# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021154
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## Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence

 College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

* Corresponding author

Received  February 2021 Revised  May 2021 Early access September 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11861065, 11771373, 11702237), the Open Project of Key Laboratory of Applied Mathematics of Xinjiang Province (Grant No. 2021D04014), the Natural Science Foundation of Xinjiang Province of China (Grant Nos. 2019D01C076, 2017D01C082), the Scientific Research Programmes of Colleges in Xinjiang (Grant No. XJEDU2021I002, XJEDU2021Y001), The Tianshan Youth Program-Training Program for Excellent Young Scientific and Technological Talents of Xinjiang (Grant No. 2019Q017)

A diffusion SEIAR model with Beddington-DeAngelis type incidence is proposed to characterize the spread of COVID-19 with spatial transmission. First, the well-posedness of solution is studied. Second, the basic reproduction number $\mathcal R_{0}$ is derived and served as a threshold value to determine whether COVID-19 will spread. Meanwhile, we consider the effect of diffusion on the spread of COVID-19 in spatial homogenous environment, by which we can obtain that if $\mathcal R_{0}<1$, then the infection-free steady state is globally asymptotically stable, while if $\mathcal R_{0}>1$, then the endemic steady state is globally asymptotically stable. Furthermore, according to the official reporting data about COVID-19 in Wuhan, China, the actual value of $\mathcal R_{0}$ is estimated, and comparing with other types of incidence, we find that the estimated peak with Beddington-DeAngelis type incidence is more close to the cases in reality. Finally, by numerical simulations, we can see that the diffusion behavior has evident impact on the spread of COVID-19 in spatial heterogeneity than homogeneity of environment.

