q., 19 (2011), 255-273. [24] P. J. Witbooi, An SEIRS epidemic model with stochastic transmission, Adv. Difference Equ., 2017 (2017), 16pp. doi: 10.1186/s13662-017-1166-6. [25] P. J. Witbooi, Stability of a stochastic model of an SIR epidemic with vaccination, Acta Biotheoretica, 65 (2017), 151-165. [26] P. J. Witbooi, G. J. Abiodun, G. J. van Schalkwyk and I. H. I. Ahmed, Stochastic modeling of a mosquito-borne disease, Adv. Difference Equ., 2020 (2020), 15pp. doi: 10.1186/s13662-020-02803-w. [27] P. J. Witbooi, C. Africa, A. Christoffels and I. H. I. Ahmed, (2020)., A population model for the 2017/18 listeriosis outbreak in South Africa, PLoS ONE, 15 (2020). doi: 10.1371/journal.pone.0229901. [28] World Health Organization Cholera, WHO Fact sheet N° 107, 2008, Available: http://www.who.int/mediacentre/factsheets/fs107/en/index.html. (Accessed January 2020). [29] Wo rld Health Organization, Global task force on cholera control, Weekly Epidemiological Record. Cholera Articles: WHO, 85 (2010), 293-308. [30] X. Zhang and H. Peng, Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching, Appl. Math. Lett., 102 (2020), 7pp. doi: 10.1016/j.aml.2019.106095.

show all references

##### References:
 [1] CHOLERA EPIDEMIOLOGY AND RESPONSE FACTSHEET NIGERIA, at https://www.unicef.org/cholera/files/UNICEF-Factsheet-Nigeria-EN-FINAL.pdf, (Accessed April 2020). [2] City Population, 2020. Accessed from: https://citypopulation.de/php/nigeria-admin.php?adm1id=NGA002. [3] M. M. Dalhat, N. A. Isa, P. Nguku, S. G. Nasir, K. Urban, M. Abdulaziz, R. S. Dankoli, P. Nsubuga and G. Poggensee, Descriptive characterization of the 2010 cholera outbreak in Nigeria, BMC Public Health, 14 (2014). doi: 10.1186/1471-2458-14-1167. [4] C. C. Dan-Nwafor, U. Ogbonna and P. Onyiah et al, A cholera outbreak in a rural north central Nigerian community: An unmatched case-control study, BMC Public Health, 19 (2019). doi: 10.1186/s12889-018-6299-3. [5] W. Feller, An Introduction to Probability Theory and its Applications, Volume II. John Wiley and Sons, Inc. New York, 1966. [6] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [7] D. M. Hartley, J. G. Morris Jr and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLOS Medicine, 3 (2005). doi: 10.1371/journal.pmed.0030007. [8] K. Hattaf, M. Mahrouf, J. Adnani and N. Yousfi, Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Physica A: Statistical Mechanics and its Applications, 490 (2018), 591-600.  doi: 10.1016/j.physa.2017.08.043. [9] D. J. Higham, Stochastic ordinary differential equations in applied and computational mathematics, IMA J. Appl. Math., 76 (2011), 449-474.  doi: 10.1093/imamat/hxr016. [10] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115. [11] T. O. Lawoyin, N. A. Ogunbodede, E. A. A. Olumide and M. O. Onadeko, Outbreak of cholera in Ibadan, Nigeria, European J. Epidemiology, 15 (1999). [12] G. Kolaye et al., Mathematical assessment of the role of environmental factors on the dynamical transmission of cholera, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 203-222.  doi: 10.1016/j.cnsns.2018.06.023. [13] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamical behavior of a stochastic epidemic model for cholera, J. Franklin Inst., 356 (2019), 7486-7514.  doi: 10.1016/j.jfranklin.2018.11.056. [14] Y. M. Marwa, I. S. Mbalawata, S. Mwalili and W. M. Charles, Stochastic dynamics of cholera epidemic model: Formulation, analysis and numerical simulation, J. Appl. Math. Physics, 7 (2019), 1097-1125.  doi: 10.4236/jamp.2019.75074. [15] V. A. Meszaros, M. Miller-Dickson, F. Baffour-Awuah Jr, S. Almagro-Moreno and C.B. Ogbunugafor, Direct transmission via households informs models of disease and intervention dynamics in cholera, PLoS ONE, 15 (2020). doi: 10.1371/journal.pone.0229837. [16] National Monthly Update for Cholera in Nigeria: NCDC Situation, Report (October, 2019), 01 November 2019, https://reliefweb.int/report/nigeria/national-monthly-update-cholera-nigeria-ncdc-situation-report-october-2019 [17] , Nigeria life expectancy, (2020), 1950-2020, Accessed from: https://www.macrotrends.net/countries/NGA/nigeria/life-expectancy. [18] Nigeria - Regional Cholera Platform., https://plateformecholera.info/index.php/country-monitoring/nigeria, (Accessed January 2020). [19] Nigeria Demographics., https://www.worldometers.info/demographics/nigeria-demographics/#life-exp, (Accessed January 2020). [20] M. Pascual, M. Bouma, A. King and E. L. Ionides, Inapparent infections and cholera dynamics, Nature, 454 (2008). [21] R. Piarroux and B. Faucher, Cholera epidemics in 2010: Respective roles of environment, strain changes, and human-driven dissemination, Clin Microbiol Infect, 18 (2012), 231-238.  doi: 10.1111/j.1469-0691.2012.03763.x. [22] T. Shelton, E. K. Groves and S. Adrian, A Model of the transmission of cholera in a population with contaminated water, CODEE Journal, 12 (2019), Available at: https://scholarship.claremont.edu/codee/vol12/iss1/5. doi: 10.5642/codee.201912.01.05. [23] J. Wang and C. Modnak, Modeling cholera dynamics with controls, Can. Appl. Math. Q., 19 (2011), 255-273. [24] P. J. Witbooi, An SEIRS epidemic model with stochastic transmission, Adv. Difference Equ., 2017 (2017), 16pp. doi: 10.1186/s13662-017-1166-6. [25] P. J. Witbooi, Stability of a stochastic model of an SIR epidemic with vaccination, Acta Biotheoretica, 65 (2017), 151-165. [26] P. J. Witbooi, G. J. Abiodun, G. J. van Schalkwyk and I. H. I. Ahmed, Stochastic modeling of a mosquito-borne disease, Adv. Difference Equ., 2020 (2020), 15pp. doi: 10.1186/s13662-020-02803-w. [27] P. J. Witbooi, C. Africa, A. Christoffels and I. H. I. Ahmed, (2020)., A population model for the 2017/18 listeriosis outbreak in South Africa, PLoS ONE, 15 (2020). doi: 10.1371/journal.pone.0229901. [28] World Health Organization Cholera, WHO Fact sheet N° 107, 2008, Available: http://www.who.int/mediacentre/factsheets/fs107/en/index.html. (Accessed January 2020). [29] Wo rld Health Organization, Global task force on cholera control, Weekly Epidemiological Record. Cholera Articles: WHO, 85 (2010), 293-308. [30] X. Zhang and H. Peng, Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching, Appl. Math. Lett., 102 (2020), 7pp. doi: 10.1016/j.aml.2019.106095.
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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