    March  2020, 13(3): 957-974. doi: 10.3934/dcdss.2020056

## 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, 2020, 13 (3) : 957-974. doi: 10.3934/dcdss.2020056
##### References:
  E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R ossler, Chua and Chen systems, Phy. Letters A, 385 (2006), 1-4.  doi: 10.1016/j.physleta.2006.04.087.  Google Scholar  M. Alquran, K. Al-Khaled, M. Ali and O. A. Arqub, Bifurcations of the time-fractional generalized coupled Hirota-Satsuma KdV system, Waves Wavelets and Fractals, 3 (2017), 31-39.  doi: 10.1515/wwfaa-2017-0003. Google Scholar  A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 89 (2016), 763-769.   Google Scholar  A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar  A. Atangana and J. F. Gmez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus., 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar  R. P. Beasley, C. C. Lin, K. Y.Wang, F. J. Hsieh, L. Y. Hwang, C. E. Stevens, T. S. Sun and W. Szmuness, Hepatocellular carcinoma and hepatitis B virus, distributions, The Lancet, 2 (1981), 1129-1133.   Google Scholar  World Health Organization Media Centre. Available: , 2017. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/. Accessed 2018. Google Scholar  M. Caputo and M. Fabrizio, A new definition of fractional derivative with-out singular kernel, Progr. Fract. Differ. Appl., 85 (2015), 73-85.   Google Scholar  F. F. F. Chenar, Y. N. Kyrychko and K. B. Blyuss, Mathematical model of immune response to hepatitis B, Jour. of Theo. Bio., 447 (2018), 98-110.  doi: 10.1016/j.jtbi.2018.03.025.  Google Scholar  D. Copot, R. De Keyser, E. Derom, M. Ortigueira and C. M. Ionescu, Reducing bias in fractional order impedance estimation for lung function evaluation, Biomed. Signal Proc. and Cont., 39 (2018), 74-80.  doi: 10.1016/j.bspc.2017.07.009. Google Scholar  H. Delavari, D. Baleanu and J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dyn., 67 (2012), 2433-2439.  doi: 10.1007/s11071-011-0157-5.  Google Scholar  K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Non linear Dyn., 71 (2013), 613-619.  doi: 10.1007/s11071-012-0475-2.  Google Scholar  P. V. D. Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar  M. A. Khan, S. Islam, M. Arif and Z. Haq, Transmission model of hepatitis B virus with the migration effect, Bio. Res. Int., 2013 (2013), 1-10.   Google Scholar  M. A. Khan, S. Islam and G. Zaman, Media coverage campaign in Hepatitis B transmission, App.math. and comp., 331 (2018), 378-393.  doi: 10.1016/j.amc.2018.03.029.  Google Scholar  Y. Li, Y. Q. Chen and I. Podlubny, Mittag Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.  Google Scholar  R. M. Lizzy, K. Balachandran and J. J. Trujillo, Controllability of nonlinear stochastic fractional neutral systems with multiple time varying delays in control, Chao. Soliton. Fract., 102 (2017), 162-167.  doi: 10.1016/j.chaos.2017.04.024.  Google Scholar  J. Pang, J. Cui and X. Zhou, Dynamical behavior of a hepatitis B virus transmission model with vaccination, Nat. Med., 265 (2010), 572-578.  doi: 10.1016/j.jtbi.2010.05.038.  Google Scholar  C. M. A. Pinto and A. R. M. Carvalho, Pinto CMA, Carvalho ARM. The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains, Ecol. Complex, 32 (2017), 1-20.   Google Scholar  I. Podlubny, Fractional Differential Equations, Academic Press, 1999.   Google Scholar  S. Sakulrang, E. J. Moore, S. Sungnul and A. Gaetano, A fractional differential equation model for continuous glucose monitoring data, Adv. Diff. Equ., 2017 (2017), Paper No. 150, 11 pp. doi: 10.1186/s13662-017-1207-1.  Google Scholar  S. M. Salman and A. M. Yousef, On a fractional-order model for HBV infection with cure of infected cells, Jour. Egyp. Math. Soci., 25 (2017), 445-451.  doi: 10.1016/j.joems.2017.06.003.  Google Scholar  S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, London: Gordon and Breach Science Publishers, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar  C. W. Shepard and E. P. Simard, Hepatitis B virus infection: epidemiology and vaccination, Epid. Rev., 28 (2006), 112-125.  doi: 10.1093/epirev/mxj009. Google Scholar  J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A novel numerical approach for a nonlinear fractional dynamical model of interpersonal and romantic relationships, Entropy, 19 (2017), 375-392.   Google Scholar  J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Diff. Equ., 2017 (2017), Paper No. 88, 16 pp. doi: 10.1186/s13662-017-1139-9.  Google Scholar  S. Thornley, C. Bullen and M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy, Nat. Med., 254 (2008), 599-603.  doi: 10.1016/j.jtbi.2008.06.022.  Google Scholar  C. Vargas-De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simulat., 24 (2015), 75-85.  doi: 10.1016/j.cnsns.2014.12.013.  Google Scholar  S. Zhang and Y. Zhou, The analysis and application of an HBV model, Appl. Math. Modell., 36 (2012), 1302-1312.  doi: 10.1016/j.apm.2011.07.087. Google Scholar  L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis B, Nat. Med., 262256 (2010), 330-338.  doi: 10.1016/j.jtbi.2009.09.035.  Google Scholar