Citation: Tao Zheng, Yantao Luo, Xinran Zhou, Long Zhang, Zhidong Teng. Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021154
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##### References:
An illustration of the model (1.2) describing the transmission of COVID-19 infection
The fitting results of cumulative confirmed cases from Jan 23rd to Feb 11rd, 2020. The blue curve denotes fitting curve of model (8.1). The star denotes the real data of cumulative confirmed cases
. Fig. (b) and (c) is the effects of different recovery rate and transmission rate on disease dynamics in symptomatic infected $I(t)$, respectively">Figure 3.  Fig. (a) is a simulation result for the outbreak size in Wuhan using model (8.1), the parameters from Table 1. Fig. (b) and (c) is the effects of different recovery rate and transmission rate on disease dynamics in symptomatic infected $I(t)$, respectively
The effect of different incidence rates on the peak number of asymptomatic infected persons $I(t)$ in the short term
Spatial distribution of the number of symptomatic infections $I(x, t)$ in the short term for $d = 1.25\times 10^{-2}$
">Figure 6.  The short time behaviour of the solution $I(x, t)$ of model (7.1) with $d = 0, 1.25\times 10^{-3}$, all other parameters as Table 1
$(\mathcal{R}_{0} = 4.1062>1)$. COVID-19 disease is finally reaching persistence">Figure 7.  The long time behaviour of the solution $I(x, t)$ of model (7.1) with $(\beta, \alpha, \eta) = (6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$ and $d = 1.25\times 10^{-2}$, all other parameters are shown in Table 1 $(\mathcal{R}_{0} = 4.1062>1)$. COVID-19 disease is finally reaching persistence
$(\mathcal{R}_{0} = 0.9551<1)$. COVID-19 disease is finally heading towards extinction">Figure 8.  The long time behaviour of the solution $I(x, t)$ of model (7.1) with $(\beta, \alpha, \eta) = (1.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$ and $d = 1.25\times 10^{-2}$, all other parameters are shown in Table 1 $(\mathcal{R}_{0} = 0.9551<1)$. COVID-19 disease is finally heading towards extinction
Spatial distribution of the number of symptomatic infections $I(x, t)$ in the short term for model (1.2)
">Figure 10.  The short time behaviour of the solution $I(x, t)$ of model (1.2) with $(\beta, \alpha, \eta) = (1+0.5\cos2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$ and $d = 0, 1.25\times 10^{-3}, 1.25\times 10^{-2}$, all other parameters are shown in Table 1
$(\mathcal{R}_{0} = 4.2476>1)$. COVID-19 disease is finally reaching persistence. Fig (b) is the relation between $\mathcal{R}_{0}$ and $c$ in $(\beta(x), \alpha(x), \eta(x)) = (1+c\cos 2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$">Figure 11.  The long time behaviour of the solution $I(x, t)$ of model (1.2) with $(\beta(x), \alpha(x), \eta(x)) = (1+0.5\cos2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$ and $d = 1.25\times 10^{-2}$, all other parameters are shown in Table 1 $(\mathcal{R}_{0} = 4.2476>1)$. COVID-19 disease is finally reaching persistence. Fig (b) is the relation between $\mathcal{R}_{0}$ and $c$ in $(\beta(x), \alpha(x), \eta(x)) = (1+c\cos 2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$
$(\mathcal{R}_{0} = 0.7231<1)$. COVID-19 disease is finally heading towards extinction. Fig (b) is the relation between $\mathcal{R}_{0}$ and $c$ in $(\beta(x), \alpha(x), \eta(x)) = (1+c\cos 2\pi x)(1.10\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$">Figure 12.  The long time behaviour of the solution $I(x, t)$ of model (1.2) with $(\beta(x), \alpha(x), \eta(x)) = (1+0.5\cos2\pi x)(1.10\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$ and $d = 1.25\times 10^{-2}$, all other parameters are shown in Table 1 $(\mathcal{R}_{0} = 0.7231<1)$. COVID-19 disease is finally heading towards extinction. Fig (b) is the relation between $\mathcal{R}_{0}$ and $c$ in $(\beta(x), \alpha(x), \eta(x)) = (1+c\cos 2\pi x)(1.10\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$
">Figure 13.  The basic reproduction number $\mathcal{R}_{0}$ of model (1.2) for $0\leq c\leq1$ and $k = 2, 7, 13$, where $(\beta(x), \alpha(x), \eta(x)) = (1+c\cos k\pi x)(1.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$ and $d = 1.25\times 10^{-2}$, all other parameters are shown in Table 1
