Application of the Homotopy perturbation method to solve influenza A (H1N1) epidemic model with vaccination

The current research, intends to offer a new approximate method which is based on the Homotopy perturbation method for solving the nonlinear mathematical model corresponding to the spread of influenza A (H1N1) problem. Supposedly, the susceptibles become infected by direct contact with infectives. The present method(HPM) will be shown that as vaccination efficacy increases, the infective population decreases.


Introduction
Influenza is an illness caused by a virus that affects mainly the nose, throat, bronchi and, occasionally, lungs infection usually lasts for about a week, and is characterized by fever, myalgia, headache, pharyngitis, cough and prostration.Influenza is highly contagious and is easily transmitted through contact with droplets from the nose and throat of an infected person who is coughing and sneezing.The disease infects the nose, throat or lungs.It often breaks out as an epidemic which quickly spreads from town to town and country to country.In order to consider the effect of influenza A, we will consider an influenza A model with vaccination [1].There are numerous numerical methods which have been focusing on the solution of the above problem.In this paper, we want to employ the homotopy perturbation technique to evaluate the approximate solution of model.The homotopy perturbation technique does not depend upon a small parameter in the equation.By the homotopy technique in topology, a homotopy is constructed with an imbedding parameter p ∈ [0, 1]; which is considered as a "small parameter".

THE MODEL
Let S(t) be the number of population members who are susceptible to an infection at time t, V (t) be the number of members who are vaccinated at time t, E(t) be the number of members who are latened at time t, and I(t) be the number of members who are infective at time t.Hence, the total population at the time t, N(t) is now given by N(t) = S(t)+E(t)+I(t)+V (t) Considering the above notations, the model for the transmission dynamics of influenza A (H1N1) in a population is given by the following system of nonlinear differential equations: with the initial conditions: where A is birth or immigrant rate, β is infection rate, η is modification parameter, ϕ is vaccination rate of susceptible individuals, µ is natural death rate, σ is factor by which the vaccine reduces infection d is disease induced death rate, k 1 is transfer rates between the exposed and the infectious and δ is rate of recovery from the disease.

MAIN IDEA
A considerable amount of research work has been invested recently in applying the homotopy perturbation method to a wide class of linear and nonlinear mathematical problems [2].The zeroth approximations S 0 , E 0 ,V 0 , I 0 can be any selective functions.However, the initial conditions are preferably used to select these approximations S 0 (t),V 0 (t), E 0 (t), I 0 (t) as will be seen later.After integrating the both sides of the above equations in the resulting system from 0 to s, replacing s with t, leads to the following relations: By the homotopy technique given by [3] we construct a homotopy for (2.3) which satisfies: and p ∈ [0, 1] is an embedding parameter.We assume that the solution of (2.3) can be expressed in a series of p as follows: as n → ∞ and p → 1 then the series is approach to the exact solution.Substituting equation (2.8) into (2.4),we get Where Equating coefficients of like powers of p in (2.9) yields p 0 : S 0 (t) = S(0) − At,V 0 (t) = V (0), E 0 (t) = E(0), I 0 (t) = I(0) Using the above equations, we assume that the approximate solution of (2.3) is given by

Main results
• As vaccination efficacy σ increases, the infective population decreases.This signifes that only by increasing the vaccination efficacy , spread of infectious disease cannot be signifcantly controlled.
• As infection rate β decreases, the infective population decreases.This signifes that the spread of infectious disease can be controlled by decreasing the infection rate.We can decrease the infection rate through vaccination.