# American Institute of Mathematical Sciences

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June  2019, 12(3): 455-474. doi: 10.3934/dcdss.2019030

## Modeling the transmission dynamics of avian influenza with saturation and psychological effect

 1 Department of Mathematics, City University of Science and Information, Technology, Peshawar, KP, 25000, Pakistan 2 Department of Mathematics, Abdul Wali Khan, University Mardan, KP, 23200, Pakistan 3 Department of Information Technology Education, University of Education, Winneba (Kumasi campus), Ghana

* Corresponding author: altafdir@gmail.com, makhan@cusit.edu.pk

Received  July 2017 Revised  November 2017 Published  September 2018

The present paper describes the mathematical analysis of an avian influenza model with saturation and psychological effect. The virus of avian influenza is not only a risk for birds but the population of human is also not safe from this. We proposed two models, one for birds and the other one for human. We consider saturated incidence rate and psychological effect in the model. The stability results for each model that is birds and human is investigated. The local and global dynamics for the disease free case of each model is proven when the basic reproduction number $\mathcal{R}_{0b}<1$ and $\mathcal{R}_0<1$. Further, the local and global stability of each model is investigated in the case when $\mathcal{R}_{0b}>1$ and $\mathcal{R}_0>1$. The mathematical results show that the considered saturation effect in population of birds and psychological effect in population of human does not effect the stability of equilibria, if the disease is prevalent then it can affect the number of infected humans. Numerical results are carried out in order to validate the theoretical results. Some numerical results for the proposed parameters are presented which can reduce the number of infective in the population of humans.

Citation: Muhammad Altaf Khan, Muhammad Farhan, Saeed Islam, Ebenezer Bonyah. Modeling the transmission dynamics of avian influenza with saturation and psychological effect. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 455-474. doi: 10.3934/dcdss.2019030
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##### References:
The behavior of infected individuals $I_h$, keeping $\alpha = m = 0.001$. Figure 1(a): varying $\beta_a$ and $\beta_h = 8\times 10^{-7}$ is fixed. Figure 1(b): varying $\beta_h$ and $\beta_a = 3\times 10^{-6}$ is fixed
The behavior of infected individuals $I_h$ when $\mathcal{R}_{0}>1$. Figure 2(a): $\alpha = m = 0$, Figure 2(b): $\alpha = m = 0.001$
The behavior of infected individuals $I_h$ and $\mathcal{R}_{0}>1$: Figure 3(a) when $\alpha = 0.001,~0.001,~0.01$ and $m = 0.001$ fixed. Figure 3(b) when $m = 0.001,~0.001,~0.01$ and $\alpha = 0.001$ fixed
The behavior of infected individuals $I_h$ and $\mathcal{R}_{0}<1$: Figure 4(a) when $\alpha = 0.001,~0.001,~0.01$ and $m = 0.001$ fixed. Figure 3(b) when $m = 0.001,~0.001,~0.01$ and $\alpha = 0.001$ fixed
The behavior of infected individuals $I_h$ and $\mathcal{R}_{0}<1$: $\alpha = 0.001,~0.01,~0.1$, $m = 0.001,~0.01,~0.1$
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