Spatiotemporal dynamics of periodic waves in SIR model with driving factors

The world faces Covid-19 waves, and the overall pattern of confirmed cases shows periodic oscillations. In this paper, we investigate the spatiotemporal spread of Covid-19 in the network-organized SIR model with an extrinsic incubation period of the driving factors. Firstly, Our analysis shows the occurrences of Hopf bifurcation and periodic outbreaks consistent with the actual spread of Covid-19. And we investigate periodic waves on spatial scales using Turing instability, and the spread of infected individuals increases the localized hot spots. We study the effect of the incubation period, and more incubation periods generate Turing instability resulting in periodic outbreaks. There is an occurrence of bursting states at peaks of periodic waves due to small diffusion of infected and susceptible, which means stable and unstable areas try to convert each other due to high competition among nodes. Also, We note the disappearance of these bursts when infected and susceptible individuals’ movements are easier; thus, the dominance of infected individuals prevails everywhere. Effective policy interventions and seasonality can cause periodic perturbations in the model, and therefore we study the impact of these perturbations on the spread of Covid-19. Periodic perturbations on the driving factors, infected individuals show co-existing spatial patterns. Chaotic outbreak becomes periodic outbreaks through alternating periodic or period-2 outbreaks as we regulate the amplitude and frequency of infected individuals. In short, regulations can erase period-2 and chaotic spread through policy interventions.


Introduction
The spread of infectious diseases, such as EBOLA, SARS, HIV, NIPAH, Monkeypox, and Covid-19, is a significant concern for public health policy over the last few decades [1][2][3]. Spatiotemporal analysis of the spread of pandemics is essential to reduce human mortality and health care expenditure [4,5]. The SIR model with behavioural changes and non-pharmaceutical interventions shows that the global stability of disease-free equilibrium is determined by the value of a certain threshold parameter, which is R 0 , and the inclusion of the delay parameter promotes the occurrence of Hopf bifurcation [6]. The basic reproduction number in the SIR network model with the contact network provides a threshold for global stability for disease-free equilibrium (R 0 < 1) [7,8]. The basic reproductive number R 0 is one of the parameters to quantify the evolution of the endemic and can be treated as a critical point for mutations, which determines the prevalence of infectious diseases at temporal scale [4,5,[9][10][11][12][13]. Globalization results movement of infected individuals across boundaries, and therefore, spatial dynamics along with temporal dynamics of spread become critical [14]. The SIR-Network model enables us to study spatiotemporal dynamics, and it can replicate the propagation of epidemics in populated cities, where neighbourhoods are considered nodes [15]. The movement of individuals is represented as directed edges, which helps investigate the impact of time-varying neighbourhood status on dynamic behaviours of infectious diseases [15][16][17]. However, the effect of the incubation period, periodic perturbations, and the interactions among infected and susceptible individuals on spatiotemporal dynamics still need to be improved and focused on in our investigation.
Interactions among the nodes of the SIR network model represent the movement of the infected and susceptible individual in space, and these theoretical references can provide insights for better mitigation strategies [16]. This method is implemented in other biological systems to investigate the emergence of patterns due to node interactions [18][19][20][21]. Alternative methods were proposed to indicate the generation mechanisms of Turing instability and patterns [22][23][24]. In general, the topology determines and affects the properties of the Turing system [25]; however, it is hard to tell the difference between the topology [26]. The distribution of eigenvalues of the network matrix is a tool to study the stability of these network-organized systems, which could explain the mechanism of Turing pattern genesis [27]. The mathematical analysis of a SIR disease network model admits that the whole dynamics of the disease transmission depends on the eigenvalue [12]. Furthermore, the concept of negative wavenumber is introduced to present a novel interaction between nodes, which shows the chaotic state of pattern formation results from the mutual effect of the negative and positive wave-numbers [28]. This complex interplay between the coupling parameter and the eigenvectors' properties determines the type of pattern formation [29]. In spite of the presence of different co-existing patterns, no general explanation exists for mechanisms of the interaction between individuals, especially the emergent properties [30,31].
The evolution process of infectious diseases is affected by external factors. And some individuals are susceptible to becoming infected after a delay, which is the extrinsic incubation period. Also, the interaction of the network nodes (cities or countries) plays a role in the spread and evolution of the epidemic, especially the emergent properties. This paper investigates the bifurcation analysis and Turing instability to show the epidemic's periodic behaviours and emergent properties in a modified SIR model. The SIR model with the driving factors due to the incubation period is analyzed through Hopf bifurcation to show the effect of the driving factors and susceptibility rate on the evolution of the epidemic, consistent with the newly confirmed cases of Covid-19 in some areas. Then the SIR model with both the driving factors and network is investigated to illustrate the importance of the node interaction. Also, the delay is considered to demonstrate the effect of the driving factors on the function of the spread of infectious diseases, which makes a Hopf bifurcation occur. Numerical results reveal that Turing, Hopf bifurcation, and periodic perturbation lead to the generation mechanism of emergent properties, which can be proved qualitatively by the newly confirmed cases of Covid-19.

