Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method.

As the COVID-19 epidemic has entered the normalization stage, the task of prevention and control remains very arduous. This paper constructs a time delay reaction-diffusion model that is closer to the actual spread of the COVID-19 epidemic, including relapse, time delay, home quarantine and temporal-spatial heterogeneous environment that affect the spread of COVID-19. These factors increase the number of equations and the coupling between equations in the system, making it difficult to apply the methods commonly used to discuss global dynamics, such as the Lyapunov function method. Therefore, we use the global exponential attractor theory in the infinite-dimensional dynamic system to study the spreading trend of the COVID-9 epidemic with relapse, time delay, home quarantine in a temporal-spatial heterogeneous environment. Using our latest results of global exponential attractor theory, the global asymptotic stability and the persistence of the COVID-19 epidemic are discussed. We find that due to the influence of relapse in the in temporal-spatial heterogeneity environment, the principal eigenvalue λ * can describe the spread of the epidemic more accurately than the usual basic reproduction number R 0 . That is, the non-constant disease-free equilibrium is globally asymptotically stable when λ * < 0 and the COVID-19 epidemic is persisting uniformly when λ * > 0 . Combine with the latest official data of the COVID-19 and the prevention and control strategies of different countries, some numerical simulations on the stability and global exponential attractiveness of the spread of the COVID-19 epidemic in China and the USA are given. The simulation results fully reflect the impact of the temporal-spatial heterogeneous environment, relapse, time delay and home quarantine strategies on the spread of the epidemic, revealing the significant differences in epidemic prevention strategies and control effects between the East and the West. The results of this study provide a theoretical basis for the current epidemic prevention and control.

Introduction

In the Spring Festival of 2020, a sudden outbreak of a new coronavirus (COVID-19) spread rapidly in China, centered on Wuhan, and diffuse extremely fast [8], [20], [34]. The Chinese government has quickly taken active public health interventions, including blocking Wuhan and surrounding cities, closely tracking contacts across the country and advising contacter to be quarantined at home. These measures are unprecedented. The WHO also highly recognizes the contributions made by the Chinese government and the Chinese people to control the COVID-19 epidemic. Recently, the COVID-19 epidemic in the United States, Brazil, and India have become intensified, with new confirmed cases and deaths all remaining high [5], [17], [19]. The global epidemic prevention and control work is still very difficult. The COVID-19 epidemic is raging around the world, and medical workers and researchers from all over the world are also working hard to prevent and control the COVID-19 epidemic.

The main reason for the rapid spread of the COVID-19 epidemic in China is the conditions of climate and temperature from January to March are suitable for the spread of coronavirus. The return visitor flow and gatherings during the Spring Festival caused a rapid increase in population density and contact in a small area. Hence, both spatial and temporal factors have a significant impact on the spread of the COVID-19 epidemic. At the end of July, the emergence of multiple confirmed cases of Dalian in Liaoning Province and Urumqi in Xinjiang Province once again sounded the alarm for the prevention and control. Recently, the epidemic in Brazil has been spreading obviously, and the epidemic in Japan has also appeared to recur. When we carefully observe the above-mentioned countries and regions, we can find that Dalian and Japan have relatively high latitudes. Urumqi has a large temperature difference between day and night. Most of Brazil’s territory is located in the southern hemisphere. The tropical rainforest climate, savanna climate and monsoon climate determine that the average temperature in Brazil is not high. The temperature in these places is still suitable for the propagation of coronavirus. These once again show that the temporal-spatial heterogeneous environment has a great influence on the spread and diffusion of the COVID-19 epidemic.

According to the latest data released by the Chinese Health Commission, the cure rate of COVID-19 epidemic in China is as high as 80%. China has adopted effective prevention and control measures such as home quarantine, blockade of traffic, and wearing masks, the COVID-19 epidemic has been effectively controlled 30 days after the outbreak. The prevention and control of COVID-19 have achieved preliminary results in China. Among them, home quarantine played a vital role. Some other countries such as Germany, Italy and France have also begun to learn this method of prevention and control in China. In contrast, the United States and India have not home quarantine and other prevention and control measures, and the number of new confirmed cases and deaths is very high every day. Therefore, home quarantine is the second key factor affecting the large-scale spread of the COVID-19 epidemic.

