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

• 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.

Mathematics Subject Classification: 92D25, 93D20.

 Citation: • • Figure 1.  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

Figure 2.  The behavior of infected individuals $I_h$ when $\mathcal{R}_{0}>1$. Figure 2(a): $\alpha = m = 0$, Figure 2(b): $\alpha = m = 0.001$

Figure 3.  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

Figure 4.  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

Figure 5.  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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