# Generalized SEIR Epidemic Model for COVID-19 in a Multipatch Environment

Authors: Lan Meng; Wei Zhu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, ann-patch SEIR epidemic model for the coronavirus disease 2019 (COVID-19) is presented. It is shown that there is unique disease-free equilibrium for this model. Then, the dynamic behavior is studied by the basic reproduction number. The transmission of COVID-19 is fitted based on actual data. The influence of quarantined rate and population migration rate on the spread of COVID-19 is also discussed by simulation.


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## Body

## 1. Introduction

The coronavirus disease 2019 (COVID-19) has spread to more than 200 countries worldwide, which has a significant impact on economy development, social stability, and people’s daily life. According to the data on August 24, 2021, of World Health Organization, the number of confirmed cases has increased to 211,730,035 and the number of confirmed deaths cases has increased to 4,430,697 [1].In fact, COVID-19 is one of the most serious viruses for human beings. How to reduce or control the spread of this epidemic has attracted much attention from various communities [2–8]. Zhu et al. [9] isolated COVID-19 from pneumonia patient samples and found that it belongs to the seventh member of the coronavirus family. Linton et al. [10] obtained that an average incubation period is 5 days for COVID-19 by statistical analysis of epidemic data. Chen et al. [11] presented a transmission network model and calculated the basic reproduction number of the model, which showed that COVID-19 has a higher infection rate than SARS but less than MERS. Saha et al. [12] proposed an SEIRS epidemic model to explore coronavirus infection and suggested that susceptible individuals can avoid infection by taking appropriate precautions. Quaranta et al. [13] performed a multiscale dynamic analysis of COVID-19 outbreak in Italy to show changes at different geographic scales. Leung et al. [14] established a Susceptible-Exposed-Infectious-Recovered (SEIR) model, and the basic reproduction was estimated. Based on heart rate and sleep data collected from wearables, Zhu et al. [15] proposed a framework to predict the prevalence of COVID-19 in different countries and cities. In [16], the Susceptible-Infectious-Recovered (SIR) epidemic model was proposed and explained by logistic equation. Msmali et al. [17] used a mathematical model to study the role of behavior change in slowing the spread of COVID-19 in Saudi Arabia. Hou et al. [18] investigated the urban resilience level, spatial differentiation, and dominant elements in the middle reaches of the Yangtze River under the COVID-19 pandemic. However, it is rare to establish a generalized multipatch SEIR epidemic model to study the impact of population migration and quarantine on the basic reproduction number.With the rapid development of transportation, population migration among different regions has been one of the important factors to study the spread of epidemic. Hethcote [19] presented a model for migration between the two patches and studied the impact for migration on infectious diseases. Sattenspiel and Herring [20] presented a population flow model with n-patch and studied the dynamic behavior of the model. Driessche and Arino [21] established an epidemic patch model for n cities and analyzed the impact of population mobility on spatial distribution. Cui et al. [22] proposed a spatial infectious disease model with migration which has the potential value for disease control. Li [23] showed that pattern transition from stationary pattern to patch invasion appears to be possible in a fully deterministic parasite-host model. Arino et al. [24] found that the possibility of disease persisting in patches and spreading epidemic increased during transportation. A two-patch SEIRS epidemic model was proposed by Liu et al. [25] to study the impact of travel on the spatial spread of dog rabies between patches with different level of disease prevalence. Zhang et al. [26] focused on a delayed multigroup SIS epidemic model with nonlinear incidence rates and patch structure. Driessche and Salmani [27] gave an SEIRS model of p-patch and analyzed the stability of the disease-free equilibrium point. The effects of heterogeneity in groups, patches, and mobility patterns on the basic reproduction number R0 and disease prevalence were explored by Bichara et al. [28]. The dynamics of an SIS epidemic patch model with the asymmetric connectivity matrix was analyzed by Chen et al. [29], and it showed that the basic reproduction number R0 was strictly decreasing with respect to the dispersal rate of the infected individuals. Therefore, it is necessary to study the epidemic model of multipatch. Furthermore, how to study the spread of COVID-19 in a multipatch environment is an interesting and important topic.Inspired by above discussion, in this paper, a generalizedn-patch SEIR epidemic model is formulated to study the stability of the model and the effect of control strategies on the spread of the disease. The paper is organized as follows. In Section 2, the SEIR model with n-patch for COVID-19 is proposed. In Section 3, the basic reproduction number associated with quarantined rate and population migration rate is given, and the stability of the model is studied. In Section 4, some numerical simulations are given to study the effects of migration and quarantine strategy on disease transmission. Based on the data of Hubei province, Chongqing, and Hunan province, numerical fitting is given. The numerical simulations show that appropriate quarantine strategy and control of the migration rate are important to reduce the spread of COVID-19. Conclusions are made in Section 5.Notations:ρA≜maxi=1,2,…,nλi and sA≜maxi=1,2,…,nReλi, where λ1,λ2,…,λn are the eigenvalues of matrix A and 0m×n denotes the m×n zero matrix.