As we all know, COVID-19 has an incubation period of 7-14 days, which may be longer for some individuals. The incubation period refers to the period from when a pathogen invades the human body to the earliest clinical symptoms appear. Since the time delay effect of the new coronavirus in the human body causes the incubation period of COVID-19, and patients in the incubation period are not subject to any restrictions in their daily lives because they have not incidence, they are more likely to contact susceptible persons. Hence, it is necessary to consider the time delay effect of infected persons during the spread of the epidemic.

The fourth key factor affecting epidemic prevention and control is relapse. Since the end of February 2020, there have been cases of relapse in China. In countries and regions with high incidence, there will be more cases of relapse. At the end of August, the WHO reminded high-risk countries and regions to be alert to the emergence of relapse cases, because relapse is often accompanied by virus mutation. Hence, considering the relapse factors is helpful for normalized epidemic prevention and control.

In view of the importance of the above four factors to the spread of the COVID-19 epidemic, we are prompted to build a mathematical model based on the above factors to study the long-term dynamics of the spread of the COVID-19 epidemic.

Most of the current COVID-19 epidemic research is based on ordinary differential equations. In Çakan’s study [3], a new SEIR epidemic model formed by taking into account the impact of health care capacity has been examined and local and global stability of the model has been analyzed. Wang et al. [24] used mathematical models to study the pathogenic features of SARS-CoV-2 infection by examining the interaction between the virus, cells and immune responses. Using the model they also evaluate the effect of several potential therapeutic interventions for SARS-CoV-2 infection. The authors of [2] used actual data to study the spread of the COVID-19 epidemic in Ghana, calculated the basic reproduction number of the model, and analyzed the global asymptotic stability of the model by constructing the Lyapunov function. An optimal control was formulated based on the sensitivity analysis and numerical simulation of the model was also done to verify the analytic results. A similar method of calculating the basic reproduction number and discussing global stability also appeared in Reference [9]. Feng et al. conducted a numerical simulation of the outbreak in the UK and gave corresponding prevention and control recommendations. Zhou [27] paid attention to the influence of the media on the spread of the epidemic. They discussed the local stability of the model with the help of the Hurwitz discriminant method and analyzed the sensitivity of some important parameters. The advantage of the above-mentioned ordinary differential models is that they intuitively describe the overall trend of epidemic spread. However, key details such as the spatial heterogeneous environment and time delay that affect the epidemic are not considered. In addition, for ordinary differential systems, Lyapunov function method and Hurwitz discriminant method are the most commonly used methods to discuss stability. Among them, the Hurwitz discriminant method can only determine the local asymptotic stability of the system, and the Lyapunov function method is used to determine the global asymptotic stability. However, when there are too many equations in the system or the coupling between equations is enhanced, the construction of Lyapunov functions will become more difficult. There are also many researchers who are good at data analysis and epidemic forecasting [6], [11], [14], [19]. There are also many researchers that share their data on the Internet [13], [15]. However, the support of theoretical research in these data analysis literatures is relatively small, and the accuracy of long-term prediction of the development of the epidemic will be reduced. Altan and Karasu [1] developed a hybrid model consisting of two-dimensional (2D) curvelet transformation, chaotic salp swarm algorithm (CSSA) and deep learning technique in order to determine the patient infected with coronavirus pneumonia from X-ray images. Their experimental results show that the proposed hybrid model can diagnose COVID-19 disease with high accuracy from chest X-ray images. This research perspective combining mathematical theory and medical technology is novel. This work provide a more accurate and effective method for detecting COVID-19 infection. However, the long-term dynamics of COVID-19 prevention and control are not considered.

