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Stability Analysis and Approximate Solution of SIR Epidemic Model with Crowley-Martin type Functional Response and Holling type-II Treatment Rate by Using Homotopy Analysis Method
Jian Zu,Parvaiz Ahmad Naik,Mohammad Ghoreishi
Keywords:Modified SIR epidemic model, Homotopy analysis method, Stability analysis, Reproduction number , Crowley-Martin type incidence rate, Holling type-II treatment rate, Auxiliary parameter .
      In this paper, the susceptible-infected-recovered model with Crowley-Martin type functional response and Holling type-II treatment rate is investigated as a nonlinear system of ordinary differential equations. The Crowley-Martin type functional response is taken to interpret the psychological or inhibitory effect on the population and Holling type-II treatment rate is considered to incorporate the limitation in treatment availability to infective individuals. This novel combination of the Crowley-Martin incidence rate and Holling type-II treatment rate is applied here into a susceptible-infected-recovered epidemic model to incorporate these important aspects. The mathematical analysis shows that the model has two equilibrium points, namely disease-free and endemic. We investigate the existence of equilibria and prove the local as well as global asymptotical stable results of the disease-free and endemic equilibrium by using LaSalle’s invariant principle and Lyapunov function. A threshold value has been found to ensure the extinction or persistence of the infection. Furthermore, homotopy analysis method is employed to obtain the series solution of the proposed model as it is a flexible method which contains the auxiliary parameters and functions. We apply this technique for a strongly nonlinear system as it utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter and allows to obtain a one-parametric family of explicit series solutions. By using the homotopy solutions, firstly, several -curves are plotted to demonstrate the regions of convergence, then the residual and square residual errors are obtained for different values of these regions. Secondly, the numerical solutions are presented for various iterations and the absolute error functions are applied to show the accuracy of the applied homotopy analysis method. The results obtained show the effectiveness and strength of the homotopy analysis method. Some numerical simulations are given to illustrate the analytical results.