Prediction of bifurcations by varying critical parameters of COVID-19

Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The propagation of COVID-19 through various sub-groups of society is studied. Some critical parameters, such as the background of mortality without considering the disease state and the speed of moving people from infected to resistance, affect the conditions of society. In this paper, early warning indicators are used to predict the bifurcation points in the system. In the interaction of various sub-groups of society, each sub-group can have various parameters. Six cases of the sub-groups interactions are studied. By coupling these sub-groups, various dynamics of the whole society are investigated.

Predicting bifurcation points of the dynamical disease are important [18,19]. Many types of research have been done to predict the bifurcation points of biological systems [19,20]. There are some evidences that near the bifurcation points of a dynamical system, the dynamic becomes slower. It means the system needs more time to pass the transients and reach its final state [21]. Various methods have been used in the prediction of bifurcation points. Some of the wellknown indicators are autocorrelation at lag-1 and variance [22]. They bear some issues in predicting bifurcations in more complex dynamics. Recently, a new method has been introduced to improve the conventional indicators [20]. Lyapunov exponent is another interesting predictor of bifurcation points [23,24].
The study of networks of systems can help to understand the collective behavior of systems. Complex networks have a great impact on human life [25]. They have many components which are interacting with various connections [26,27]. Synchronization is one of the most interesting properties in the dynamics of networks [28,29]. Many biological systems, such as neurons, do their task in a network [30,31]. Many studies have been done on networks [32,33]. A chaotic model of epilepsy based on the neural network was proposed in [34]. Various dynamics of a multilayer network was discussed in [35]. In this paper, various dynamics of the SEIR system, as a model for the development of COVID-19, are investigated. Bifurcation diagrams of the system are studied to show its various behaviors by changing parameters. Bifurcation points of the system are discussed, and they are predicted using autocorrelation. Also, the interaction of five cities is studied with various connections and parameters to show the effect of traveling in the outbreak of the disease.

Studied model
In this paper, the SEIR model is used as a model for the development of COVID-19. It is a compartmental model of infectious disease and its epidemic outbreak [11]. It contains four classes of people as susceptible (S), exposed (E), infected (I), and resistant (R). The SEIR model is as follows: In this model S, E, I, R are the number of susceptible, exposes, infected, and resistant people. Parameters l; b; m; r; c are the background of mortality without considering the disease state, the speed of moving people from susceptible to exposed, the vaccination rate, the speed of moving people from exposed to infected, and the speed of moving people from infected to resistance. It should be noted that the vaccination causes people to move from a susceptible group to the resistant one directly.

Dynamical properties
To study the dynamical properties of the model, various bifurcations are investigated. Part (a) of Fig. 2 shows the bifurcations of Model (1) by changing the parameter of the background of mortality without considering the disease state. It should be considered that in each parameter, the variables approach to an equilibrium point by passing enough time. The bifurcation diagram is plotted with the forwarding continuation method. Dynamics of the system have various bifurcation points. To predict bifurcation points of the model, autocorrelation (AC) is used.
The absolute value of autocorrelation determines the slowness of dynamics by approaching to ''one'' near the bifurcation points. Autocorrelation is calculated from the time series of susceptible, exposed, infected,   Fig. 2. It can be seen that the AC approaches one in bifurcation points l ¼ 0:01 and l ¼ 0:34. Also, the population of exposed and infected people has a change of concavity in approximately l ¼ 0:1, and the AC can predict this bifurcation. The AC of the resistant population is fixed to one in l [ 0:4 because of a computational error. There are not any transients in the time series of l [ 0:4 with a forward continuation method.
The speed of moving people from infected to resistance is another parameter that is investigated in the following. Figure 3 shows the bifurcation diagram of Model (1) concerning changing parameter c. In part (a) of that figure, bifurcation diagram of four variables of Model (1) by changing parameter c is plotted. To have a better view of the three smaller variables, their zoomed view is shown in part (b). AC is used to predict the bifurcation points of the population variables as part (c). AC predicts bifurcation point in c ¼ 0:105 and also shows a small peak in the change of concavity of resistant population. Predicting these bifurcation points can help the world from being surprised by variations of COVID-19 epidemic states.