For the coupled reaction-diffusion equations of spatial heterogeneity, the Lyapunov function method is almost invalid. At this time, we need to use the global attractor theory of infinite-dimensional dynamical systems to discuss the global dynamics of the system. It is well-known that the global attractor is an important theory to describe the long-term dynamic behavior of dissipative infinite dimensional dynamical systems [12], [18], [28], [29], [30], [31]. In [32], we discussed a class of reaction-diffusion SVIR model with relapse and a varying external source in spatial heterogeneous environment. We studied the long-term dynamic behavior of this model by means of global exponential attractor theory and gradient flow method, but the diffusion coefficient and are constant number. In 2020, we introduce a method of global exponential attractor in the reaction-diffusion epidemic model in spatial heterogeneous environment to study the spread trend and long-term dynamic behavior of the COVID-19 epidemic [33].

In summary, the research on the global dynamics of the COVID-19 model in a heterogeneous temporal-spatial environment is still almost blank. Research on the COVID-19 model in a spatially heterogeneous environment is also in its infancy, and there are not many research results. In the temporal-spatial heterogeneous environment, what long-term dynamics will the reaction-diffusion COVID-19 model with home quarantine, time delay and relapse has? What is the difference between the dynamical behavior in a temporal-spatial heterogeneous environment and a spatial heterogeneous environment? These are the questions to be studied in this manuscript. In this manuscript, we first study a reaction-diffusion COVID-19 model with home quarantine, time delay and relapse in the temporal-spatial heterogeneous environment. Except for the diffusion coefficient, all coefficients of this model are temporal-spatial heterogeneous. Then, apply the result of the global exponential attractor we got recently in [32] and gradient flow method, the global stability of the disease-free equilibrium of the model and the persistence of the COVID-19 epidemic are discussed. We find that due to the role of temporal-spatial heterogeneity, especially the effect of relapse in temporal-spatial heterogeneity environment, the usual basic reproduction number has not been able to fully describe the spread of disease. However, we find that the principal eigenvalue has threshold characteristics, which can better describe the spread of the COVID-19 epidemic in temporal-spatial heterogeneity environment.

The organization of this paper is as follows. In Section 2, we first construct a time delay reaction-diffusion COVID-19 model with home quarantine and relapse in the temporal-spatial heterogeneous environment under some assumption and prove the existence of the positive solutions. Secondly, we give the uniform boundedness and the existence of global solutions to our system. Third, we discuss the global stability and the persistence of the COVID-19 epidemic by using the existence theorem of global exponential attractor. That is, we prove that the non-constant disease-free equilibrium is globally asymptotically stable when and the COVID-19 epidemic is persisting uniformly when where is the principal eigenvalue. In Section 3, we simulate the spread trend and the impact on the temporal-spatial heterogeneous environment of the time delay COVID-19 epidemic by the official data in China and the USA. By adjusting the temporal-spatial heterogeneous parameters, we can see the impact of the temporal-spatial heterogeneous environment on the diffusion of the COVID-19 epidemic. Combining the respective prevention and control strategies of China and the United States, and comparing the cumulative number of confirmed cases and cumulative deaths between the two countries, we can find that home quarantine measures are very effective in preventing and controlling the COVID-19 epidemic. In particular, under the assumption that 30% of Americans adopt home quarantine measures, we also perform a simulation and the results show that four months are sufficient to effectively control the COVID-19 epidemic in the United States. In Section 4, we give our conclusions and some discussions.

Long-term dynamic behavior of a time delay reaction-diffusion COVID-19 epidemic model with home quarantine and relapse in temporal-spatial heterogeneous environment

In this section, we consider a time delay reaction-diffusion COVID-19 model with home quarantine and relapse in temporal-spatial heterogeneous environment. Our model is divided into six compartments, namely susceptible individuals (), exposed individuals (), quarantined individuals in home (), infected individuals (), quarantined individuals in hospital () and temporary restorers (). The parameters description and transfer diagram as shown in Table 1 and Fig. 1 .

Table 1

State variables and parameters of COVID-19 () model.

