| Citation: |
Parvaiz Ahmad Naik, Jian Zu, Mohammad Ghoreishi. 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[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1482-1515. doi: 10.11948/20190239
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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
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Abstract
In this paper, SIR epidemic model with Crowley-Martin type functional response and Holling type-Ⅱ treatment rate is investigated. The analysis of the model shows that it has two equilibria, namely disease-free and endemic. We investigate the existence and stability results of equilibria by using LaSalle's invariant principle and Lyapunov function. $\mathfrak{R}_{0}$ 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. By using the homotopy solutions, firstly, several $\hbar$-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. -
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References
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Parvaiz Ahmad Naik, Jian Zu, Mohammad Ghoreishi. 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[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1482-1515. doi: 10.11948/20190239
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- Figure 1. Schematic diagram of the modified SIR epidemic model
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Figure 2. Samples of
$ \hbar $ -curves for$ x(t) $ ,$ y(t) $ and$ z(t) $ under$ 8^{th} $ -order approximation for$ t = 0.5 $ -
Figure 3. The residual error function for
$ x(t) $ ,$ y(t) $ and$ z(t) $ at$ t = 0.5 $ under$ 8^{th} $ -order approximation versus auxiliary parameter$ \hbar $ -
Figure 4. The square residual error function for
$ x(t) $ ,$ y(t) $ and$ z(t) $ at$ t = 0.5 $ under$ 8^{th} $ -order approximation versus auxiliary parameter$ \hbar $ -
Figure 5. The absolute error function for
$ x(t) $ ,$ y(t) $ and$ z(t) $ at$ t = 0.5 $ under$ 8^{th} $ -order approximation