# American Institute of Mathematical Sciences

October  2020, 13(10): 2761-2787. doi: 10.3934/dcdss.2020168

## Numerical study of an influenza epidemic dynamical model with diffusion

 Center for Applied Mathematics & Bioinformatics, Department of Mathematics & Natural Sciences, Gulf University for Science & Technology, P.O. Box 7207, Hawally 32093, Kuwait

* Corresponding author: M. Ben-Romdhane

Received  November 2018 Revised  May 2019 Published  October 2020 Early access  December 2019

In this paper, a deterministic model is formulated in the aim of performing a thorough investigation of the transmission dynamics of influenza. The main advantage of our model compared to existing models is that it takes into account the effects of hospitalization as well as the diffusion. The proposed model consisting of a dynamical system of partial differential equations with diffusion terms is numerically solved using fast and accurate numerical techniques for partial differential equations. Furthermore, the basic reproduction number that guarantees the local stability of disease-free steady state without diffusion term is calculated. Various numerical simulation for different values of the model input parameters are finally presented in order to show the effect of the effective contact rate on the steady state of the different population compartments.

Citation: Mudassar Imran, Mohamed Ben-Romdhane, Ali R. Ansari, Helmi Temimi. Numerical study of an influenza epidemic dynamical model with diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2761-2787. doi: 10.3934/dcdss.2020168
##### References:
 [1] F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, BioSystems, 113 (2013), 155-164.  doi: 10.1016/j.biosystems.2013.06.004. [2] M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM Journal of Applied Dynamical Systems, 3 (2004), 503-524.  doi: 10.1137/030600370. [3] J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. H. Wue, A model for influenza with vaccination and antiviral treatment, Journal of Theoretical Biology, 253 (2008), 118-130.  doi: 10.1016/j.jtbi.2008.02.026. [4] F. Brauer, P. van den Driessche and J. H. Wu, Mathematical Epidemiology, Lecture Notes in Mathematics, 1945. Mathematical Biosciences Subseries, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78911-6. [5] C. B. Bridges, S. A. Harper, K. Fukuda, T. M. Uyeki, N. J. Cox and J. A. Singleton, Prevention and control of influenza: Recommendations of the advisory committee on immunization practice (ACIP), Morbid Mortal Weekly Report, 52 (2003), 1-36. [6] K. F. Cheng and P. C. Leung, What happened in China during the 1918 influenza pandemic, International Journal of Infectious Diseases, 11 (2007), 360-364.  doi: 10.1016/j.ijid.2006.07.009. [7] N. S. Chong, J. M. Tchuenche and R. J. Smith., A mathematical model of avian influenza with half-saturated incidence, Theory in Biosciences, 133 (2014), 23-38.  doi: 10.1007/s12064-013-0183-6. [8] G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Transmission dynamics of the great influenza pandemic of 1918 In Geneva, Switzerland: Assessing the effects of hypothetical iInterventions, Journal of Theoretical Biology, 241 (2006), 193-204.  doi: 10.1016/j.jtbi.2005.11.026. [9] C. Cosner, Reaction-diffusion equations and ecological modeling, Tutorials in Mathematical Biosciences. IV, Lecture Notes in Math., Math. Biosci. Subser., Springer, Berlin, 1922 (2008), 77-115.  doi: 10.1007/978-3-540-74331-6_3. [10] P. van den Driessche and J. Watmough., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [11] G. Gonzàlez-Parra, A. J. Arenas, D. F. Aranda and L. Segovia, Modeling the epidemic waves of AH1N1/09 influenza around the world, Spatial and Spatio-temporal Epidemiology, 2 (2011), 219-226. [12] A. B. Gumel, Global dynamics of a two-strain avian influenza model, International Journal of Computer Mathematics, 86 (2009), 85-108.  doi: 10.1080/00207160701769625. [13] D. Hoff, Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations, SIAM J. Numer. Anal., 15 (1978), 1161-1177.  