">Figure 14.  The solution of $I(x, t)$ of model (7.1) with initial condition (8.2) and $d = 0, \ 1.25\times 10^{-3}, \ 1.25\times 10^{-2}$, where $(\beta, \alpha, \eta) = (1.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7})$, all other parameters are shown in Table 1
The values of parameters for model (8.1)
 Parameter Description Value Reference $\lambda$ Influx rate $\frac{8, 999, 990}{76.79\times365}$ [52] $\mu$ Natural mortality $\frac{1}{76.79\times365}$ [25] $\beta$ Infection rate between $S$ and $E$ $6.51\times10^{-7}$ Fitted $\alpha$ Infection rate between $S$ and $I$ $3.11\times10^{-7}$ [41] $\eta$ Infection rate between $S$ and $A$ $1.56\times10^{-7}$ [41] $\omega$ Incubation period 1/7 [41] $\theta$ Incubation period 1/7 [41] $\delta$ Asymptomatic infection rate 0.2412 Fitted $\gamma$ Recovery rate of $I$ 1/15 [41] $\varpi$ Recovery rate of $A$ 0.8613 Fitted $m_{1}$ Inhibition rate $1.00\times10^{-6}$ Fitted $m_{2}$ Inhibition rate $7.11\times10^{-4}$ Fitted $n_{1}$ Inhibition rate $8.92\times10^{-4}$ Fitted $n_{2}$ Inhibition rate $1.00\times10^{-3}$ Fitted $e_{1}$ Inhibition rate $1.00\times10^{-3}$ Fitted $e_{2}$ Inhibition rate $7.83\times10^{-4}$ Fitted $S(0)$ Initial value 8998187 [52] $E(0)$ Initial value 845.6 Fitted $I(0)$ Initial value 475 [52] $A(0)$ Initial value 472.4 Fitted $R(0)$ Initial value 10 [52]
 Parameter Description Value Reference $\lambda$ Influx rate $\frac{8, 999, 990}{76.79\times365}$ [52] $\mu$ Natural mortality $\frac{1}{76.79\times365}$ [25] $\beta$ Infection rate between $S$ and $E$ $6.51\times10^{-7}$ Fitted $\alpha$ Infection rate between $S$ and $I$ $3.11\times10^{-7}$ [41] $\eta$ Infection rate between $S$ and $A$ $1.56\times10^{-7}$ [41] $\omega$ Incubation period 1/7 [41] $\theta$ Incubation period 1/7 [41] $\delta$ Asymptomatic infection rate 0.2412 Fitted $\gamma$ Recovery rate of $I$ 1/15 [41] $\varpi$ Recovery rate of $A$ 0.8613 Fitted $m_{1}$ Inhibition rate $1.00\times10^{-6}$ Fitted $m_{2}$ Inhibition rate $7.11\times10^{-4}$ Fitted $n_{1}$ Inhibition rate $8.92\times10^{-4}$ Fitted $n_{2}$ Inhibition rate $1.00\times10^{-3}$ Fitted $e_{1}$ Inhibition rate $1.00\times10^{-3}$ Fitted $e_{2}$ Inhibition rate $7.83\times10^{-4}$ Fitted $S(0)$ Initial value 8998187 [52] $E(0)$ Initial value 845.6 Fitted $I(0)$ Initial value 475 [52] $A(0)$ Initial value 472.4 Fitted $R(0)$ Initial value 10 [52]
Peak values of symptomatic infection $I(t)$ with different incidence functions
 Transmission rate Forms Fitted parameters Peak value Bilinear incidence $\beta SE$, $\alpha SI$, $\eta SA$ $\beta$, $\delta$, $\varpi$ 674950 Saturated incidence for the susceptible $\frac{\beta SE}{1+m_{1}S}$, $\frac{\beta SI}{1+n_{1}S}$, $\frac{\beta SA}{1+e_{1}S}$ $\beta$, $m_{1}$, $n_{1}$, $e_{1}$, $\delta$, $\varpi$ 3357600 Saturated incidence for the infected $\frac{\beta SE}{1+m_{2}E}$, $\frac{\beta SI}{1+n_{2}I}$, $\frac{\beta SA}{1+e_{2}A}$ $\beta$, $m_{2}$, $n_{2}$, $e_{2}$, $\delta$, $\varpi$ 71357 Beddington-DeAngelis type incidence $\frac{\beta SE}{1+m_{1}S+m_{2}E}$, $\frac{\beta SI}{1+n_{1}S+n_{2}I}$, $\frac{\beta SA}{1+e_{1}S+e_{2}A}$ $\beta$, $m_{1}$, $m_{2}$, $n_{1}$, $n_{2}$, $e_{1}$, $e_{2}$, $\delta$, $\varpi$ 67599
 Transmission rate Forms Fitted parameters Peak value Bilinear incidence $\beta SE$, $\alpha SI$, $\eta SA$ $\beta$, $\delta$, $\varpi$ 674950 Saturated incidence for the susceptible $\frac{\beta SE}{1+m_{1}S}$, $\frac{\beta SI}{1+n_{1}S}$, $\frac{\beta SA}{1+e_{1}S}$ $\beta$, $m_{1}$, $n_{1}$, $e_{1}$, $\delta$, $\varpi$ 3357600 Saturated incidence for the infected $\frac{\beta SE}{1+m_{2}E}$, $\frac{\beta SI}{1+n_{2}I}$, $\frac{\beta SA}{1+e_{2}A}$ $\beta$, $m_{2}$, $n_{2}$, $e_{2}$, $\delta$, $\varpi$ 71357 Beddington-DeAngelis type incidence $\frac{\beta SE}{1+m_{1}S+m_{2}E}$, $\frac{\beta SI}{1+n_{1}S+n_{2}I}$, $\frac{\beta SA}{1+e_{1}S+e_{2}A}$ $\beta$, $m_{1}$, $m_{2}$, $n_{1}$, $n_{2}$, $e_{1}$, $e_{2}$, $\delta$, $\varpi$ 67599
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