## 2. Then-Patch SEIR Model for COVID-19

The population in patchi is stratified as susceptible (Si), exposed (Ei), infectious (Ii), quarantined (Qi), hospitalized (Hi), and recovered (Ri). By tracing close contacts, some individuals exposed to the virus are quarantined. It is assumed that the self-quarantined susceptible individuals who stay in safe areas have no contact with infected individuals. The disease transmission in each patch is shown in Figure 1.Figure 1
The disease transmission in patchi.Based on the epidemiology of COVID-19 and the control measures taken by the government, the generalizedn-patch SEIR epidemic model for COVID-19 is formulated as follows:(1)dSidt=bi−βiSi1−ziviEi+Ii−μiSi+∑j≠1naijSj−ajiSi,dEidt=βiSi1−ziviEi+Ii−qi+σi+μiEi+∑j≠1nbijEj−bjiEi,dIidt=σiEi−δi+γi+θiIi+∑j≠1ncijIj−cjiIi,dQidt=qiEi−αi+μiQi,dHidt=αiQi+δiIi−ηi+giHi,dRidt=γiIi+ηiHi−μiRi+∑j≠1ndijRj−djiRi,i=1,2,…,n,where βi,zi,qi,σi,δi,γi, and ηi represent the disease transmission coefficient, the self-quarantined rate of susceptible individuals, the quarantined rate of exposed individuals, the transition rate of exposed individuals to infected individuals, the transition rate of infected individuals to hospitalized individuals, the recovery rate of the infectious individuals q, and the recovery rate of hospitalized individuals, respectively. Since COVID-19 differs from traditional diseases in which the exposed individuals are also infectious, the disease transmission coefficient regulator of exposed individuals vi is added to the model. bi, μi, θi, and gi are the number of births per unit time, the natural death rate, the death rate of infected individuals, and the death rate of hospitalized individuals in patch i, respectively. The migration rates of susceptible, exposed individuals, infectious, and removed individuals from patch j to patch i are denoted by aij,bij,cij, and dij, respectively. It is clear that all these parameters are nonnegative.Remark 1.
Different with the model in [30], the natural birth rate, the number of births per unit time, the death rate of hospitalized individuals, and the self-quarantined rate of susceptible are taken into consideration. Moreover, people can migrate between any two patches.


