# American Institute of Mathematical Sciences

## A fractional order HBV model with hospitalization

 1 Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, 25120, Pakistan 2 Department of Mathematics, City University of Science and Information Technology, Peshawar, Khyber Pakhtunkhwa, 25000, Pakistan 3 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Khyber Pakhtunkhwa, Pakistan

* Corresponding author: altafdir@gmail.com

Received  April 2018 Revised  June 2018 Published  March 2019

Hepatitis B is a viral infection that can cause both acute and chronic disease and mainly attacks the liver. The present paper describes the dynamics of HBV with hospitalization. Due to the fatal nature of this disease, it is necessary to formulate a new mathematical model in order to reduce the burden of HBV. Therefore, we formulate a new HBV model with fractional order derivative. The fractional order model is formulated in Caputo sense. Two equilibria for the model exist: the disease-free and the endemic equilibriums. It is shown, that the disease-free equilibrium is both locally and globally asymptotically stable if $\mathcal{R}_0<1$ for any $\alpha\in(0,1)$. The sensitivity analysis of the model parameters are calculated and their results are depicted. The numerical results for the stability of the endemic equilibrium are presented. The complex dynamics of the disease can be best described by using the fractional derivative and this is illustrated through many graphical results.

Citation: Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Taza Gul, Fawad Hussain. A fractional order HBV model with hospitalization. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020056
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The effect of $\delta$ and $h_1$ on $\mathcal{R}_0$
Contour plot of $\delta$ and $h_1$
The effect of $\delta$ and $h_2$ on $\mathcal{R}_0$
Contour plot of $\delta$ and $h_2$
The effect of $\mu$ and $h_2$ on $\mathcal{R}_0$
Contour plot of $\mu$ and $h_2$
The effect of $\mu$ and $h_1$ on $\mathcal{R}_0$
Contour plot of $\mu$ and $h_1$
The plot shows the susceptible individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
The plot shows the exposed individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
The plot shows the acute individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
The plot shows the carrier individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
The plot shows the hospitalized individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
The plot shows the recovered individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
The plot shows the total number of infected individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$
The plot shows the total number of infected individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$ and $h_1$
The plot shows the total number of infected individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$ and $h_2$.
Values of parameters used for numerical simulations
 parameters description of parameter Values $b$ Birth rate 0.4 $d$ Natural death rate 0.01 $h_1$ The acute individuals to be hospitalized 0.01 $h_2$ Flow rate from carrier class to the hospitalized class 0.01 $\beta$ The transmission coefficient 0.0002 $\delta$ Rate of flow from exposed to carrier 0.01 $d_A$ mortality rate due to acute infection 0.001 $d_C$ carrier individuals death rate 0.002 $\gamma$ the rate by which acute individuals move to carries class 0.01 $\xi$ The rate of recovery 0.02 $\psi$ Un-immunized children born to carrier mothers 0.2 $\mu$ Carriers infectiousness related to acute infection 0.2
 parameters description of parameter Values $b$ Birth rate 0.4 $d$ Natural death rate 0.01 $h_1$ The acute individuals to be hospitalized 0.01 $h_2$ Flow rate from carrier class to the hospitalized class 0.01 $\beta$ The transmission coefficient 0.0002 $\delta$ Rate of flow from exposed to carrier 0.01 $d_A$ mortality rate due to acute infection 0.001 $d_C$ carrier individuals death rate 0.002 $\gamma$ the rate by which acute individuals move to carries class 0.01 $\xi$ The rate of recovery 0.02 $\psi$ Un-immunized children born to carrier mothers 0.2 $\mu$ Carriers infectiousness related to acute infection 0.2
Sensitivity indices of $\mathcal{R}_0$ with respect to the model parameters
 Parameter Sensitivity index $\beta$ +0.7165 $\delta$ +0.5000 $\gamma$ +0.3680 $h_1$ -0.0455 $h_2$ -0.3739 $\mu$ +0.0161 $\psi$ +0.8064 $d_A$ -0.0454 $d_C$ -0.0748
 Parameter Sensitivity index $\beta$ +0.7165 $\delta$ +0.5000 $\gamma$ +0.3680 $h_1$ -0.0455 $h_2$ -0.3739 $\mu$ +0.0161 $\psi$ +0.8064 $d_A$ -0.0454 $d_C$ -0.0748
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