Model description
SIR network model [4,5] with driving factors and delay are following where S i , I i , R i (i = 1, . . . , n) in ith node are the density of susceptible, infected, and recovery, respectively. n is the number of nodes. k 1 is the susceptibility rate. k 2 is the infection rate induced by the driving factors which can be imported cases, abnormal diffusion, etc), and it directly impacts the infected rate. τ is the delay from the susceptible to the infected. k 3 , k 4 , k 5 are infected, death and recovery rates, respectively. d 1 , d 2 is the diffusion rate (i.e. the strength of coupling) of susceptible and infected, respectively. The adjacent matrix A represents the interplay of network nodes, h 1 (S k , S i ), h 2 (I k , I i ) is the kinetic equation of the interaction of network nodes(in general, they are linear [26][27][28]). We do not include recovery for the qualitative analysis as it does not affect the system's stability (1). A schematic model diagram is shown in figure 1(a), where we can see the k 2 S factor from the susceptible to infected individuals, a driving factor caused by the external factors. We investigate the dynamic behaviour of system (1) with initial stability analysis of simplified version (2) Three forms of the infectious disease exist, including disease-free, endemic, and periodic outbreaks. The dynamics of the disease-free and endemic are investigated before [4,5]; therefore, we focus our study on the periodic outbreak, and give the following theories, (2) is stable for µ > µ 0 , system (2) has periodic oscillations for µ < µ 0 ; If da(µ) dµ | µ0 < 0, system (2) has periodic oscillations for µ > µ 0 , system (2) is stable for µ < µ 0 . Proof. Suppose (S 0 , I 0 ) is the equilibrium point of system (2), We get the Jacobian matrix at (S 0 , I 0 ), and the corresponding characteristic equation is where k j is the control parameter when µ = k j (j = 1, 2, 3, 4, 5, we only consider j = 1, 2 in this paper), and According to the definition of Hopf bifurcation [33,34], Hopf bifurcation occurs in system (2) when a(µ 0 ) = 0, b(µ 0 ) > 0. Meanwhile, the characteristic value is λ 1,2 = ±i b(µ 0 ), µ 0 is the critical value µ = k j (j = 1, 2) of Hopf bifurcation. Namely, If da(µ) dµ | µ0 > 0, system (2) is stable for µ > µ 0 , system (2) has periodic oscillations for µ < µ 0 ; If da(µ) dµ | µ0 < 0, system (2) has periodic oscillations for µ > µ 0 , system (2) is stable for µ < µ 0 . Theorem 2 ([33, 34]). If d 1 , d 2 could make b min < 0 hold, Turing instability occurs in system, and the Turing bifurcation point satisfies b min = 0; If a(µ) = 0 and b min < 0, Turing-Hopf bifurcation occurs in system (4).
Proof. We consider system (1) without delay, and its linearized equation at (S 0 , I 0 ) is where S k − S i , I k − I i is the linear part of h 1 (S k , S i ), h 2 (I k , I i ), respectively. Also, The general solution [22] of system (4) is Substituting the general solution (6) into system (4), we have the Jacobian matrix Because Λ i ⩽ 0, a i > 0, the stability of system (4) is determined by the sign of b i . And Turing bifurcation occurs when Further, it can be obtained that If d 1 , d 2 could make b min < 0 hold, Turing instability occurs in system, and the Turing bifurcation point satisfies b min = 0; If a(µ) = 0 and b min < 0, Turing-Hopf bifurcation occurs in system (4).

and the characteristic equation is
where and we have where Suppose x 1 , x 2 , x 3 are the positive real root of equation (7), ω j = √ x j (j = 1, 2, 3). One has We define and the corresponding ω i of the critical value τ 0 is ω 0 .

