Numerical study of an influenza epidemic dynamical model with diffusion

Mudassar Imran; Mohamed Ben-Romdhane; Ali R. Ansari; Helmi Temimi

doi:10.3934/dcdss.2020168

Article Contents

2020, Volume 13, Issue 10: 2761-2787. Doi: 10.3934/dcdss.2020168

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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
^* Corresponding author: M. Ben-Romdhane

Received: November 2018

Revised: May 2019

Early access: December 2019

Published: July 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B35, 65M22, 92B99; Secondary: 37M05.

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Figure 1. 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

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Figure 2. 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

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Figure 3. 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

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Figure 4. 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

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Figure 5. 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

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Figure 6. 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

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Figure 7. 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

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Figure 8. 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

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Figure 9. 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

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Figure 10. 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 $

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Figure 11. 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

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Figure 12. 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

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Figure 13. 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

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Table 1. 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

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Table 2. 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

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Table 3. 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]

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