Modelling for the COVID-19 with the contacting distance

Background: The COVID-19, which belongs to the family of Coronaviridae and is large-scale outbreak in the whole world, is a public health emergency for human beings and brings some very harmful consequences in social and economic ﬁelds. For COVID-19, the contacting distance between the healthy individuals and the asymptomatic or symptomatic infected individuals is an important factor in its spread. Therefore, the contacting distance must be considered in order to modelling the COVID-19 and develop the eﬃcient control method corresponding to the contacting distance, Methods: This paper proposes an SEIR-type epidemic model and incorporates the contacting distance between the healthy individuals and the asymptomatic or symptomatic infected individuals into the proposed model based on the spread mechanism of the COVID-19. The proposed model with the contacting distance is analysed by the methods of diﬀerential dynamical systems and the realistic meanings are revealed. Methods: Results: Firstly, the results show that the COVID-19 will be controlled while the contacting distance between the healthy individuals and the symptomatic infected individuals is larger than the threshold value ˜ d ∗ and the immigration rate is smaller than the threshold value ˜ A ∗ . Secondly, the results show that the contacting distance and the immigration rate play an important role in controlling the COVID-19 by the sensitivity analysis. Finally, the results show that the extinct lag decreases as the the contacting distance increase or the immigration rate decrease by the numerical test for Wuhan city. Conclusions: This paper considered some critical factors which impact the spread of the COVID-19 and the threshold values for these critical factors are obtained to control the the COVID-19. Our study could give some reasonable suggestions for the health oﬃcials and the public.

cases in Europe, 265164 cases in Eastern Mediterranean, and so all. The whole world has conducted many effective measures to control the COVID-19 and it is reasonable to think that the COVID-19 will be completely controlled under the effective work of the centers for disease control and prevention, the medical workers.
The COVID-19 has already caused a huge economic loss due to national lockdowns, travel restrictions, and global distractions of trade and manufacturing chains, and a large number of deaths due to the shortage of medical resources. Therefore, the COVID-19 is a seriously public health emergency for the whole world and brings some very harmful consequences in social and economic fields (Chen et al., 2020; WHO Situation Report-112, 2020; Zhao et al., 2020). In order to prevent and control the COVID-19 effectively, lots of researchers have conducted to study the COVID-19 based on medicine, epidemiology, biology and mathematics (Cheng et (Li et al., 2020) collected the confirmed information on demographic characteristics, exposure history, and illness timelines of laboratory-confirmed cases that had been reported on January 22 th , 2020. They described characteristics of the cases and estimated the key epidemiologic time-delay distributions (the 95th percentile of the distribution at 12.5 days), the epidemic doubling time (DT = 7.4 days) and the basic reproductive number (R 0 = 2.2 (95% CI, 1.4-3.9)). Fortunately, in the early period, this research have successfully characterized and predicted some critical properties of the COVID-19. However, as the development of the COVID-19 and the adoptions of some effective prevention and control measures, the COVID-19 should be re-estimated. Tang et al. (Tang et al., 2020) presented a general SEIR-type epidemiological model and estimated the basic reproduction number by means of mathematical modeling. Their results showed that the control reproduction number might be as high as 6.47 (95% CI, 5.71-7.23). As recognized by the World Health Organization (WHO), the mathematical models, which are employed to describe the COVID-19, play an important role for the disease prevention and control, and the policy formation.
The published researches have shown that the main transmission route of the COVID-19, including all epidemic diseases induced by the coronaviruses, is the respiratory transmission (Kahn and McIntosh, 2005; Hui and Zumla, 2019; Tang et al., 2020; Annas et al., 2020; Boukanjime et al., 2020). Hence, keeping the essential contacting distance between the healthy groups and the infected groups is very important for controlling the COVID-19. contacting distance refers to the extent to which people experience a sense of familiarity (nearness and intimacy) or unfamiliarity (farness and difference) between themselves and people belonging to different social, ethnic, occupational, and religious groups from their own (Bogardus, 1925;Hodgetts et al., 2011;Ma et al., 2020). the contacting distance describes the distance between different groups, and this notion includes all differences such as social class, race/ethnicity or sexuality, but also the fact that the different groups do not mix (Bogardus, 1925;Hodgetts et al., 2011). Therefore, in the mathematically modelling processes, the contacting distance between the healthy groups and the symptomatic infected ones is a critical parameter which determines the control measures for the COVID-19. Motivated by these, this paper will present an SEIRtype model incorporating the contacting distance between the healthy groups and the infected groups, and mainly focus on exploiting the control measures for the COVID-19 based on the contacting distance. This paper will determine the threshold contacting distance between the healthy groups and the infected groups, and the COVID-19 will be ultimately controlled when the contacting distance between the healthy groups and the infected groups is larger than this threshold contacting distance. Otherwise, the COVID-19 may not be extinct. Here, the threshold contacting distance is a critical value, and the COVID-19 will be extinct under this threshold contacting distance and the COVID-19 may be endemic above this threshold contacting distance.