Parameter	Description
S(x,t)	Density of susceptible individuals at location x and time t.
E(x,t)	Density of exposed individuals at location x and time t.
H(x,t)	Density of quarantined individuals in home at location x and time t.
I(x,t)	Density of infected individuals at location x and time t.
Q(x,t)	Density of quarantined individuals in hospital at location x and time t.
R(x,t)	Density of temporary restorers at location x and time t.
Λ(x,t)	Total recruitment scale into this homogeneous social mixing community at location x and time t.
β_i(x,t),i=1,2	Contact rate at location x and time t.
α(x,t),ω(x,t)	Incidence rate at location x and time t.
δ(x,t)	Home quarantine rate at location x and time t.
γ(x,t)	Quarantined rate at location x and time t.
ρ_i(x,t),i=1,2	Relapse rate at location x and time t.
ϕ(x,t),σ(x,t)	Per-capita recovery (treatment) rate at location x and time t.
μ(x,t)	Natural mortality rate at location x and time t.
η_i(x,t),i=1,2,3	Disease-related death rate at location x and time t.
d_S(x),d_I(x),d_R(x)	Diffusion rate at location x.

Fig. 1

Transfer diagram for the time delay COVID-19 () model with relapse in temporal-spatial heterogeneous environment.

From Fig. 1, the following system with the initial-boundary-value conditions is constructed by:Here, is a bounded domain in with smooth boundary (when ), are positive, continuous and uniformly bounded diffusion coefficients depend on space, are positive Hölder continuous functions on accounting for the total recruitment scale, rates of contact, relapse, incidence, quarantined, recovery, natural death and disease-related death respectively. Neumann boundary conditions denotes that the change rate on the boundary of the region is equal to 0. It is straightforward to verify that is a Lipschitz continuous function of and in the open first quadrant. Therefore, we can extend it to the entire first quadrant by defining it to be zero whenever or . Throughout the paper, we assume that the initial value and are nonnegative continuous functions on and the number of infected individuals is positive, i.e., . Since the population in compartments and are quarantined, so we do not consider the diffusion of them in this article. Specific parameters described in Table 1.

Threshold dynamics of the time delay reaction-diffusion COVID-19 model with home quarantine and relapse in temporal-spatial heterogeneous environment

In this section, we will analyze the qualitative behavior of system (2.1). It is clearly see that system (2.1) admits a disease-free equilibrium .

In order to further study the long-term dynamic behavior of the time delay diffusive COVID-19 model with home quarantine and relapse in temporal-spatial heterogeneous environment, we need to prove the existence of principal eigenvalues of system (2.1). If linearizing the second, the third, the forth, the fifth and the sixth equations of (2.1) at disease-free equilibrium, we getLet Eqs. (2.2) can be rewritten asDenote andwhereand . Therefore, Eqs. (2.3) can be rewritten asBy Krein–Rutman theorem, we can obtain that there exists a real eigenvalue of Eq. (2.4) and a corresponding eigenvector satisfying for all in the case of Neumann boundary conditions. To further discuss the principal eigenvalue of time delay systems, by a similar argument in [21, Theorem 2.2], we give the following lemma.

There exists a principal eigenvalueof(2.1)associated with a strictly positive eigenvector, and for anyhas the same sign as.

The proof of Lemma 2.1 is similar to the proof of [21, Theorem 2.2]. By Lemma 2.1, we can obtain that there exists a principal eigenvalue of Eq. (2.1) and a corresponding eigenvector satisfying for all in the case of Neumann boundary conditions.

Existence and dimensions of global exponential attractor and stability analysis of system (2.1)

In this section we use the global exponential attractor theory to discuss the long-term dynamic behavior of the time delay reaction-diffusion COVID-19 model with home quarantine and relapse in temporal-spatial heterogeneous environment.

From now on, we denote that and . Note that and are Banach spaces equipped with normandFor any given continuous function on we denote

In the following, we adiscuss the existence, positivity and boundedness of the global solution of the system (2.1).

For each, , system(2.1)exists a positive and bounded global solution.

Since , is a symmetrical sectorial operator and all eigenvalues of are wherebe quasimonotone and satisfy the locally Lipschitz conditions. Sinceso by [22, Theorem 11.3.5] and [26, Theorem 2.3], we can guarantees that system (2.1) exists a global solutionAccording to the method in [32, Lemma 2.1 and Theorem 2.2], we can easily discuss the positivity of the global solution of the system (2.1). We next consider the following total population at time . DefineFrom system (2.1), it is easy to see thatBy Gronwall’s inequality in differential form [32, Lemma 2.2], we can obtain thatSo whereThis shows that is bounded. By the positivity of the solution of the system (2.1), we obtain thatWe denote that then we knowIn view of [10, Theorem 1 and Corollary 1], there exists a positive constant depending on such thatThus, we can obtain that are uniformly bounded on . This implies the global solution of system (2.1) is positive and uniformly bounded. □

System(2.1)exists a global exponential attractorit exponential attracts any bounded set in.