## 3. Main Results

In order to discuss the dynamic behavior of system (1), we firstly introduce the following definition.Definition 1.
(see [31]). Let A=aij∈Rn×n, where aij<0 if i≠j, and the sum of each column element of the matrix is positive, then the matrix is a nonsingular M matrix and A−1≥0.Lemma 1.
System (1) has unique disease-free equilibrium.Proof.
By the definition of the disease-free equilibrium, substitutingIi=0i=1,2,…,n into (1), we have(2)bi−βiSi1−ziviEi−μiSi+∑j≠1naijSj−ajiSi=0,βiSi1−ziviEi−qi+σi+μiEi+∑j≠1nbijEj−bjiEi=0,σiEi=0,qiEi−αi+μiQi=0,αiQi−ηi+giHi=0,ηiHi−μiRi+∑j≠1ndijRj−djiRi=0,i=1,2,…,n.
Then,(3)bi−μiSi+∑j=1naijSj−ajiSi=0,−μiRi+∑j≠1ndijRj−djiRi=0,i=1,2,…,n,that is,(4)G1S=B,G2R=0,where(5)S=S1,S2,…,Sn′,B=B1,B2,…,Bn′,R=R1,R2,…,Rn,0=0n×1,G1=μ1+∑j≠1naj1−a12…−a1n−a21μ2+∑j≠2naj2…−a2n⋮⋮⋱⋮−an1−an2…μn+∑j≠nnajn,G2=μ1+∑j≠1ndj1−d12…−d1n−d21μ2+∑j≠2ndj2…−d2n⋮⋮⋱⋮−dn1−dn2…μn+∑j≠nndjn.
SinceG1 and G2 are nonsingular M-matrix, then G1−1≥0 and G2−1≥0. Therefore, (3) has unique solution S∗=S10,S20,…,Sn0′=G1−1B and R∗=0.
Thus, system (1) has unique disease-free equilibrium C0=S∗′,0′,0′,0′,0′,0′.
The proof is completed.Lemma 2.
Γ=S1,E1,I1,Q1,H1,R1,…,Sn,En,In,Qn,Hn,Rn∈ℝ+6n|Nt≤B¯/μ∗,0≤Si≤Si0,i=1,2,…,n, is a positively invariant set for system (1), where B¯=∑i=1nbi,μ∗=minμi,θi,gi and Nt=∑i=1nSi+Ei+Ii+Qi+Hi+Ri.Proof.
In term of system (1), we have(6)N˙t=B¯−∑i=1nμiSi+μiEi+θiIi+μiQi+giHi+μiRi+∑i=1n∑j=1naijSj−ajiSi+∑j=1nbijEj−bjiEi+∑j=1ncijIj−cjiIi+∑j=1ndijRj−djiRi=B¯−∑i=1nμiSi+μiEi+θiIi+μiQi+giHi+μiRi≤B¯−μ∗N.
Therefore,(7)Nt≤N0−B¯μ∗e−μ∗t+B¯μ∗,where N0 is the initial population.
Therefore,Nt≤B¯/μ∗ if and only if N0≤B¯/μ∗.
From the first equation of system (1), we have(8)dSidt≤bi−μiSi+∑j=1naijSj−ajiSi=G1S0−G1Si,where G1S0−G1Si is the i-th element of G1S0−G1S.
Therefore,dSi/dt≤0 if Si=Si0,Sj≤Sj0,i,j=1,2,…,n,andi≠j.
The proof is completed.
Based on Lemma2, the dynamical properties of system (1) are studied only in Γ. Now, we are in the position to discuss the basic reproduction number of system (1).