Method
In this section, an epidemic SEIR-type model with the contacting distance between the healthy groups and the symptomatic infected groups are presented to describe the spread of the COVID-19 based the following two steps.
Step 1. Proposing an SEIR-type epidemic model for the COVID-19 with the incidence rate which will be presented in the later step 2. (A 11 ) Based on the spreading mechanism of the COVID-19, the total populations are divided into five subclasses: the susceptible subpopulation (S(t)), the incubation subpopulation (E(t)), the asymptomatic infected subpopulation (I A (t)), the symptomatic infected subpopulation (I(t)) and the recovered subpopulation (R(t)). (A 12 ) The mean migrating rate of susceptible population for a local region, which describes the local outdoor-movements of the susceptible individuals during the spread of the COVID-19, is a constant recruitment A, which is defined in the interval [0, +∞). If the mean migrating rate is larger than zero and smaller than one, it means that individuals can only immigrate into the objective streets from their own home in their own local region. If the mean migrating rate is larger than one, it implies that some parts of individuals immigrate to the objective streets of the local region from the other regions. (A 13 ) The natural death rates of all subpopulation are proportional to their existing densities, and the proportional coefficient is µ. The death rates of the asymptomatic infected subpopulation and symptomatic infected subpopulation are proportional to their existing densities, and the proportional coefficients are µ 21 and µ 22 , respectively. The death rates of the incubation subpopulation could be omitted according to the report of the National Health Commission of the People's Republic of China (NHCPRC) and the World Health Organization (WHO). (A 14 ) The conversion rates from the incubation subpopulation to the asymptomatic infected subpopulation and the symptomatic infected subpopulation are proportional to the former's density and the proportional coefficients are γ 11 and γ 12 , respectively. These conversion rates are mainly determined by the incubation period of the COVID-19. (A 15 ) The recovery rate of the asymptomatic infected subpopulation and symptomatic infected subpopulation are proportional to their densities, and the proportional coefficients are γ 21 and γ 22 , respectively. These are mainly determined by their autoimmunity and the medical level of the local hospitals. At present, the recovery rate of the symptomatic infected individuals prominently increase under the effective medical measures being adopted by the local government.
(A 16 ) The conversion rates from the susceptible subpopulation to the asymptomatic infected subpopulation and the asymptomatic infected subpopulation are defined as the incidence rate which are proposed in the following step 2.
Step 2. According to our published work (Ma et al., 2020) which was done by some authors of this paper, we present the two incidence functions with the contacting distance explicitly for the asymptomatic and symptomatic infected populations induced by the COVID-19 as the following form, respectively.
(IF (I A )) The incidence functions with the contacting distance explicitly for the asymptomatic infected populations where β 1 is the natural infectious rate of the asymptomatic infected indi- (IF (I)) The incidence functions with the contacting distance explicitly for the symptomatic infected populations where β 2 is the intrinsically infectious rate of the symptomatic individuals infected by the COVID-19, h 2 is the average outdoor-time of each symptomatic individuals infected by the COVID-19.
It is reasonable to assumed that β 1 < β 2 and h 1 < h 2 . Based on the above two steps, the epidemic model for the COVID-19 incorporating the contacting distance is presented as the following form: with the initial conditions: Note that the recover variable R(t) does not appear in the first four equations of model (3), this paper focuses on the following subsystem which mainly determine the dynamical behaviors of the original model system: with the initial conditions: 3 Results

The extinct threshold value
In this section, the importantly interesting thing is to consider the dynamical behavior of the disease-free equilibrium since the main focus is to develop the controlling methods for the COVID-19.
Clearly, the disease-free equilibrium point of model (3) P 0 is ( A µ , 0, 0, 0), and we mainly consider the dynamical behavior of the disease-free equilibrium point E 0 .
then the disease-free equilibrium point P 0 is unstable, then the disease-free equilibrium point P 0 is locally asymptotically stable. This threshold value decreases with the decrease in the immigration rate of susceptible population.