doi: 10.1137/0715077. [14] M. Imran, T. Malik, A. R. Ansari and A. Khan, Mathematical analysis of swine influenza epidemic model with optimal control, Japan Journal of Industrial and Applied Mathematics, 33 (2016), 269-296.  doi: 10.1007/s13160-016-0210-3. [15] M. Imran, M. Usman, M. Dur-e-Ahmad and A. Khan, Transmission dynamics of zika fever: A SEIR based model, Differ. Equ. Dyn. Syst., (2017), 1–24. doi: 10.1007/s12591-017-0374-6. [16] S. Islam and R. Zaman, A computational modeling and simulation of spatial dynamics in biological systems, Applied Mathematical Modelling, 40 (2016), 4524-4542.  doi: 10.1016/j.apm.2015.11.025. [17] A. Khan, M. Waleed and M. Imran, Mathematical analysis of an influenza epidemic model, formulation of different controlling strategies using optimal control and estimation of basic reproduction number, Mathematical and Computer Modelling of Dynamical Systems, 21 (2015), 432-459.  doi: 10.1080/13873954.2015.1016975. [18] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898717839. [19] B. Lina, Chapter 12: History of Influenza Pandemics, Paleomicrobiology: Past Human Infections, Springer-Verlag, 2008. [20] J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [21] M. Nuño, G. Chowell and A.B. Gumel, Assessing transmission control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK, and the Netherlands, Proceedings of the Royal Society Interface, 4 (2007), 505-521. [22] S. G. Ruan and W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [23] P. L. Salceanu, Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov Exponents, Math. Biosci. Eng., 8 (2011), 807-825.  doi: 10.3934/mbe.2011.8.807. [24] M. Samsuzzoha, M. Singh and D. Lucy, Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient, Applied Mathematical Modelling, 35 (2007), 5507-5523.  doi: 10.1016/j.apm.2011.04.029. [25] M. Samsuzzoha, M. Singh and D. Lucy, Numerical study of an influenza epidemic model with diffusion, Applied Mathematics and Computation, 217 (2010), 3461-3479.  doi: 10.1016/j.amc.2010.09.017. [26] M. Samsuzzoha, M. Singh and D. Lucy, Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient, Applied Mathematical Modelling, 35 (2011), 5507-5523.  doi: 10.1016/j.apm.2011.04.029. [27] M. Samsuzzoha, M. Singh and D. Lucy, A numerical study on an influenza epidemic model with vaccination and diffusion, Applied Mathematics and Computation, 219 (2012), 122-141.  doi: 10.1016/j.amc.2012.04.089. [28] M. Derouich and A. Boutayeb, An avian influenza mathematical model, Applied Mathematical Sciences, 2 (2008), 1749-1760. [29] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011. [30] N. I. Stilianakis, A. S. Perelson and F. G. Hayden, Emergence of drug resistance during an influenza epidemic: Insights from a mathematical model, The Journal of Infectious Diseases, 177 (1998), 863-873.  doi: 10.1086/515246. [31] J. K. Taubenberger and D. M. Morens, 1918 influenza: The mother of all pandemics, Emerging Infectious Diseases, 12 (2006), 15-22.  doi: 10.3201/eid1209.05-0979. [32] J. K. Taubenberger, A. H. Reid and T. G. Fanning, The 1918 influenza virus: A killer comes into view, Virology, 274 (2000), 241-245.  doi: 10.1006/viro.2000.0495. [33] S. Toubaei, M. Garshasbi and M. Jalalvand, A numerical treatment of a reaction-diffusion model of spatial pattern in the embryo, Computational Methods for Differential Equations, 4 (2016), 116-127. [34] M. Usman, G. Flora, C. Yakopcic and M. Imran, A computational study and stability analysis of a mathematical model for in vitro inhibition of cancer cell mutation, Int. J. Appl. Comput. Math., 3 (2017), 1861-1878.  doi: 10.1007/s40819-016-0201-8. [35] World Health Organization, Influenza: Data and Statistics, 2018, http://www.euro.who.int/en/health-topics/communicable-diseases/influenza/data-and-statistics. [36] World Health Organization, Influenza Virus Infections in Humans, 2018, http://www.who.int/influenza/GIP_InfluenzaVirusInfectionsHumans_Jul13.pdf. [37] World Health Organization, Pandemic (H1N1) 2009-update 81, 2018, http://www.who.int/csr/don/2010_03_05/en/index.html.

