JAMP  Vol.7 No.5 , May 2019
Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation
In this paper, we describe the two different stochastic differential equations representing cholera dynamics. The first stochastic differential equation is formulated by introducing the stochasticity to deterministic model by parametric perturbation technique which is a standard technique in stochastic modeling and the second stochastic differential equation is formulated using transition probabilities. We analyse a stochastic model using suitable Lyapunov function and Itô formula. We state and prove the conditions for global existence, uniqueness of positive solutions, stochastic boundedness, global stability in probability, moment exponential stability, and almost sure convergence. We also carry out numerical simulation using Euler-Maruyama scheme to simulate the sample paths of stochastic differential equations. Our results show that the sample paths are continuous but not differentiable (a property of Wiener process). Also, we compare the numerical simulation results for deterministic and stochastic models. We find that the sample path of SIsIaR-B stochastic differential equations model fluctuates within the solution of the SIsIaR-B ordinary differential equation model. Furthermore, we use extended Kalman filter to estimate the model compartments (states), we find that the state estimates fit the measurements. Maximum likelihood estimation method for estimating the model parameters is also discussed.
Cite this paper: Marwa, Y. , Mbalawata, I. , Mwalili, S. and Charles, W. (2019) Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation. Journal of Applied Mathematics and Physics, 7, 1097-1125. doi: 10.4236/jamp.2019.75074.

[1]   World Health Organization (2016) World Health Statistics. Monitoring Health for the SDGs Sustainable Development Goals.

[2]   Roberts, M., Andreasen, V., Lloyd, A. and Pellis, L. (2015) Nine Challenges for Deterministic Epidemic Models. Epidemics, 10, 49-53.

[3]   Edward, S. and Nyerere, N. (2015) A Mathematical Model for the Dynamics of Cholera with Control Measures. Applied and Computational Mathematics, 4, 53-63.

[4]   King, A.A., Ionides, E.L., Pascual, M. and Bouma, M.J. (2008) Inapparent Infections and Cholera Dynamics. Nature, 454, 877-880.

[5]   Koelle, K., Rod, X., Pascual, M., Yunus, M. and Mostafa, G. (2005) Refractory Periods and Climate Forcing in Cholera Dynamics. Nature, 436, 696-700.

[6]   Obeng-Denteh, W., Andam, E.A., Obiri-Apraku, L. and Agyeil, W. (2015) Modeling Cholera Dynamics with a Control Strategy in Ghana. British Journal of Research, 2, 30-41.

[7]   Marwa, Y.M., Mwalili, S. and Mbalawata, I.S. (2018) Markov Chain Monte Carlo Analysis of Cholera Epidemic. Journal of Mathematical and Computational Science, 8, 584-610.

[8]   Dimi, J.L., Bissila, J.F. and Mbaya, T. (2016) Some Stochastic Properties of Cholera Model. Nonlinear Analysis and Differential Equations, 4, 779-792.

[9]   Gani, J. and Swift, R.J. (2009) Deterministic and Stochastic Models for the Spread of Cholera. The ANZIAM Journal, 51, 234-240.

[10]   Sandro, A., Maritan, A., Bertuzzo, E., Rodriguez-Iturbe, I. and Rinaldo, A. (2010) Stochastic Dynamics of Cholera Epidemics. Physical Review E, 81, Article ID: 051901.

[11]   Wang, X. and Wang, J. (2017) Modeling the Within-Host Dynamics of Cholera: Bacterial Viral Interaction. Journal of Biological Dynamics, 11, 484-501.

[12]   Cohn, J.N., Johnson, G., Ziesche, S., Cobb, F., Francis, G., Tristani, F. and Bhat, G. (1991) A Comparison of Enalapril with Hydralazine-Isosorbide Dinitrate in the Treatment of Chronic Congestive Heart Failure. New England Journal of Medicine, 325, 303-310.

[13]   Van den Driessche, P. and Watmough, J. (2002) Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48.

[14]   Oksendal, B. (2003) Stochastic Differential Equations. Springer, Berlin, 65-84.

[15]   Rezaeyan, R. and Farnoosh, R. (2010) Stochastic Differential Equations and Application of the Kalman-Bucy Filter in the Modeling of RC Circuit. Applied Mathematical Sciences, 4, 1119-1127.

[16]   Allen, E. (2007) Modeling with Ito Stochastic Differential Equations. Springer Science, Business Media, Berlin, 22.

[17]   Allen, L.J. (2008) An Introduction to Stochastic Epidemic Models. In: Brauer, F., van den Driessche, P. and Wu, J., Eds., Mathematical Epidemiology, Springer, Berlin, 81-130.

[18]   Borodin, A.N. (2017) Stochastic Processes. Springer, Berlin.

[19]   Bahar, A. and Mao, X. (2004) Stochastic Delay Lotka-Volterra Model. Journal of Mathematical Analysis and Applications, 292, 364-380.

[20]   Shaikhet, L. (1996) Stability of Stochastic Hereditary Systems with Markov Switching. Theory of Stochastic Processes, 2, 180-184.

[21]   Burrage, K., Burrage, P.M. and Tian, T. (2004) Numerical Methods for Strong Solutions of Stochastic Differential Equations: An Overview. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460, 373-402.

[22]   Mbalawata, I.S. (2014) Adaptive Markov Chain Monte Carlo and Bayesian Filtering for State Space Models. Doctoral Dissertation, Acta Universitatis Lappeenrantaensis.

[23]   Akman, O., Corby, M.R. and Schaefer, E. (2016) Examination of Models for Cholera: Insights into Model Comparison Methods. Letters in Biomathematics, 3, 93-118.

[24]   Khan, M.A., Ali, A., Dennis, L.C. and Gui, T. (2015) Dynamical Behavior of Cholera Epidemic Model with Non-Linear Incidence Rate. Applied Mathematical Sciences, 9, 989-1002.

[25]   Neilan, R.L., Schaefer, E., Gaff, H., Fister, K.R. and Lenhart, S. (2010) Modeling Optimal Intervention Strategies for Cholera. Bulletin of Mathematical Biology, 72, 2004-2018.

[26]   Njagarah, J.B. and Nyabadza, F. (2015) Modelling Optimal Control of Cholera in Communities Linked by Migration. Computational and Mathematical Methods in Medicine, 2015, Article ID: 898264.