Sensitivity analysis
In this section, the numerical test for model (17) is conducted based on the reported cases induced by the COVID-19 in Wuhan in Hubei province and the estimated parameters of the published works (Li et al., 2020;Tang et al., 2020). All parameter values are given in the following Table 1. In order to identify the key factors affecting the change of infection variables, we used Latin Hypercube Sampling (LHS) for parameter sampling and Paranoid Rank Correlation Coefficient (PRCC) method for sensitivity analysis of the parameters in the model (17). When performing parameter sampling, we choose uniform distribution as the prior distribution, sampling times N = 500, the reference values of the parameters in the model are given in Table 1, and the value range is 20% up and down the reference value. The parameters with p value greater than 0.01 are considered to have no effect on the model. From Figure 1(a) and Table 2, it can be seen that parameter A and E(t) are positively correlated, that is, as parameter A increases, the population of E(t) subgroups will increase. µ 1 , γ 12 , d are negatively correlated with E(t), that is, increasing µ 1 , γ 12 , d will cause E(t) to decrease. It can be seen from the Figure   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 2 that the monotonicity of A, µ 1 , γ 12 , d is the most significant, that is, E(t) is mainly affected by these parameters. Figure 1(b) and Table 3 shows that A and γ 12 are positively correlated with I(t), that is, as A and γ 12 increase, I(t) will increase. µ 1 , γ 22 , d and I(t) are negatively correlated, that is, as µ 1 , γ 22 , d increase, the population of I(t) will decrease. It can be seen from the Figure 3 that the monotonicity of A, µ 1 , γ 12 ,γ 22 , d is the most significant, in other words, I(t) is mainly affected by these parameters.
(a) (b) Figure 1: PRCC values of parameters in E(t) and I(t) at t=4000.
It can be seen from Figure 4 that the PRCC value of each parameter changes significantly in the early stage of the outbreak, especially before t = 500. It can be seen from (b) that the parameters A, µ 1 , γ 12 , γ 22 and d have undergone significant changes relative to I(t). The PRCC values of A and d show a trend of first increasing, then decreasing and then increasing over time. This is due to the rapid increase in the number of patients after the outbreak, and the susceptible subpopulation S(t) is more likely to be infected under the same d as the initial outbreak. At this time, social distance d will have a greater impact on the number of patients. With the improvement of people's awareness of the prevention of epidemics and medical conditions, people's behavior of reducing overtime and using masks will lead to less and less impact of d on I(t). Due to the phased results of the epidemic prevention  work, people's awareness of epidemic prevention will become weaker than before. At this time, the influence of A and d on I(t) will increase. Figure 5 and Table 4 show the influence of each parameter on the basic reproduction number R 0 . A, γ 12 , β 2 and R 0 are positively correlated, and γ 12 has a greater impact on R 0 . As γ 12 increases, R 0 will increase rapidly, and more susceptible people will be infected. Among them, µ, γ 22 and d are negatively correlated with R 0 , and the maximum value of PRCCs is |P RCC(d)| = 0.9094, indicating that d has the greatest impact on R 0 . In other words, as long as d is controlled within a certain range, R 0 can be minimized.  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65

Numerical test
According to Proposition 3.3.1 and Proposition 3.3.2, the threshold values of the contacting distanced * and the immigration rate A * are 0.8767 and 0.1849, respectively. That is to say, the COVID-19 will be completely controlled when the contacting distance is larger than the threshold value 0.8767, or the immigration rate smaller than the threshold value 0.1849. Here, the immigration rate measures the density of population who immigrates to the streets from their own home in Wuhan city .   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 Figure 6: The disease-free equilibrium pointP0 is globally asymptotically stable with d = 0.89 which is larger than the threshold value 0.8767.  Figure 7: The time-series diagrams for E and I with the different contacting distances while it is larger than the threshold value 0.8767. Figure 6- Figure 8 show that the COVID-19 will be completely controlled when the contacting distance is larger than the threshold value 0.8767. Figure 6 selects the contacting distance between the susceptible individuals and the symptomatic infected individuals as 0.89 which is larger than the threshold value 0.8767, and hence that the COVID-19 will be vanished. Noting that the parameter of the contacting distance is normalized in the modeling processes, hence, for the COVID-19, it will be completely controlled only if the susceptible individuals and the symptomatic infected individuals keep their contacting distance to be larger than 0.8767 × D, where D is the spread distance of the coronavirus of the COVID-19 when people are breathing and/or sneezing. Generally, individuals are require to keep away at least one meter. Furthermore, Figure 7 reveals that the extinct lag of the COVID-19 closely relates to the contacting distance when the peoples' immigrating number is 30 percent of the total population in Wuhan city after January 23 t h, 2020, and the 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65  extinct lag decreases as the the contacting distance increase. Figure 7(a) reveals that the minimum extinct lag of the incubation infected individuals is about 18 days, Figure 7(b) shows that the minimum extinct lag of the symptomatic infected individuals is about 28 days, since the symptomatic infected individuals must spend at least 7-14 days on being cured in hospitals and recovering in the isolated points. Figure 8 reveals that the extinct lag of the COVID-19 has the close relationship with the peoples' immigration to Whan city, and the extinct lag decreases with the peoples' immigrating number to the streets of Wuhan city decreasing. Figure 8(a) and Figure 8(b) show that the minimum extinct lag of the incubation and symptomatic infected individuals are 18 days and 24 days respectively when the peoples' immigrating number to the streets from their own home in Wuhan city is 29 percent of the total populations of Wuhan city. Moreover, the extinct lag of the COVID- 19   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 (a) (b) Figure 10: The time-series diagrams for E and I with the different contacting distance while it is smaller than the threshold value 0.8767.
(a) (b) Figure 11: The time-series diagrams for E and I with the different immigration rate while the contacting distance is smaller than the threshold value 0.8767.
will be become longer when some individuals immigrate to the streets of Wuhan city from the other cities (A = 1.1 and A = 1.5 in Figure 8(a) and Figure 8(b) respectively). Most interestingly, the outdoor-time of the active individuals has not influence on the extinction of the COVID-19, this implies that some parts of peoples could move outdoors only when they can keep just far enough contacting distance and the density of immigrating population to Wuhan city is smaller enough. Hence, it is reasonable to believe that the necessary and safe outdoor activities of the healthy individuals could not make the COVID-19 to outbreak again. Figure 9- Figure 11 show that the COVID-19 may become endemic disease when the contacting distance smaller than the threshold value 0.8767. Figure 9 reveals that the COVID-19 will become endemic after 70 days (five times of the mean incubation period) when the contacting distance between the susceptible individuals and the symptomatic infected individuals is smaller than the threshold value 0.8767. The endemic lag decreases as the contacting distance decreases, and hence the decrease in the contacting distance is must be encouraged. Figure 10(a) and Figure  10(b) show the minimum endemic lag of the incubation and symptomatic infected individuals are 28 days (two times of the longest incubation period). Figure 11(a) and Figure 11(b) show that the COVID-19 may become endemic disease after 38 days when the immigrating individuals to the streets from their own home in Wuhan   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 city is 29 percent of the total populations of Wuhan city. The endemic densities of the incubation and symptomatic infected subpopulations increase and the outbreak lag of the COVID-19 decreases with the immigrating number of the susceptible individuals increasing, especially. This public events will be getting worse when some parts of peoples immigrate to Wuhan city from the other cities (A = 1.1 and A = 1.5 in Figure 11(a) and Figure 11(b) respectively).