show all references

##### References:
  E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R ossler, Chua and Chen systems, Phy. Letters A, 385 (2006), 1-4.  doi: 10.1016/j.physleta.2006.04.087.  Google Scholar  M. Alquran, K. Al-Khaled, M. Ali and O. A. Arqub, Bifurcations of the time-fractional generalized coupled Hirota-Satsuma KdV system, Waves Wavelets and Fractals, 3 (2017), 31-39.  doi: 10.1515/wwfaa-2017-0003. Google Scholar  A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 89 (2016), 763-769.   Google Scholar  A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar  A. Atangana and J. F. Gmez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus., 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar  R. P. Beasley, C. C. Lin, K. Y.Wang, F. J. Hsieh, L. Y. Hwang, C. E. Stevens, T. S. Sun and W. Szmuness, Hepatocellular carcinoma and hepatitis B virus, distributions, The Lancet, 2 (1981), 1129-1133.   Google Scholar  World Health Organization Media Centre. Available: , 2017. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/. Accessed 2018. Google Scholar  M. Caputo and M. Fabrizio, A new definition of fractional derivative with-out singular kernel, Progr. Fract. Differ. Appl., 85 (2015), 73-85.   Google Scholar  F. F. F. Chenar, Y. N. Kyrychko and K. B. Blyuss, Mathematical model of immune response to hepatitis B, Jour. of Theo. Bio., 447 (2018), 98-110.  doi: 10.1016/j.jtbi.2018.03.025.  Google Scholar  D. Copot, R. De Keyser, E. Derom, M. Ortigueira and C. M. Ionescu, Reducing bias in fractional order impedance estimation for lung function evaluation, Biomed. Signal Proc. and Cont., 39 (2018), 74-80.  doi: 10.1016/j.bspc.2017.07.009. Google Scholar  H. Delavari, D. Baleanu and J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dyn., 67 (2012), 2433-2439.  doi: 10.1007/s11071-011-0157-5.  Google Scholar  K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Non linear Dyn., 71 (2013), 613-619.  doi: 10.1007/s11071-012-0475-2.  Google Scholar  P. V. D. Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar  M. A. Khan, S. Islam, M. Arif and Z. Haq, Transmission model of hepatitis B virus with the migration effect, Bio. Res. Int., 2013 (2013), 1-10.   Google Scholar  M. A. Khan, S. Islam and G. Zaman, Media coverage campaign in Hepatitis B transmission, App.math. and comp., 331 (2018), 378-393.  doi: 10.1016/j.amc.2018.03.029.  Google Scholar  Y. Li, Y. Q. Chen and I. Podlubny, Mittag Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.  Google Scholar  R. M. Lizzy, K. Balachandran and J. J. Trujillo, Controllability of nonlinear stochastic fractional neutral systems with multiple time varying delays in control, Chao. Soliton. Fract., 102 (2017), 162-167.  doi: 10.1016/j.chaos.2017.04.024.  Google Scholar  J. Pang, J. Cui and X. Zhou, Dynamical behavior of a hepatitis B virus transmission model with vaccination, Nat. Med., 265 (2010), 572-578.  doi: 10.1016/j.jtbi.2010.05.038.  Google Scholar  C. M. A. Pinto and A. R. M. Carvalho, Pinto CMA, Carvalho ARM. The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains, Ecol. Complex, 32 (2017), 1-20.   Google Scholar  I. Podlubny, Fractional Differential Equations, Academic Press, 1999.   Google Scholar  S. Sakulrang, E. J. Moore, S. Sungnul and A. Gaetano, A fractional differential equation model for continuous glucose monitoring data, Adv. Diff. Equ., 2017 (2017), Paper No. 150, 11 pp. doi: 10.1186/s13662-017-1207-1.  Google Scholar  S. M. Salman and A. M. Yousef, On a fractional-order model for HBV infection with cure of infected cells, Jour. Egyp. Math. Soci., 25 (2017), 445-451.  doi: 10.1016/j.joems.2017.06.003.  Google Scholar  S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, London: Gordon and Breach Science Publishers, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar  C. W. Shepard and E. P. Simard, Hepatitis B virus infection: epidemiology and vaccination, Epid. Rev., 28 (2006), 112-125.  doi: 10.1093/epirev/mxj009. Google Scholar  J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A novel numerical approach for a nonlinear fractional dynamical model of interpersonal and romantic relationships, Entropy, 19 (2017), 375-392.   Google Scholar  J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Diff. Equ., 2017 (2017), Paper No. 88, 16 pp. doi: 10.1186/s13662-017-1139-9.  Google Scholar  S. Thornley, C. Bullen and M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy, Nat. Med., 254 (2008), 599-603.  doi: 10.1016/j.jtbi.2008.06.022.  Google Scholar  C. Vargas-De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simulat., 24 (2015), 75-85.  doi: 10.1016/j.cnsns.2014.12.013.  Google Scholar  S. Zhang and Y. Zhou, The analysis and application of an HBV model, Appl. Math. Modell., 36 (2012), 1302-1312.  doi: 10.1016/j.apm.2011.07.087. Google Scholar  L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis B, Nat. Med., 262256 (2010), 330-338.  doi: 10.1016/j.jtbi.2009.09.035.  Google Scholar 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
  Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432  Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269  A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441  Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466  Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457  Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219  Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118  Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275  Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319  Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276  Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251  Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453  Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450  Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346  Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446  Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078  Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449  Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348  Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440  Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 1.233