Methods and results
Our system has three equilibrium conditions and periodic waves emerging due to Hopf bifurcation. The driving factors and delay are analogous to abnormal infections and incubation period. Here, We show the effect of the driving factors and network on the dynamics of the epidemic. We fix parameters k 3 = 1, k 4 = 0.001, k 5 = 1, n = 100, and k 1 , k 2 , τ, d 1 , d 2 are the control parameters. We use a random initial conditions S(i, 0), and I mean is the mean of all nodes representing the behaviour of aggregates of individuals. Also, the scale-free network [32] is constructed with initial node m 0 = 10(no links), the maximum number of nodes n = 100 and the new number of edges m = 1. Then, one node is added per unit of time, and one edge is linked with an old node according to Gillespie algorithm [36]. Finally, the scale-free network is completed if the number of nodes is n = 100. All data used in this paper can be found on https://covid19.who.int and https://usafacts.org/visualizations/coronaviruscovid-19-spread-map.

Driving factors causing periodic Covid-19 waves
The system (1) has a disease-free, endemic, periodic outbreak, and chimeras states. The chimera state represents the co-existence of multiple states, and the periodic state shows that the number of infected is periodically oscillating. Conventionally, the disease-free and endemic dynamics are investigated [4,5], and the other two states are rarely discussed. However, we can see periodic outbreaks of Covid-19; therefore, we focus on periodic waves and chimera state (9).
An increase in infected individuals has induced epidemics, and the periodic waves are largely determined by the susceptible [12] (figure 2). The infected individuals are small in numbers but exist in the system (2) (figure 2(a)) when k 1 = 0.1, k 2 = 0.1. The periodic outbreaks (figure 2(b)) emerge due to the supercritical Hopf bifurcations for higher susceptibility rate ( figure 2(d)). Further increase in susceptibility rate induce subcritical Hopf bifurcation (figure 2(d)) leading to endemic state (figure 2(c)). This endemic state suppresses the periodic outbreaks [39]. The evolution of the epidemic initially shows outbreaks with small amplitude and then a periodic explosion as (figure 2(d)) for k 2 = 0.1, and eventually, large outbreaks lead to endemic. This trend has been seen in the new confirmed covid cases presented in figure 1 and is currently in a periodic outbreak state (figure 1). A massive outbreak may occur if the number of susceptible individuals cannot be effectively reduced, and finally, the endemic state of Covid-19 will be like a common cold.
We consider the effect of driving factors where susceptible become infected after the incubation period, and in general, these factors k 2 are small and increase infected individuals after a delay (τ ). We draw a bifurcation diagram to show the combined effect of susceptibility and driving factors ( figure 3(d)). The driving factor becomes primary when the susceptibility rate is low (k 1 = 0.3). For low k 2 = 0.01 driving factors are influential, leading to periodic outbreaks ( figure 3(b)), and as we use mitigation strategies to lower the driving factors, we observe stable endemic state ( figure 3(a)). For higher k 2 = 0.07, the infected individuals become prevalent, and an endemic equilibrium occurs. We have a disease free state for k 2 = 0, a Hopf bifurcation occur at k 2c = 0.065 ( figure 3(c)), and we observe a periodic oscillations ( figure 3(b)).