For the system (2.1), we first verify the [33, condition (2.3)]. Since From Theorem 2.2, we know is uniformly bounded, hence, [33, condition (2.3)] holds. In addition, there exists constants such thatSimilarly, If we chooseand denote thenHence, Lipschitz condition is well verified. As we know that is a symmetrical sectorial operator and all eigenvalues of aretherefore, by [33, Lemma 2.5], system (2.1) exists a invariant set, it exponential attracts any bounded set in . By [33, Theorem 2.7], we can obtain that system (2.1) has a global exponential attractor and . □

After getting the global exponential attractor, we discuss the stability and persists uniformly of the epidemic disease.

The following statements are valid.

Iftheninand hence, the disease-free equilibrium is globally asymptotically stable.

Ifthen there exists a functionindependent of the initial data, such that any solutionsatisfiesforand hence, the disease persists uniformly

Suppose . We will use the comparison principle to show that as for every . First, we observe from the system (2.1) thatNext, let us define where are the eigenvalue and eigenvectors in Eqs. (2.3) and is chosen so large that for every . It can be shown that satisfiesBy the comparison principle [23, Lemma 5.2.1], for every and . Since as for every we also have that as for every

Next we claim uniformly on as . Given any small constant there exists a large time such that for all . From the first equation in system (2.1), it is observed that is a super-solution toand a sub-solution toDenote by and the solution of system (2.5) and system (2.6), respectively. The parabolic comparison principle givesFor system (2.5), we can verify thatthis means that system (2.5) satisfies [33, condition (2.3)]for . It is similar to the proof of Theorem 2.3, we can obtain that system (2.5) exists a global exponential attractor . In addition, system (2.5) has a variational structure, the corresponding functional of the variational structure iswhereThenso is a gradient type operator. From [12, Theorem A.2.2], we can prove thatwhere is the unique positive steady state of problems (2.5). Similarly, for system (2.6), we can obtainwhere is the unique positive steady state of problems (2.6). Furthermore, due to the arbitrariness of , it is easily checked thatThus, our analysis implies that the uniformly as .

Since it is observed that the solution ofis a sub-solution of the first equation in (2.1). Similar to the proof of conclusion (1), system (2.7) is also a gradient type equation. From [12, Theorem A.2.2], we can prove thatBy weak maximum principle, we know that for all Next, let us definewhere and is a sufficiently small constant. Substituting into the second, the third, the forth, the fifth and the sixth equations of system (2.1), we know andTherefore, is the sub-solution of the second, the third, the forth and the fifth equations of system (2.1). We choose , we can obtain thatfor then it shows that the disease persists. □

The synthesis of Theorem 2.3 and Theorem 2.4 can give the following results:

If and then the positive solution of system (2.1) is globally exponential attractive and the attraction domain is . Medically speaking, the disease persists in this situation. The epidemic is globally asymptotically stable or attracted by the global exponential attractor.

Numerical simulation and comparison

Numerical simulation of the COVID-19 epidemic in China

The COVID-19 was named by the World Health Organization on January 12, 2020. During the Spring Festival in 2020, the epidemic broke out in China with Wuhan as the center and began to diffuse. The Chinese government quickly adopted a strong response. Since January 23, 2020, Wuhan has implemented the strategy of closing the city, and various regions have adopted strategies such as restricting travel to control the spread of the epidemic. On January 30, 2020, WHO released a public health emergency of international concern: COVID-19 pneumonia. The WHO emphasized that travel and trade are not recommended during the epidemic and affirmed the control measures of the Chinese government. Home quarantine is a very effective method. China’s rapid control of the diffusion of the epidemic has benefited formally from this measure.

