# American Institute of Mathematical Sciences

• Previous Article
Schrödinger–Kirchhoff–Hardy $p$–fractional equations without the Ambrosetti–Rabinowitz condition
• DCDS-S Home
• This Issue
• Next Article
Design of intelligent retrieval algorithm for literature information resources in digital library

## 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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020168
##### References:

show all references

##### References:
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]
 [1] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [2] Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826 [3] Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 [4] Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014 [5] Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074 [6] Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064 [7] Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119 [8] G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783 [9] Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045 [10] Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93 [11] Jianquan Li, Yicang Zhou, Jianhong Wu, Zhien Ma. Complex dynamics of a simple epidemic model with a nonlinear incidence. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 161-173. doi: 10.3934/dcdsb.2007.8.161 [12] T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277 [13] Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157 [14] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [15] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 [16] Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure & Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038 [17] G. Leugering, Marina Prechtel, Paul Steinmann, Michael Stingl. A cohesive crack propagation model: Mathematical theory and numerical solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1705-1729. doi: 10.3934/cpaa.2013.12.1705 [18] Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1 [19] Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51 [20] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191

2018 Impact Factor: 0.545

## Tools

Article outline

Figures and Tables