TITLE:
Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation
AUTHORS:
Yohana Maiga Marwa, Isambi Sailon Mbalawata, Samuel Mwalili, Wilson Mahera Charles
KEYWORDS:
Stochastic Differential Equations, Stability Condition, Extended Kalman Filter, Itô Formula, Lyapunov Function, Euler-Maruyama Scheme
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.7 No.5,
May
23,
2019
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.