DefineF=F11F1200 and V=V110V21V22, where(9)F11=diagv1′,v2′,…,vn′,vi′=1−ziβiviSi0,F12=diagu1,u2,…,un,ui=1−ziβiSi0,V11=q1+σ1+μ1+∑j≠1nbj1−b12…−b1n−b21q2+σ2+μ2+∑j≠2nbj2…−b2n⋮⋮⋰⋮−bn1−bn2…qn+σn+μn+∑j≠nnbjn,V21=diag−σ1,−σ2,…,−σn,V22=δ1+γ1+θ1+∑j≠1ncj1−c12…−c1n−c21δ2+γ1+θ1+∑j≠2ncj2…−c2n⋮⋮⋱⋮−cn1−cn2…δn+γn+θn+∑j≠nncjn.
By using the method in Driessche and Watmough [31], the basic reproduction number R0 can be determined by computing the spectral radius of the matrix as follows:(10)FV−1=F11V11−1−F12V22−1V21V11−1F12V22−100,that is, R0=ρFV−1=ρF11V11−1−F12V11−1V21V22−1.Lemma 3 (see [32]).
IfJ1 is a nonnegative matrix and J2 is a nonsingular matrix, then sJ1−J2<0⇔ρJ1J2−1<1 and sJ1−J2>0⇔ρJ1J2−1>1.Theorem 1.
The disease-free equilibriumC0 is globally asymptotically stable if R0<1, and the disease-free equilibrium C0 is unstable if R0>1.(11)X˙=F−VX.Proof.
From the second equations and the third equations of system (1), we have, that is, X=E1,E2,…,En,I1,I2,…,In′.
SinceF is a nonnegative matrix and V is a nonsingular matrix, it follows from Lemma 3 that(12)SF−V>0,ifR0=ρFV−1>1.
Thus, the disease-free equilibriumC0 is unstable if R0>1.
Next, we proveC0 is globally asymptotically stable if R0<1.
Since(13)V−1F=V−1FV−1V,then V−1F is similar to FV−1. Therefore,(14)ρV−1F=ρFV−1=R0.
SinceV−1F is a nonnegative irreducible matrix, by Perron–Frobenius theorem [31], V−1F has a positive left eigenvector x and(15)xV−1F=ρFV−1x.
Letx=e1,e2,…,en,r1,r2,…,rn, then(16)e1,e2,…,en,r1,r2,…,rnV−1F=R0e1,e2,…,en,r1,r2,…,rn.
LetW=∑i=1nkiEi+∑i=1nliIi be a candidate Lyapunov function, where(17)k1,k2,…,kn,l1,l2,…,ln=e1,e2,…,en,r1,r2,…,rnV−1.
Then, calculating the derivative of W along the solution of system (1), we have(18)W˙=∑i=1nkidEidt+lidIidt=∑i=1nkiβiSi1−ziviEi+Ii−qi+σi+μiEi+∑j=1nbijEj−bjiEi+∑i=1nliσiEi−δi+γi+θiIi+∑j=1ncijIj−cjiIi≤∑i=1nkiβiSi01−ziviEi+Ii−qi+σi+μiEi+∑j=1nbijEj−bjiEi+∑i=1nliσiEi−δi+γi+θiIi+∑j=1ncijIj−cjiIi=k1,k2,…,knF11−V11E+F12I−l1,l2,…,lnV21E+V22I=k1,k2,…,kn,l1,l2,…,lnF−VX=e1,e2,…,en,r1,r2,…,rnV−1F−VX=e1,e2,…,en,r1,r2,…,rnV−1F−EX=e1,e2,…,en,r1,r2,…,rnR0−1X,where E=E1,E2,…,En′ and I=I1,I2,…,In′.
IfR0<1, then W˙≤0, and W˙=0 if and only if Si=Si0orEi=0, Ii=0,i=1,2,…,n.
Thus,C0 is globally asymptotically stable if R0<1.
The proof is completed.