The official website of the National Health Committee of the People’s Republic of China has updated the relevant data of the COVID-19 epidemic [16] from January 24, 2020. The official website of the WHO can check the relevant data from January 21 to the present [4]. Based on these official data, with the help of simple data analysis and calculation, we can get some important parameters in Table 2 .

Table 2

The parameters description of the COVID-19 epidemic in China.

Parameter	Data estimated	Data sources
Λ	80000	Estimate
β_1	0.6	Estimate
β_2	0.3	Estimate
α+ω	0.423	References[16]
γ	0.798	References[16]
ρ_1	0.001	Estimate
ρ_2	0.002	Estimate
ϕ	0.8	References[16]
μ	0.1595	References[25]
η_1	0.021	References[16]
η_2	0.021	References[16]
η_3	0.157	References[16]
δ	0.7	Estimate
σ	0.6	Estimate
τ	1	Estimate
d_S	2	Estimate
d_E	1	Estimate
d_I	0.3	Estimate
d_R	2	Estimate

According to the data in Table 3 and our system (2.1), we first simulate the spread of the COVID-19 epidemic in China (Fig. 2 ).

Table 3

The parameters description of the COVID-19 epidemic.

Date	Total	Total confirmed	Total deaths	Total
	confirmed cases	new cases		new deaths
2020.7.8	2923432	46194	129963	320
2020.7.7	2877238	43686	129643	235
2020.7.6	2833522	57186	129408	182
2020.7.5	2776366	51993	129226	745
2020.7.4	2724433	53213	128481	623
2020.7.3	2671220	54271	127858	725
2020.7.2	2616949	43556	127113	560
2020.7.1	2573393	35757	126573	370

Fig. 2

The spread of the COVID-19 epidemic in China.

From Fig. 2, we can see that numerical simulation and official data is relatively consistent, and that the spread of the disease is spatially dependent. In addition, we also find that COVID-19 showed a high incidence and extremely low cure rate at the beginning of the outbreak. In such a period, the number of quarantines in the hospital will be large, and the sickbeds in the hospital will be very tight.

The Chinese government has tried to curb the spread of the epidemic by blocking cities and traffic. The essence is to control the exposure of the disease. If we choose in Table 3, then we can get the image in Fig. 3 .

Fig. 3

The spread of the COVID-19 epidemic when .

From the Fig. 3 we can see that merely controlling the contact rate between susceptible and infected people can only delay the time of getting sick, and cannot obviously reduce the number of people infected. If we hope to obtain a better control, we need to control the contact rate between the susceptible and the exposed individuals, because there are more people in the exposed period and their activities are more frequent. If we choose in Table 3, then we can get a simulated image of the disease-free equilibrium of constant coefficient model (Fig. 4 ).

Fig. 4

The global stability of disease-free equilibrium of constant coefficient model.

We can see from the Fig. 4 that population density depends on spatial location before disease dies and if people’s travel is restricted and the contact rate between susceptible and incubation period patients is controlled, the epidemic will be effectively controlled and eventually die out. At this time, the disease-free equilibrium of the COVID-19 epidemic is globally asymptotically stable. Therefore, it is very correct for the Chinese government to restrict people’s access through methods such as traffic blocking and forced quarantine. From another point of view, everyone actively takes home quarantine and reduces contact with each other, which can effectively control the spread of the epidemic.

Next, we discuss the dynamics of the epidemic model with a temporal-spatial heterogeneous environment and relapse, so we then simulate the stability and persistence of the temporal-spatial system (2.1). From the system (2.1) we can see that all the parameters are temporal-spatial related functions, so we choose different functions will directly lead to different stability results. If we choose and select other parameters from Table 3, then we can clearly see that the disease-free equilibrium of the temporal-spatial heterogeneity epidemic is globally asymptotically stable (Fig. 5 ).

Fig. 5

The global stability of disease-free equilibrium of temporal-spatial heterogeneity system (2.1) when .

According to our discussion in the previous Section, the diffusion coefficient is highly dependent on space and time. The spread of viruses and human activities are subject to conditions such as population density, climate and temperature. From this perspective, the diffusion coefficient should have periodic characteristics. Hence, from now on, we choose . If we then choose and select other parameters from Table 3, then we can clearly see that the temporal-spatial heterogeneity epidemic is persists uniformly (Fig. 6 ).