Discussion
In this paper, an SEIR-type epidemic model for the COVID-19 incorporating the contacting distance between the susceptible and asymptomatic/symptomatic infected individuals is presented. The contacting distance between the asymptomatic and symptomatic infected individuals should be explicitly incorporated into the incidence function to estimate the COVID-19. Our results show that the contacting distance between the susceptible and asymptomatic/symptomatic infected individuals plays an important role in controlling the COVID-19, and the corresponding threshold value, which can induce the COVID-19 to be extinct, is obtained by mathematical analysis based on the ordinary differential equation and the numerical simulation based on the published data. Meanwhile, the threshold value of the immigration rate is also obtained by the similar methods. The COVID-19 will be completely extinct when the contacting distance is larger than the corresponding threshold valued * = 1 − µ(µ + µ 1 + γ 12 )(µ + µ 22 + γ 22 ) β 2 Aγ 12 or the immigration rate is smaller than the corresponding threshold value A * = µ(µ + µ 1 + γ 12 )(µ + µ 22 + γ 22 ) β 2 (1 − d) under some certain conditions. Researchers could estimate these threshold values based on the data of the other cities and give some positive proposals for the disease control. Furthermore, our results reveal that the average outdoor-time of the asymptomatic/symptomatic infected individuals has no influence on the theoretical control for the COVID-19. Our study gives some physical and theoretical methods to control the COVID-19 and some reasonable suggestions for the policy-makers. However, our paper use the average contacting distance in the modelling processes, and does not consider the temporal and spatial effects in the contacting distance. Meanwhile, our model considers the respiratory transmission and the contacting transmission in the modelling processes only, and does not take into account the aerosol transmission. These are open issues and could be considered in the future work.
Similarly, the positivity of the other solutions E(t), I A (t), I(t) and R(t) can be shown.
Thus, the solution (S(t), E(t), I A (t), I(t), R(t)) of model (3) with the initial conditions (4) are positive on the finite region for all t > 0.
Therefore, the positivity of all solutions of model (5) with the initial conditions (6) is also guaranteed since model (3) is a special case of model (4).
Therefore, the boundness of all solutions of model (5) with the initial conditions (6) is also guaranteed since model (3) is a special case of model (4).
Proof of Proposition 3.3.1. We will prove the globally asymptotical stability of the disease-free equilibrium point E 0 . Define the following matrices: Based on the above definitions and notations, model (3) nearing the disease-free equilibrium point E 0 can be rewritten as: According to the local stability of the disease-free equilibrium point E 0 and all characteristic roots of matrix J E0 have negative real parts.