Interaction of various sub-groups of society
In this section, the connections of five cities are considered to show the effect of intercity traffic  restrictions. To reach this goal, System (1) is considered as the model of development of COVID-19 in each city. Then, the five cities are connected in a Watts-Strogatz network [36,37]. The connection graph is shown in Fig. 4. So, each of the four variables of System (1) is connected to the variables of other cities as Eq. 2. x i is the state variables of each city, and f is their functions. The nodes are connected by coupling matrix C, and k is the strength of coupling. An undirected network is used. So, we consider if the coupling between node A and node B is one, it means that people can travel from city A to B and vice versa.
Now, various cases are studied. The initial values of variables are set constant as Table 1.
In the first case, we consider that no people can travel from one city to another one (k ¼ 0). Parameters   Fig. 5. The results show that each city has its own evolution, and they have no effect on each other. The number of exposed and infected people approaches zero by passing the time, and the number of susceptible and resistant people becomes constant but different for each city.
In the second case, the cities are considered to have some connections to each other (k ¼ 0:1). Parameters are set constant for all five cities as ðl; b; m; r; cÞ ¼ ð0:4; 0:9; 0:2; 0:5; 0:2Þ. The results which are shown in Fig. 6 present that this small connection makes all the cities become the same in the evolution of COVID-19. However, their approaches to the same value are slow. In the third case, the cities are considered to have more significant connections to each other (k ¼ 0:2). Parameters are set constant for all five cities as ðl; b; m; r; cÞ ¼ ð0:4; 0:9; 0:2; 0:5; 0:2Þ. The results of this network are shown in Fig. 7. In a comparison of this network with the previous one, the current network is faster in approaching their same constant values. In other words, by passing enough time, the five cities become synchronized.
In the fourth case, the cities are considered to have some connections to each other (k ¼ 0:2). In the study of a single system, the results show that parameter l can change the dynamic of the evolution of COVID-19. Parameters of the network are set constant for all five cities as ðl; b; m ; r; cÞ ¼ ð0:1; 0:9; 0:2; 0:5; 0:2Þ in this case. The parameter l is different from the previous network. The results (which are presented in Fig. 8) show that decreasing the parameter l makes an undershoot in susceptible and resistant population. Also, the lower l parameter makes the susceptible population constant to a lower number and resistant population to a higher number. To have a more realistic viewpoint, the parameters of cities are considered to be different. In the next two cases, parameters for each city are considered as, In the fifth case, the cities with various parameters are considered to have some connections to each other (k ¼ 0:2). The results of Fig. 9 show that in the case with various parameters for each city, the population of the susceptible and resistant group in the cities cannot be synchronized, and they are different in all the times. However, the population of exposed and infected people approaches to zero in all the cities. Also, the lower l parameter makes the susceptible population constant to a lower number and resistant population to a higher number In the sixth case, the cities with various parameters are considered to have more connections to each other (k ¼ 11). In other words, the coupling strength is increased to reveal the parameter in which the dynamics of various cities become synchronized. The results of the network are shown in Fig. 10. It shows that the population of exposed and infected people in all cities approaches zero by passing the time. The population of susceptible and resistant people of various cities is approximately synchronized; however, the zoomed view of the susceptible population shows they are not completely synchronized.
So, in cities with the same parameters, increasing the coupling strength causes the dynamics of five cities to become the same by passing enough time. However, in cities with various parameters, reaching the same dynamics needs a much bigger coupling strength. Also, decreasing parameter l changes the dynamics of COVID-19 outbreak.

Conclusion
In this paper, the SEIR model was used as the model for the development of COVID-19 outbreak. In the first step of this study, various bifurcations of the model by changing some critical parameters such as the background of mortality without considering the disease state and the speed of moving people from Fig. 9 Variations of state variables of five cities with different parameters for cities and coupling strength k ¼ 0:2; The population of the susceptible and resistant group in the cities cannot be synchronized, and they are different in all the times. However, the population of exposed and infected people approaches to zero in all the cities infected to resistance were discussed. Then, autocorrelation was used to predict bifurcation points of the model. To visualize the collective behavior of COVID-19 by the effect of traveling between various cities, a network consisting of five cities was studied. The effect of various parameters and the coupling strength between cities was discussed. Watts-Strogatz network was used to simulate the connection of five cities. In the interaction of various cities, each city can have various parameters. By coupling these cities, various dynamics of the whole society were investigated. Six cases were defined to study various cases of cities interactions. The results showed that in cities with the same parameters, by increasing the coupling strength from zero, the dynamics of five cities become the same by passing enough time. However, in cities with various parameters, the same dynamics cannot be reached very quickly, and it needs a much bigger coupling strength. Also, the results showed that decreasing the parameter l can change the dynamics of COVID-19 outbreak in cases with the same parameters for five cities.