Fig. 6

The global stability of endemic of temporal-spatial heterogeneity system (2.1) when .

In Fig. 6, we find that the fluctuation of the image shows an irregular and unsmooth state. This irregular and unsmooth phenomenon is caused by the spatial heterogeneity of the diffusion coefficient. However, this irregular fluctuation is still in a controllable range. And this range is also the scope of the global exponential attractor.

From all above simulation we can clearly see that both the disease-free equilibrium and endemic equilibrium of the constant-coefficient model are global asymptotic stability. For temporal-spatial heterogeneous models, the stability of the model is dependent on the temporal-spatial parameters and diffusion coefficients. In addition, we also see that the image whether rising or falling is very fast in the initial stage, and then fast tend to stability. It is also confirmed that the solutions of the system (2.1) are global exponentially attractive.

Numerical simulation of the COVID-19 epidemic in the USA

The United States is a country that has not adopted any home quarantine measures. The following table is the relevant data of the COVID-19 epidemic situation in the United States since July (from July 1st to July 10th) extracted from the WHO official website [4].

From Table 4 , we can see that the cumulative number of confirmed cases and cumulative deaths in the USA are very large. By summarizing WHO data, we found that the outbreak of the USA epidemic occurred on March 4, 2020. Using these official data, we have plotted the total confirmed cases and total deaths in the USA from March 4th to July 10th.

Table 4

The parameters description of the COVID-19 epidemic in the USA.

Parameter	Data estimated	Data sources
Λ	18000000	Estimate
β_1	0.75	Estimate
β_2	0.6	Estimate
α	0.088	References [7]
ω	0.3	Estimate
γ	0.001072	References [7]
ρ_1	0.1	Estimate
ρ_2	0.2	Estimate
ϕ	0.35	Estimate
μ	0.1595	References [25]
η_1	0.055	References [7]
η_2	0.055	References [7]
η_3	0.049	References [4]
δ	0.3	Estimate
σ	0.6	Estimate
d_S	2	Estimate
d_E	1	Estimate
d_I	0.3	Estimate
d_R	2	Estimate

We can see from Fig. 7 that the epidemic has not been effectively controlled after spreading in the United States for more than four months. The number of confirmed cases and deaths is still increasing. Although the growth trend seen on the graph is relatively stable, the number of daily increases is quite large. The reason for this phenomenon is because the USA government has not taken effective measures to control the spread of the epidemic. In the USA, in addition to the diagnosed patients receiving treatment in hospitals, other people’s travel, work, and life are not subject to any restrictions, and defensive measures to wear masks are not taken in daily activities. This greatly increases the probability of contact between incubation patients and susceptible persons, making the incidence rate high.

Fig. 7

Total confirmed cases and total deaths in the USA from March 4th to July 10th.

The Centers for Disease Control and Prevention publishes weekly summary of the COVID-19 epidemic in the USA [7]. The latest weekly summary shows that the incidence of the USA epidemic has dropped to 8.8%, the mortality rate has dropped to 5.5%, and the rate of isolation treatment in hospitals is 107.2/100000. If the USA also has some measures to quarantine the house, can the epidemic be alleviated accordingly? We select the home quarantine rate of 0.3 and get the following data table.

According to the data in Table 4, we simulate the spreading trend of infected persons. We can see from Fig. 8 that if the United States also adopts some measures for home quarantine, the epidemic will not be as difficult to control as it is now. As shown in Fig. 8, if 30% of Americans are quarantined at home, the infected person will tend to stabilize in about 120 days. The COVID-19 epidemic was effectively controlled at this time.

Fig. 8

Total confirmed cases and total deaths in the USA from March 4th to July 10th.

Comparison of the spread of the epidemic in China and the United States—the effectiveness of temporal-spatial heterogeneity and home quarantine

We know that China implemented strict home quarantine measures during the outbreak period. And the life of American residents is the same as usual. Comparing the spread of the COVID-19 epidemic in these two countries, we can find the effect of home quarantine measures. The outbreak of the COVID-19 in China began on January 2020. With the help of official data from the National Health Committee of the People’s Republic of China [16], we draw the total confirmed cases and total deaths in China from January 24th to March 31st (Fig. 9 ).

