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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 957-974. doi: 10.3934/dcdss.2020056
References:
[1]

E. AhmedA. 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.

[2]

M. AlquranK. Al-KhaledM. 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.

[3]

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. 

[4]

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.

[5]

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.

[6]

R. P. BeasleyC. C. LinK. Y.WangF. J. HsiehL. Y. HwangC. E. StevensT. S. Sun and W. Szmuness, Hepatocellular carcinoma and hepatitis B virus, distributions, The Lancet, 2 (1981), 1129-1133. 

[7]

World Health Organization Media Centre. Available: , 2017. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/. Accessed 2018.

[8]

M. Caputo and M. Fabrizio, A new definition of fractional derivative with-out singular kernel, Progr. Fract. Differ. Appl., 85 (2015), 73-85. 

[9]

F. F. F. ChenarY. 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.

[10]

D. CopotR. De KeyserE. DeromM. 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.

[11]

H. DelavariD. 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.

[12]

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.

[13]

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.

[14]

M. A. KhanS. IslamM. Arif and Z. Haq, Transmission model of hepatitis B virus with the migration effect, Bio. Res. Int., 2013 (2013), 1-10. 

[15]

M. A. KhanS. 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.

[16]

Y. LiY. 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.

[17]

R. M. LizzyK. 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.

[18]

J. PangJ. 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.

[19]

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. 

[20] I. Podlubny, Fractional Differential Equations, Academic Press, 1999. 
[21]

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.

[22]

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.

[23]

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.

[24]

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.

[25]

J. SinghD. KumarM. 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. 

[26]

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.

[27]

S. ThornleyC. 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.

[28]

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.

[29]

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.

[30]

L. ZouW. 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.

show all references

References:
[1]

E. AhmedA. 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.

[2]

M. AlquranK. Al-KhaledM. 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.

[3]

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. 

[4]

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.

[5]

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.

[6]

R. P. BeasleyC. C. LinK. Y.WangF. J. HsiehL. Y. HwangC. E. StevensT. S. Sun and W. Szmuness, Hepatocellular carcinoma and hepatitis B virus, distributions, The Lancet, 2 (1981), 1129-1133. 

[7]

World Health Organization Media Centre. Available: , 2017. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/. Accessed 2018.

[8]

M. Caputo and M. Fabrizio, A new definition of fractional derivative with-out singular kernel, Progr. Fract. Differ. Appl., 85 (2015), 73-85. 

[9]

F. F. F. ChenarY. 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.

[10]

D. CopotR. De KeyserE. DeromM. 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.

[11]

H. DelavariD. 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.

[12]

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.

[13]

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.

[14]

M. A. KhanS. IslamM. Arif and Z. Haq, Transmission model of hepatitis B virus with the migration effect, Bio. Res. Int., 2013 (2013), 1-10. 

[15]

M. A. KhanS. 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.

[16]

Y. LiY. 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.

[17]

R. M. LizzyK. 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.

[18]

J. PangJ. 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.

[19]

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. 

[20] I. Podlubny, Fractional Differential Equations, Academic Press, 1999. 
[21]

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.

[22]

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.

[23]

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.

[24]

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.

[25]

J. SinghD. KumarM. 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. 

[26]

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.

[27]

S. ThornleyC. 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.

[28]

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.

[29]

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.

[30]

L. ZouW. 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.

Figure 1.  The effect of $ \delta $ and $ h_1 $ on $ \mathcal{R}_0 $
Figure 2.  Contour plot of $ \delta $ and $ h_1 $
Figure 3.  The effect of $ \delta $ and $ h_2 $ on $ \mathcal{R}_0 $
Figure 4.  Contour plot of $ \delta $ and $ h_2 $
Figure 5.  The effect of $ \mu $ and $ h_2 $ on $ \mathcal{R}_0 $
Figure 6.  Contour plot of $ \mu $ and $ h_2 $
Figure 7.  The effect of $ \mu $ and $ h_1 $ on $ \mathcal{R}_0 $
Figure 8.  Contour plot of $ \mu $ and $ h_1 $
Figure 9.  The plot shows the susceptible individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
Figure 10.  The plot shows the exposed individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
Figure 11.  The plot shows the acute individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
Figure 12.  The plot shows the carrier individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
Figure 13.  The plot shows the hospitalized individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
Figure 14.  The plot shows the recovered individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$.
Figure 15.  The plot shows the total number of infected individuals when $ \mathcal{R}_0 = 1.0248>1 $ for different values of $ \alpha $
Figure 16.  The plot shows the total number of infected individuals when $ \mathcal{R}_0 = 1.0248>1 $ for different values of $ \alpha $ and $ h_1 $
Figure 17.  The plot shows the total number of infected individuals when $\mathcal{R}_0 = 1.0248>1$ for different values of $\alpha$ and $h_2$.
Table 1.  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
Table 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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