Turing instability captures spatial dynamics of Covid-19 cases
Without node interactions, all nodes have the same dynamical behaviour ( figure 4(a)). However, countries, cities or any spatial entity have different measures to mitigate the spread; thus, we may observe different spatial patterns. The presence and absence of endemic are two alternative stable states that can occur in the spatial domain ( figure 4(b)), which can reveal other spatial patterns as well [37]. Meanwhile, the different spatial patterns of coexistence states can occur in the evolution of Covid-9 ( figure 4(c)). Our simulation results match the Covid-19 cases from the 35 western Pacific countries from 1.3.2020 to 5.27.2022. The epidemic is induced by the diffusion of infected individuals in space, and the periodic behaviours are determined by the diffusion of the susceptible figure 4(d). Therefore, the diffusion of the susceptible induces  periodic outbreaks through Turing instability; the diffusion of the infected will lead to the endemic equilibrium for all nodes.
System (9) without network presents the periodic behaviour (figure 3(c)) when k 1 = 0.3, k 2 = 0.05, k 3 = 1, k 4 = 0.001, k 5 = 1. The endemic, disease-free and periodic outbreak coexist in the spatial pattern when the diffusion of the susceptible is involved (i.e. d 1 = 1.0) ( figure 5(a)), and the overall pattern shows irregular periodicity ( figure 5(b)). Increasing the diffusion of the infected (d 1 ) leads to the spread of the diseases in the regions which were untouched before in ( figure 5(a)). However, a few areas with very high infected individuals (red regions) are negligible. The final solution is to reduce the susceptibility rate through vaccination ( figure 2(d)). When the diffusion of the infected and susceptible is involved, the spatial pattern of the system (9) is an asynchronous period (figures 5(c) and (d)). Although the burst intensity of each node is reduced, the outbreak is widespread (figures 5(c), (d) and 6(a)). Meanwhile, system (9) presents an alternating pattern ( figure 5(c)), which means the intensity of each outbreak periodically varies (figures 5(d)  and 6(b)). Our results show that the simulated pattern is qualitatively consistent with the real situation (figures 5 and 6).

High incubation period promotes periodic outbreak
The delay or incubation period is an essential factor that influences the spread of disease. A low incubation period shows lower numbers of infected individuals ( figure 7(a)), and as the incubation period is increased, periodic outbreaks occur ( figure 7(b)). The extended incubation period makes detecting the infected hard, leading to more infections. The mean incubation period of Covid-19 is 6.38 days; however, it may vary from 2.33 to 17.60 days [38]. We choose a critical value τ 0 = 8.038 when k 2 = 0.01 and τ 0 = 13.66. when k 2 = 0.1 (figure 7), this critical values can not be changed but τ 0 can be controlled through policy interventions. (figures 7(c) and (d)) show the occurrence of Turing bifurcation and periodic outbreaks beyond a threshold value of τ 0 , which means effective measures can minimize the incubation period, and therefore periodic outbreaks can be avoided. Controlling the driving factors can reduce the incubation period, and therefore it may also avoid periodic outbreaks ( figure 8(a)). Lowering the incubation period and reducing the effect of driving factors lower the infected, and therefore k 1 , k 2 are varied to plot the critical values of the τ 0 ( figure 8(b)).
Further, we investigate the effect of delay on the spatial dynamics of the spread with the variation in the diffusion of infected and susceptible (τ > τ 0 ) (figure 9). We see spatial periodic outbreaks even when nodes have no interactions and the incubation period is large ( figure 9(a)). We observe bursting at peaks ( figure 9(b)), which is followed by a period of preparation before the next burst. These bursts are mutual effects between driving factors and delay, and they can have different types of oscillations (figures 9(c) and (e)). The burst pattern is highly disordered and chaotic (figures 9(c) and (e)) when d 1 d 2 is  very small, which create unpredictable states (figures 9(d) and (f)). Ultimately, the regular vertical stripe pattern occurs when d 1 d 2 is increased (figure 9(g)). Therefore, the bursting state is difficult to control, and data of Covid cases in the Eastern Mediterranean from 1.3.2020 and 5.27.2022 shows this bursting behaviour at the peaks of the outbreaks (figures 10(a) and (b)). However, the mechanism of these bursting is still not very clear, and to investigate this mechanism, system (1) can be expressed as a mean-field system [8], Λ i is treated as the control parameter to determine the stability of the system (1) and (10) and here Λ i ∈ (−8, 0]. The Hopf bifurcation curve is presented in figures 10(c) and (d), where the dotted line represents τ = 9, to evaluate the distribution of τ 0 with a change of Λ i . Hopf bifurcation occurs at a different value for different nodes for (10) (figures 10(c) and (d)), here any node i is stable when τ i > τ = 9 and unstable when τ i < τ = 9. The co-existence of stable and unstable nodes creates competition, and that causes the genesis of bursting or emergence (figures 9(c) and (d)). In other words, all the unstable nodes will pressure stable nodes to change their stability and thus induces a chaotic state. Unstable nodes are less (figure 10(d)) for large d 1 , d 2 , therefore a weaker competitions restores regular pattern state (figure 9(e)) and bursting state (figure 9(f)). In the context of Covid-19 spread, unstable and stable nodes represent infected and disease-free areas, respectively. The infected areas try to convert disease-free areas to infected areas; This competition among infected and disease-free nodes causes the bursting in the spread of Covid-19 ( figure 10).  Our simulation result (figure 10(f)) captures the qualitative behaviour of actual data of the Covid-19 spread presented in ( figure 10(b)).

