A study on numerical solutions of epidemic models

This thesis deal with three different types of deterministic epidemic models that explore the transmission dynamics of influenza. Models are represented by ordinary differential equations and reaction-diffusion equations. First two models are based on basic Susceptible-Exposed-Infectious-Recovered (SEIR) model. Third model represents an attempt on modeling the effects of vaccination on the SEIR epidemic model. First the SEIR type model is discussed. The SEIR model equations with and without diffusion have been solved numerically using different initial conditions. The effect of diffusion on the influenza transmission is investigated. The effects of some intervention strategies have also been investigated. It is shown that diffusion and initial population distribution play crucial role for disease transmission. Numerical analysis is provided to explain the multiple steady states and their bifurcation. In order to represent the transmission dynamics of influenza, SEIRS model is proposed. Sustained and damped oscillations are obtained for different type of disease transmission rate. To help controlling the influenza transmission by vaccination, vaccinated type (SV EIRS) epidemic model is proposed. First the model of SV EIRS is discussed to describe the behavior of an epidemic disease when a vaccination policy is in effect. The stability analysis of steady states and vaccination-reduced basic reproduction number is investigated. In order to determine the most influential parameters to the initial disease transmission and equilibrium disease prevalence, the sensitivity indices of the basic reproduction number, (ℜvac) and the endemic point of equilibrium, (P∗) to the parameters in the model are calculated. Sensitivity analysis of basic reproduction number is performed with two different approaches: (i) through local derivative on ℜvac, where ℜvac is estimated by specific input parameter values; (ii) sampling-based approaches, where sensitivity indices are calculated considering the effects of uncertainty in the parameter estimation. Results are obtained and compared for random and Latin hypercube sampling approaches with 1000 sample sizes. The SV EIRS model parameters are estimated afresh using the available field data. The SV EIRS model equations are solved numerically using new estimated parameters’ value. To investigate the vaccination policy to the diseases transmission, different vaccination rate, vaccine efficacy and initial vaccine coverage are introduced. The analysis of this model suggests that vaccination program can control the spread of the disease.