Fig. 9

Total confirmed cases and total deaths in China from January 24th to March 31st.

From Fig. 9 we can clearly see that the number of confirmed cases in China began to stabilize 30 days after the outbreak. The deaths also stabilized 50 days after the outbreak. This phenomenon is in sharp contrast with the current spread of the epidemic in the United States in Fig. 7. We have to admit that home quarantine measures are effective in preventing and controlling the epidemic.

As of October 2020, the epidemic in the United States has not been effectively controlled. US President Trump was once infected with the COVID-19 virus. On the other hand, in China, most people have taken off their masks and returned to normal lives. Therefore, it is recommended that American people strengthen self-protection, reduce going out as much as possible, and conduct home quarantine.

China is currently one of the countries with the best control of the COVID-19 epidemic. Home quarantine measures have played a vital role in the prevention and control of the entire epidemic in China. The model we have constructed highlights the importance of temporal-spatial heterogeneity and home quarantine. From the numerical simulation, we can see that the simulation results of the epidemic in China are highly consistent with the actual official data, which indicates that our model is in line with the actual development of the epidemic and is a successful modeling.

In [33], we performed numerical simulations of the COVID-19 model without home quarantine in a spatially heterogeneous environment, and the simulated images showed a flat state regardless of rising or falling. However, the images of system (2.1) in this article have an obvious effect of space-time heterogeneity, that is, the images are not horizontal from the perspective of the spatial axis. This phenomenon is the result of the interaction of time and space factors in the epidemic and is more consistent with the actual spread of the epidemic.

Conclusions and discussion

In China, home quarantine measures have achieved great results during the prevention and control period of the COVID-19 epidemic. This effect has also attracted widespread attention from other countries and have followed suit. We first formulate a time delay reaction-diffusion COVID-19 model with home quarantine and relapse in the temporal-spatial heterogeneous environment and prove the positivity, uniform boundedness and existence of solution of the epidemiological model with temporal-spatial heterogeneous and relapse. Then applying the global exponential attractor method for dissipative evolution system to the COVID-19 model, we can get the existence of the attractor of the epidemic model. In addition, we prove that the disease-free equilibrium is global asymptotically stable and the endemic is persisting uniformly. Due to the influence of temporal-spatial heterogeneity relapse rate, it is no longer possible to rely on the basic reproductive number to describe the spread of epidemics. Therefore, we describe the phenomenon of disease transmission through the principal eigenvalue of the system. From our proof, we can find that has threshold characteristics. Finally, we perform numerical simulations of the spread trend of the COVID-19 epidemic in China and the USA recently. With these authoritative data, we also simulate the diffusion of the epidemic in the temporal-spatial heterogeneous environment. Simulation results show that the best way to eliminate the disease is to control the contact rate, and home quarantine is a very effective measure. Choosing different temporal-spatial coefficients can produce a fundamental change of the global dynamics of the system.

Recently, due to the continuous drop in temperatures in the northern hemisphere, the COVID-19 epidemic has shown a resurgence in Europe and the United States. The UK has recently seen a large number of confirmed cases due to statistical errors. Italy increased 22,000 confirmed cases in a single day. The German Chancellor announced that some cities in Germany have implemented a light lockdown strategy. Political dignitaries and sports athletes from many countries in Europe and America were once infected with the new crown. These events just reflect that our research in this article is in line with the actual situation, but also has great practical significance. In view of the current actual situation and our research results, there are still some issues worthy of further discussion. For example, will the new coronavirus mutate according to climate change, geographical movement of population, and disease relapse? What is the impact of the virus mutation on the spread of the epidemic? What measures should be taken to predict or avoid virus mutation in advance? Will eating habits in different regions affect the spread of the COVID-19 epidemic? Can strengthening the control of ecological and food safety issues control the spread of the epidemic? How do the above factors affect the long-term dynamics of the epidemic through mathematical models?

CRediT authorship contribution statement

Cheng-Cheng Zhu: Conceptualization, Methodology, Software, Formal analysis, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization, Funding acquisition. Jiang Zhu: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.