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 JAMP  Vol.7 No.5 , May 2019
Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation
Abstract:
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.
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