## 4. Numerical Simulation of COVID-19 Transmission

In this section, the spread of COVID-19 among Hubei province, Chongqing, and Hunan province is simulated. The SEIR epidemic model withn=3 is restated is as follows:(19)dSidt=bi−βiSi1−ziviEi+Ii−μiSi+∑j≠1naijSj−ajiSi,dEidt=βiSi1−ziviEi+Ii−qi+σi+μiEi+∑j≠1nbijEj−bjiEi,dIidt=σiEi−δi+γi+θiIi+∑j≠1ncijIj−cjiIi,dQidt=qiEi−αi+μiQi,dHidt=αiQi+δiIi−ηi+giHi,dRidt=γiIi+ηiHi−μiRi+∑j≠1ndijRj−djiRi,i=1,2,…,n.It is assumed thataij=bij=cij=dij=pij,i,j=1,2,3,andi≠j, where pij is the migration rate from patch j to patch i. The migration matrix is P=pij3×3, where pii=0,i=1,2,3.


### 4.1. Fitting on the Transmission of COVID-19

Leti=1,2,and3 represent Hubei province, Chongqing, and Hunan province in system (19), respectively. The values of θi and γi are taken as the average death rate and cure rate of confirmed infected persons in patch i from January 20, 2020, to February 1, 2020. The values of gi and ηi are taken as the average death rate and cure rate of confirmed infected persons in patch i from January 25, 2020, to March 1, 2020, respectively. Based on the data from January 25, 2020, to March 15, 2020, least squares estimate (LSE) is used to estimate the parameters. The data are divided into two stages: the first stage is from January 25 to February 7 and the second stage is from February 8 to March 15, 2020. The difference between the two stages is reflected in qi,zi,δi, and αi. Moreover, qij,zij,δij, andαij represent the corresponding parameter value of the j-th stage, where j=1,2. The parameters of system (19) are shown in Table 1.Table 1
Parameter data and sources of Hubei province, Chongqing, and Hunan province, China.
i = 1i = 2i = 3Average errorSourceβi2 × 10−92 × 10−108.2 × 10−113.2 × 10−11LSEvi1.81.51.60.13LSEσi0.20.20.2—[10]µi1.94 × 10−53.62 × 10−52 × 10−5—[33–35]γi0.0050.0050.005—[36–38]θi0.010.010.01—[36–38]gi0.0040.00050.0002—[36–38]bi184111261967—[33–35]ηi0.0240.0430.055—[36–38]Si(0)59270000325420073195300—[33–35]Ei(0)28000320750323LSEIi(0)14500160310217LSEQi(0)200060120113LSEHi(0)9587569—[36–38]Ri(0)4200—[36–38]qi10.150.030.020.005LSEzi10.530.620.50.08LSEδi10.040.120.150.02LSEαi10.070.110.010.01LSEqi20.850.860.780.05LSEzi20.870.850.80.09LSEδi20.170.250.40.07LSEαi20.0050.080.0020.004LSEThe numerical fitting diagram is shown in Figure2. It can be found that the simulated curve is approximately the same as the actual data.Figure 2
Fitting on the transmission of COVID-19 in (a) Hubei province; (b) Chongqing; (c) Hunan province.
(a)(b)(c)


### 4.2. The Effects of Migration and Quarantine Measures on COVID-19 Transmission

Next, the effect of migration on COVID-19 transmission is shown in Figure3 when only migration rate is changed. As the migration rate increases, the number of cumulative confirmed cases increases in Chongqing and Hunan province. However, with the increase in migration rate, the number of cumulative confirmed cases decreases in Hubei province. In this case, since the epidemic situation is most severe in Hubei province, population migration can reduce the spread of COVID-19 in Hubei province. Nevertheless, it makes epidemic situation worse in other areas.Figure 3
Effect of migration rate on COVID-19 transmission, wherei, j = 1, 2, 3: (a) Hubei province; (b) Chongqing; (c) Hunan province.
(a)(b)(c)The effect of quarantine strategy on COVID-19 transmission is shown in Figure4 when only quarantined rate is changed. Figures 4(a)–4(c) show that if quarantine measures had not been implemented, the epidemic situation would have been worse than the actual situation, especially in Hubei province. Furthermore, as the quarantined rate increases, the number of cumulative confirmed cases will decrease. In Figure 4(d), the red solid represents self-quarantined rate of susceptible individuals where P=P0,zi=0,andi=1,2,3; black solid represents quarantined rate of exposed individuals where P=P0,qi=0,andi=1,2,3; and blue solid represents that both kinds of quarantine strategies are performed simultaneously where P=P0,qi=zi,andi=1,2,3. Figure 4(d) shows that R0 decreases with the increase in quarantined rate. The simultaneous execution of both quarantine strategies has the greatest impact on R0. Therefore, quarantine strategy plays an important and effective role in controlling the outbreak. Thus, reducing migration rate and increasing quarantine measures in Hubei can effectively reduce the spread of the disease.Figure 4
Effect of quarantined rate on COVID-19 transmission, wherei=1,2,3: (a) Hubei province; (b) Chongqing; (c) Hunan province.
(a)(b)(c)(d)