show all references

##### References:
 [1] F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, BioSystems, 113 (2013), 155-164.  doi: 10.1016/j.biosystems.2013.06.004. [2] M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM Journal of Applied Dynamical Systems, 3 (2004), 503-524.  doi: 10.1137/030600370. [3] J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. H. Wue, A model for influenza with vaccination and antiviral treatment, Journal of Theoretical Biology, 253 (2008), 118-130.  doi: 10.1016/j.jtbi.2008.02.026. [4] F. Brauer, P. van den Driessche and J. H. Wu, Mathematical Epidemiology, Lecture Notes in Mathematics, 1945. Mathematical Biosciences Subseries, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78911-6. [5] C. B. Bridges, S. A. Harper, K. Fukuda, T. M. Uyeki, N. J. Cox and J. A. Singleton, Prevention and control of influenza: Recommendations of the advisory committee on immunization practice (ACIP), Morbid Mortal Weekly Report, 52 (2003), 1-36. [6] K. F. Cheng and P. C. Leung, What happened in China during the 1918 influenza pandemic, International Journal of Infectious Diseases, 11 (2007), 360-364.  doi: 10.1016/j.ijid.2006.07.009. [7] N. S. Chong, J. M. Tchuenche and R. J. Smith., A mathematical model of avian influenza with half-saturated incidence, Theory in Biosciences, 133 (2014), 23-38.  doi: 10.1007/s12064-013-0183-6. [8] G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Transmission dynamics of the great influenza pandemic of 1918 In Geneva, Switzerland: Assessing the effects of hypothetical iInterventions, Journal of Theoretical Biology, 241 (2006), 193-204.  doi: 10.1016/j.jtbi.2005.11.026. [9] C. Cosner, Reaction-diffusion equations and ecological modeling, Tutorials in Mathematical Biosciences. IV, Lecture Notes in Math., Math. Biosci. Subser., Springer, Berlin, 1922 (2008), 77-115.  doi: 10.1007/978-3-540-74331-6_3. [10] P. van den Driessche and J. Watmough., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [11] G. Gonzàlez-Parra, A. J. Arenas, D. F. Aranda and L. Segovia, Modeling the epidemic waves of AH1N1/09 influenza around the world, Spatial and Spatio-temporal Epidemiology, 2 (2011), 219-226. [12] A. B. Gumel, Global dynamics of a two-strain avian influenza model, International Journal of Computer Mathematics, 86 (2009), 85-108.  doi: 10.1080/00207160701769625. [13] D. Hoff, Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations, SIAM J. Numer. Anal., 15 (1978), 1161-1177.  doi: 10.1137/0715077. [14] M. Imran, T. Malik, A. R. Ansari and A. Khan, Mathematical analysis of swine influenza epidemic model with optimal control, Japan Journal of Industrial and Applied Mathematics, 33 (2016), 269-296.  doi: 10.1007/s13160-016-0210-3. [15] M. Imran, M. Usman, M. Dur-e-Ahmad and A. Khan, Transmission dynamics of zika fever: A SEIR based model, Differ. Equ. Dyn. Syst., (2017), 1–24. doi: 10.1007/s12591-017-0374-6. [16] S. Islam and R. Zaman, A computational modeling and simulation of spatial dynamics in biological systems, Applied Mathematical Modelling, 40 (2016), 4524-4542.  doi: 10.1016/j.apm.2015.11.025. [17] A. Khan, M. Waleed and M. Imran, Mathematical analysis of an influenza epidemic model, formulation of different controlling strategies using optimal control and estimation of basic reproduction number, Mathematical and Computer Modelling of Dynamical Systems, 21 (2015), 432-459.  doi: 10.1080/13873954.2015.1016975. [18] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898717839. [19] B. Lina, Chapter 12: History of Influenza Pandemics, Paleomicrobiology: Past Human Infections, Springer-Verlag, 2008. [20] J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [21] M. Nuño, G. Chowell and A.B. Gumel, Assessing transmission control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK, and the Netherlands, Proceedings of the Royal Society Interface, 4 (2007), 505-521. [22] S. G. Ruan and W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [23] P. L. Salceanu, Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov Exponents, Math. Biosci. Eng., 8 (2011), 807-825.  