Generalized SEIR Epidemic Model for COVID-19 in a Multipatch Environment
In this paper, an -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.
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 .
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.  isolated COVID-19 from pneumonia patient samples and found that it belongs to the seventh member of the coronavirus family. Linton et al.  obtained that an average incubation period is 5 days for COVID-19 by statistical analysis of epidemic data. Chen et al.  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.  proposed an SEIRS epidemic model to explore coronavirus infection and suggested that susceptible individuals can avoid infection by taking appropriate precautions. Quaranta et al.  performed a multiscale dynamic analysis of COVID-19 outbreak in Italy to show changes at different geographic scales. Leung et al.  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.  proposed a framework to predict the prevalence of COVID-19 in different countries and cities. In , the Susceptible-Infectious-Recovered (SIR) epidemic model was proposed and explained by logistic equation. Msmali et al.  used a mathematical model to study the role of behavior change in slowing the spread of COVID-19 in Saudi Arabia. Hou et al.  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  presented a model for migration between the two patches and studied the impact for migration on infectious diseases. Sattenspiel and Herring  presented a population flow model with n-patch and studied the dynamic behavior of the model. Driessche and Arino  established an epidemic patch model for n cities and analyzed the impact of population mobility on spatial distribution. Cui et al.  proposed a spatial infectious disease model with migration which has the potential value for disease control. Li  showed that pattern transition from stationary pattern to patch invasion appears to be possible in a fully deterministic parasite-host model. Arino et al.  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.  to study the impact of travel on the spatial spread of dog rabies between patches with different level of disease prevalence. Zhang et al.  focused on a delayed multigroup SIS epidemic model with nonlinear incidence rates and patch structure. Driessche and Salmani  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 and disease prevalence were explored by Bichara et al. . The dynamics of an SIS epidemic patch model with the asymmetric connectivity matrix was analyzed by Chen et al. , and it showed that the basic reproduction number 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 generalized n-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: and , where are the eigenvalues of matrix and denotes the zero matrix.
2. The n-Patch SEIR Model for COVID-19
The population in patch is stratified as susceptible (), exposed (), infectious (), quarantined (), hospitalized (), and recovered (). 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.
Based on the epidemiology of COVID-19 and the control measures taken by the government, the generalized -patch SEIR epidemic model for COVID-19 is formulated as follows:where , and 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 is added to the model. , , , and 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 , respectively. The migration rates of susceptible, exposed individuals, infectious, and removed individuals from patch to patch are denoted by , and , respectively. It is clear that all these parameters are nonnegative.
Remark 1. Different with the model in , 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 ). Let , where if , and the sum of each column element of the matrix is positive, then the matrix is a nonsingular matrix and .
Lemma 1. System (1) has unique disease-free equilibrium.
Proof. By the definition of the disease-free equilibrium, substituting into (1), we haveThen,that is,whereSince and are nonsingular -matrix, then and . Therefore, (3) has unique solution and .
Thus, system (1) has unique disease-free equilibrium .
The proof is completed.
Lemma 2. , is a positively invariant set for system (1), where and .
Proof. In term of system (1), we haveTherefore,where is the initial population.
Therefore, if and only if .
From the first equation of system (1), we havewhere is the -th element of .
Therefore, if .
The proof is completed.
Based on Lemma 2, the dynamical properties of system (1) are studied only in . Now, we are in the position to discuss the basic reproduction number of system (1).
Define and , whereBy using the method in Driessche and Watmough , the basic reproduction number can be determined by computing the spectral radius of the matrix as follows:that is,
Lemma 3 (see ). If is a nonnegative matrix and is a nonsingular matrix, then and .
Theorem 1. The disease-free equilibrium is globally asymptotically stable if , and the disease-free equilibrium is unstable if .
Proof. From the second equations and the third equations of system (1), we have, that is, .
Since is a nonnegative matrix and is a nonsingular matrix, it follows from Lemma 3 thatThus, the disease-free equilibrium is unstable if
Next, we prove is globally asymptotically stable if .
Sincethen is similar to . Therefore,Since is a nonnegative irreducible matrix, by Perron–Frobenius theorem , has a positive left eigenvector andLet , thenLet be a candidate Lyapunov function, whereThen, calculating the derivative of W along the solution of system (1), we havewhere and .
If , then , and if and only if , .
Thus, is globally asymptotically stable if .
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 with is restated is as follows:
It is assumed that , where is the migration rate from patch to patch . The migration matrix is , where .
4.1. Fitting on the Transmission of COVID-19
Let represent Hubei province, Chongqing, and Hunan province in system (19), respectively. The values of and are taken as the average death rate and cure rate of confirmed infected persons in patch from January 20, 2020, to February 1, 2020. The values of and are taken as the average death rate and cure rate of confirmed infected persons in patch 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 , and . Moreover, represent the corresponding parameter value of the -th stage, where The parameters of system (19) are shown in Table 1.
The numerical fitting diagram is shown in Figure 2. It can be found that the simulated curve is approximately the same as the actual data.
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 Figure 3 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.
The effect of quarantine strategy on COVID-19 transmission is shown in Figure 4 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 black solid represents quarantined rate of exposed individuals where ; and blue solid represents that both kinds of quarantine strategies are performed simultaneously where Figure 4(d) shows that decreases with the increase in quarantined rate. The simultaneous execution of both quarantine strategies has the greatest impact on . 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.
In this paper, based on epidemiological characteristics of COVID-19 and government intervention strategy, an n-patch SEIR epidemic model is presented. It is shown that system (1) has unique disease-free equilibrium Moreover, is globally asymptotically stable , and it is unstable if . 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.
The data are available on Hubei Provincial Bureau of Statistics website, Chongqing Bureau of Statistics website, Hunan Provincial Bureau of Statistics website, Health Commission of Hubei Province website, Health Commission of Chongqing website, and Health Commission of Hunan Province website. Detailed links are given in references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported partly by National Natural Science Foundation of China under Grants 61673080, 61773321, and 61773083, partly by the Science and Technology Research Program of Chongqing Municipal Education Commission under Grant KJZD-K202000601, partly by Natural Science Foundation of Chongqing under Grant cstc2019jcyj-msxmX0102, and partly by Venture & Innovation Support Program for Chongqing Overseas Returnees under Grant cx2017099.
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