## 4.1. Fitting on the Transmission of COVID-19

Leti=1,2,and3 represent Hubei province, Chongqing, and Hunan province in system (19), respectively. The values of θi and γi are taken as the average death rate and cure rate of confirmed infected persons in patch i from January 20, 2020, to February 1, 2020. The values of gi and ηi are taken as the average death rate and cure rate of confirmed infected persons in patch i from January 25, 2020, to March 1, 2020, respectively. Based on the data from January 25, 2020, to March 15, 2020, least squares estimate (LSE) is used to estimate the parameters. The data are divided into two stages: the first stage is from January 25 to February 7 and the second stage is from February 8 to March 15, 2020. The difference between the two stages is reflected in qi,zi,δi, and αi. Moreover, qij,zij,δij, andαij represent the corresponding parameter value of the j-th stage, where j=1,2. The parameters of system (19) are shown in Table 1.Table 1
Parameter data and sources of Hubei province, Chongqing, and Hunan province, China.
i = 1i = 2i = 3Average errorSourceβi2 × 10−92 × 10−108.2 × 10−113.2 × 10−11LSEvi1.81.51.60.13LSEσi0.20.20.2—[10]µi1.94 × 10−53.62 × 10−52 × 10−5—[33–35]γi0.0050.0050.005—[36–38]θi0.010.010.01—[36–38]gi0.0040.00050.0002—[36–38]bi184111261967—[33–35]ηi0.0240.0430.055—[36–38]Si(0)59270000325420073195300—[33–35]Ei(0)28000320750323LSEIi(0)14500160310217LSEQi(0)200060120113LSEHi(0)9587569—[36–38]Ri(0)4200—[36–38]qi10.150.030.020.005LSEzi10.530.620.50.08LSEδi10.040.120.150.02LSEαi10.070.110.010.01LSEqi20.850.860.780.05LSEzi20.870.850.80.09LSEδi20.170.250.40.07LSEαi20.0050.080.0020.004LSEThe numerical fitting diagram is shown in Figure2. It can be found that the simulated curve is approximately the same as the actual data.Figure 2
Fitting on the transmission of COVID-19 in (a) Hubei province; (b) Chongqing; (c) Hunan province.
(a)(b)(c)


## 4.2. The Effects of Migration and Quarantine Measures on COVID-19 Transmission

Next, the effect of migration on COVID-19 transmission is shown in Figure3 when only migration rate is changed. As the migration rate increases, the number of cumulative confirmed cases increases in Chongqing and Hunan province. However, with the increase in migration rate, the number of cumulative confirmed cases decreases in Hubei province. In this case, since the epidemic situation is most severe in Hubei province, population migration can reduce the spread of COVID-19 in Hubei province. Nevertheless, it makes epidemic situation worse in other areas.Figure 3
Effect of migration rate on COVID-19 transmission, wherei, j = 1, 2, 3: (a) Hubei province; (b) Chongqing; (c) Hunan province.
(a)(b)(c)The effect of quarantine strategy on COVID-19 transmission is shown in Figure4 when only quarantined rate is changed. Figures 4(a)–4(c) show that if quarantine measures had not been implemented, the epidemic situation would have been worse than the actual situation, especially in Hubei province. Furthermore, as the quarantined rate increases, the number of cumulative confirmed cases will decrease. In Figure 4(d), the red solid represents self-quarantined rate of susceptible individuals where P=P0,zi=0,andi=1,2,3; black solid represents quarantined rate of exposed individuals where P=P0,qi=0,andi=1,2,3; and blue solid represents that both kinds of quarantine strategies are performed simultaneously where P=P0,qi=zi,andi=1,2,3. Figure 4(d) shows that R0 decreases with the increase in quarantined rate. The simultaneous execution of both quarantine strategies has the greatest impact on R0. Therefore, quarantine strategy plays an important and effective role in controlling the outbreak. Thus, reducing migration rate and increasing quarantine measures in Hubei can effectively reduce the spread of the disease.Figure 4
Effect of quarantined rate on COVID-19 transmission, wherei=1,2,3: (a) Hubei province; (b) Chongqing; (c) Hunan province.
(a)(b)(c)(d)


## 5. Conclusion

In this paper, based on epidemiological characteristics of COVID-19 and government intervention strategy, ann-patch SEIR epidemic model is presented. It is shown that system (1) has unique disease-free equilibrium C0. Moreover, C0 is globally asymptotically stable R0<1, and it is unstable if R0>1. A 3-patch SEIR epidemic model is formulated to explore the effect of migration rate and quarantined rate on COVID-19 transmission. The numerical results show that appropriate controls on migration and quarantined rate are necessary to prevent outbreaks.



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*Source: 5401253-2021-12-17.xml*