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In this paper, we modified the model of [23] and then applied a new semi-analytic technique namely the Homotopy Analysis Method (HAM) in solving the SEIRS Epidemic Mathematical Model. The modified SEIRS model wasfirst formulated and adequately analyzed. We investigated the basic properties of the model by proving the positivity of the solutions and establishing the invariant region. We further obtained the steady states: disease-free equilibrium (DFE) and endemic equilibrium (EE), then we went further to determine the local stability of the DEF and EE using the basic reproduction number which was calculated. We also applied Lyaponuv method to prove the global stability of endemic equilibrium, The HAM was applied to obtain an accurate solution to the model in few iterations. Finally, a numerical solution (simulation) of the model was obtained using MAPLE 15 computation software.

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This paper presents a seven-dimensional ordinary differential equation of mathematical model of zika virus between humans and mosquitoes population with non-linear forces of infection in form of saturated incidence rate. Vertical transmission is introduced into the model. These incidence rates produce antibodies in response to the presence of parasite-causing zika virus in both human and mosquito populations. The existence of region where the model is epidemiologically feasible is established (invariant set) and the positivity of the models is also established. The basic properties of the model are determined including the reproduction number of both cases, R0 and R0 |p=q=0 R respectively. Stability analysis of the disease-free equilibrium is investigated via the threshold parameter (reproduction number R0 |p=q=0) obtained using the next generation matrix technique. The special case model results shown that the disease-free equilibrium is locally asymptotical stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Under specific conditions on the model parameters, the global dynamics of the special case model around the equilibra are explored using Lyapunov functions. For a threshold parameter less than unity, the disease-free equilibrium is globally asymptotically stable. While the endemic equilibrium is shows to be globally asymptotically stable at threshold parameter greater than unity. Numerical simulations are carried out to confirm the analytic results and explore the possible behavior of the formulated model. The result shows that, horizontal and vertical transmission contributes a higher percentage of infected individuals in the population than only horizontal transmission.