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
References:
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K. HattafM. MahroufJ. 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.  Google Scholar

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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.  Google Scholar

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A. LahrouzL. 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.  Google Scholar

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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.  Google Scholar

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Q. LiuD. JiangT. 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.  Google Scholar

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Y. M. MarwaI. S. MbalawataS. 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.  Google Scholar

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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.  Google Scholar

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, Nigeria life expectancy, (2020), 1950-2020, Accessed from: https://www.macrotrends.net/countries/NGA/nigeria/life-expectancy. Google Scholar

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Nigeria - Regional Cholera Platform., https://plateformecholera.info/index.php/country-monitoring/nigeria, (Accessed January 2020). Google Scholar

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Nigeria Demographics., https://www.worldometers.info/demographics/nigeria-demographics/#life-exp, (Accessed January 2020). Google Scholar

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M. Pascual, M. Bouma, A. King and E. L. Ionides, Inapparent infections and cholera dynamics, Nature, 454 (2008). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Wang and C. Modnak, Modeling cholera dynamics with controls, Can. Appl. Math. Q., 19 (2011), 255-273.   Google Scholar

[24]

P. J. Witbooi, An SEIRS epidemic model with stochastic transmission, Adv. Difference Equ., 2017 (2017), 16pp. doi: 10.1186/s13662-017-1166-6.  Google Scholar

[25]

P. J. Witbooi, Stability of a stochastic model of an SIR epidemic with vaccination, Acta Biotheoretica, 65 (2017), 151-165.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[29]

Wo rld Health Organization, Global task force on cholera control, Weekly Epidemiological Record. Cholera Articles: WHO, 85 (2010), 293-308.   Google Scholar

[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.  Google Scholar

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). Google Scholar

[2]

City Population, 2020. Accessed from: https://citypopulation.de/php/nigeria-admin.php?adm1id=NGA002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

W. Feller, An Introduction to Probability Theory and its Applications, Volume II. John Wiley and Sons, Inc. New York, 1966.  Google Scholar

[6]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.  Google Scholar

[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.  Google Scholar

[8]

K. HattafM. MahroufJ. 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.  Google Scholar

[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.  Google Scholar

[10]

A. LahrouzL. 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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[13]

Q. LiuD. JiangT. 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.  Google Scholar

[14]

Y. M. MarwaI. S. MbalawataS. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[17]

, Nigeria life expectancy, (2020), 1950-2020, Accessed from: https://www.macrotrends.net/countries/NGA/nigeria/life-expectancy. Google Scholar

[18]

Nigeria - Regional Cholera Platform., https://plateformecholera.info/index.php/country-monitoring/nigeria, (Accessed January 2020). Google Scholar

[19]

Nigeria Demographics., https://www.worldometers.info/demographics/nigeria-demographics/#life-exp, (Accessed January 2020). Google Scholar

[20]

M. Pascual, M. Bouma, A. King and E. L. Ionides, Inapparent infections and cholera dynamics, Nature, 454 (2008). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Wang and C. Modnak, Modeling cholera dynamics with controls, Can. Appl. Math. Q., 19 (2011), 255-273.   Google Scholar

[24]

P. J. Witbooi, An SEIRS epidemic model with stochastic transmission, Adv. Difference Equ., 2017 (2017), 16pp. doi: 10.1186/s13662-017-1166-6.  Google Scholar

[25]

P. J. Witbooi, Stability of a stochastic model of an SIR epidemic with vaccination, Acta Biotheoretica, 65 (2017), 151-165.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[29]

Wo rld Health Organization, Global task force on cholera control, Weekly Epidemiological Record. Cholera Articles: WHO, 85 (2010), 293-308.   Google Scholar

[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.  Google Scholar

Figure 1.  The classes $ I(t) $ and $ B(t) $ in the deterministic model as calculated from the data over the outbreak period $ [t_0, t_2] $
Figure 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 $
Figure 3.  Evolution of the $ I(t) $ as calculated for the stochastic model from the data over the post-outbreak period $ [t_2, t_2+30] $
Figure 4.  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
Table 1.  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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