doi: 10.3934/mbe.2011.8.807. [24] M. Samsuzzoha, M. Singh and D. Lucy, Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient, Applied Mathematical Modelling, 35 (2007), 5507-5523.  doi: 10.1016/j.apm.2011.04.029. [25] M. Samsuzzoha, M. Singh and D. Lucy, Numerical study of an influenza epidemic model with diffusion, Applied Mathematics and Computation, 217 (2010), 3461-3479.  doi: 10.1016/j.amc.2010.09.017. [26] M. Samsuzzoha, M. Singh and D. Lucy, Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient, Applied Mathematical Modelling, 35 (2011), 5507-5523.  doi: 10.1016/j.apm.2011.04.029. [27] M. Samsuzzoha, M. Singh and D. Lucy, A numerical study on an influenza epidemic model with vaccination and diffusion, Applied Mathematics and Computation, 219 (2012), 122-141.  doi: 10.1016/j.amc.2012.04.089. [28] M. Derouich and A. Boutayeb, An avian influenza mathematical model, Applied Mathematical Sciences, 2 (2008), 1749-1760. [29] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011. [30] N. I. Stilianakis, A. S. Perelson and F. G. Hayden, Emergence of drug resistance during an influenza epidemic: Insights from a mathematical model, The Journal of Infectious Diseases, 177 (1998), 863-873.  doi: 10.1086/515246. [31] J. K. Taubenberger and D. M. Morens, 1918 influenza: The mother of all pandemics, Emerging Infectious Diseases, 12 (2006), 15-22.  doi: 10.3201/eid1209.05-0979. [32] J. K. Taubenberger, A. H. Reid and T. G. Fanning, The 1918 influenza virus: A killer comes into view, Virology, 274 (2000), 241-245.  doi: 10.1006/viro.2000.0495. [33] S. Toubaei, M. Garshasbi and M. Jalalvand, A numerical treatment of a reaction-diffusion model of spatial pattern in the embryo, Computational Methods for Differential Equations, 4 (2016), 116-127. [34] M. Usman, G. Flora, C. Yakopcic and M. Imran, A computational study and stability analysis of a mathematical model for in vitro inhibition of cancer cell mutation, Int. J. Appl. Comput. Math., 3 (2017), 1861-1878.  doi: 10.1007/s40819-016-0201-8. [35] World Health Organization, Influenza: Data and Statistics, 2018, http://www.euro.who.int/en/health-topics/communicable-diseases/influenza/data-and-statistics. [36] World Health Organization, Influenza Virus Infections in Humans, 2018, http://www.who.int/influenza/GIP_InfluenzaVirusInfectionsHumans_Jul13.pdf. [37] World Health Organization, Pandemic (H1N1) 2009-update 81, 2018, http://www.who.int/csr/don/2010_03_05/en/index.html.
Spatial variation of the solutions using initial condition (ⅰ) and without diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Spatial variation of the solutions using initial condition (ⅱ) and without diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Spatial variation of the solutions using initial condition (ⅲ) and without diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Spatial variation of the solutions using initial condition (ⅱ) and with diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Spatial variation of the solutions using initial condition (ⅲ) and with diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Variation of the solutions in three dimensional display using initial condition (ⅲ) and without diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Variation of the solutions in three dimensional display using initial condition (ⅲ) and with diffusion, for effective contact rate $\beta = 0.514$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Time variation of the solutions using initial condition (ⅲ) and without diffusion, at $x = 0$, for the effective contact rate values $\beta = 0.25$, $0.4$, $0.5$, and $0.75$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Time variation of the solutions using initial condition (ⅲ) and with diffusion, at $x = 0$, for the effective contact rate values $\beta = 0.25$, $0.4$, $0.5$, and $0.75$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Time variation of the solutions S, I, H, and R, using initial condition (ⅲ) and without diffusion, at $x = 0$ for various effective contact rate values: (fig.a)$\beta = 0.25$, (fig.b)$\beta = 0.4$, (fig.c)$\beta = 0.5$, (fig.d)$\beta = 0.6$, (fig.e)$\beta = 0.75$, and (fig.f)$\beta = 1$
Time variation of the solutions using initial condition (ⅲ) and without diffusion, at $x = 0$, for the effective contact rate values $\beta = 0.514$, for different values of $\eta$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Time variation of the solutions using initial condition (ⅲ) and with diffusion, at $x = 0$, for the effective contact rate values $\beta = 0.514$, for different values of $\eta$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Time variation of the solutions using initial condition (ⅲ) and without diffusion, at $x = 0$, for the effective contact rate values $\beta = 0.514$, for different values of the inhibition parameters $a$ and $b$ with $\eta = 1$: (fig.a)S, (fig.b)E, (fig.c)I, (fig.d)H, and (fig.e)R
Description of the variables of the deterministic model (3)
 Variable Description $N(x, t)$ Total Population of all individuals $S(x, t)$ Population of susceptible individuals $E(x, t)$ Population of exposed individuals $I(x, t)$ Population of infected individuals $H(x, t)$ Population of hospitalized individuals $R(x, t)$ Population of recovered individuals
 Variable Description $N(x, t)$ Total Population of all individuals $S(x, t)$ Population of susceptible individuals $E(x, t)$ Population of exposed individuals $I(x, t)$ Population of infected individuals $H(x, t)$ Population of hospitalized individuals $R(x, t)$ Population of recovered individuals
Description of the parameters of the deterministic model (3)
 Parameter Description $\beta$ Effective contact rate $\Pi$ Recruitment rate $\mu$ Natural death rate for susceptible individuals $\alpha$ Disease-induced death rate for infected individuals $\delta$ Disease-induced death rate for hospitalized individuals $\sigma$ Infection rate for exposed individuals $\tau$ Hospitalization rate for infected individuals $\gamma$ Recovery rate for infected individuals $\theta$ Recovery rate for hospitalized individuals $a$ and $b$ Half saturation constants (inhibition factors) $\eta$ fraction of inhibition effect from the hospitalized individuals
 Parameter Description $\beta$ Effective contact rate $\Pi$ Recruitment rate $\mu$ Natural death rate for susceptible individuals $\alpha$ Disease-induced death rate for infected individuals $\delta$ Disease-induced death rate for hospitalized individuals $\sigma$ Infection rate for exposed individuals $\tau$ Hospitalization rate for infected individuals $\gamma$ Recovery rate for infected individuals $\theta$ Recovery rate for hospitalized individuals $a$ and $b$ Half saturation constants (inhibition factors) $\eta$ fraction of inhibition effect from the hospitalized individuals
Values of the parameters used in the simulation
 Parameter Value Reference $\Pi$ $7.14\times 10^{-5}$ [27] $\mu$ $5.5 \times 10^{-8}$ [27] $\alpha$ $9.3\times 10^{-6}$ [27] $\delta$ $9.3\times 10^{-6}$ [27,14,21] $\sigma$ 1/3 [14,27,21] $\tau$ 1/2 [14,21] $\gamma$ 1/5 [14,27] $\theta$ 1/5 [14,27]
 Parameter Value Reference $\Pi$ $7.14\times 10^{-5}$ [27] $\mu$ $5.5 \times 10^{-8}$ [27] $\alpha$ $9.3\times 10^{-6}$ [27] $\delta$ $9.3\times 10^{-6}$ [27,14,21] $\sigma$ 1/3 [14,27,21] $\tau$ 1/2 [14,21] $\gamma$ 1/5 [14,27] $\theta$ 1/5 [14,27]
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