Pattern formation with the periodic perturbation
Covid-19 may spread like the flu in the coming years; therefore, we investigate the effect of seasonality in our study. Consequently, we introduce periodic perturbations in our analysis that arise due to epidemic policy interventions or seasonality, and it profoundly impacts the dynamics of similar systems [35]. To show the effect of the periodic perturbation on the pattern formation ( figure 5(a)) and explain the alternate phenomenon in the spread of the epidemic (figure 11), we first investigate the periodic perturbation of the birth rate of susceptible individuals through the system (11).
where ω 0 = 0.5261 is the intrinsic frequency, k 1c = 0.4224 is the Hopf bifurcation point, ν is the frequency modulation, and k 2 = 0.01, γ is the amplitude of the periodic perturbation, the frequency of periodic oscillations is close to twice the intrinsic frequency of the system [7]. Further, frequency modulation and amplitude play a vital role in the selection of pattern formation figure 12. When the interaction between nodes is weak, the emergence occurs ( figure 12(a)), and the corresponding mean evolution curve presents a chaotic state ( figure 12(b)). This chaotic behaviour disappears as the strength of coupling increases, and irregular, periodic behaviour occurs (figure 12(c)); however, the corresponding mean evolution curve shows the periodic behaviour (figures 12(d) and 1). Also, the alternating pattern, spot pattern and stripe pattern coexist (figure 12(e)) when the frequency modulation ν = 0, which shows the complexity of dynamic behaviour. Its mean evolution curve is a bursting behaviour (figure 12(f)), which will not appear in a node. Therefore, the periodic perturbation of k 1 can be used to illustrate the alternating pattern phenomenon (figures 5(a) and 11) in the spread of Covid-19. Now, we consider the periodic perturbation of the driving factors in system (13).
where ω 0 = 0.3919 is the intrinsic frequency, k 2c = 0.0649 is the Hopf bifurcation point, and k 1 = 0.1. The periodic perturbation of the driving factors creates the co-existence of vertical, wave and spot patterns (figure 13(a)) with periodic corresponding mean evolution curve ( figure 13(b)). This behaviour can suggest a periodic cycle (figure 13(b)) even countries that are in different states of the spread (figure 13(a)) at any time. The vertical and spot patterns coexist (figure 13(c)) when the amplitude becomes larger, and its behaviour manifests the alternating pattern (figure 13(c)) and a burst (figure 13(d)) as a whole. If we continue to control ν, the pattern changes to the synchronization period (figure 13(e)), which is analogous to system (2) (figure 13(f)). This behaviour shows the network-organized system can be reduced to a simple system. A simple SIR model may express the evolution of the epidemic, but more is needed to investigate the emergence, burst and coexisting states on spatial scales.
Finally, we consider the periodic perturbation of the delay.
where ω 0 = 0.1245 is the intrinsic frequency, τ 0 = 8.038 is the Hopf bifurcation point, and k 1 = 0.1, k 2 = 0.01. System (13) presents the wave pattern ( figure 14(a)) and the overall periodic behaviour ( figure 14(b)) when the periodic perturbation γ = 0.05, ν = 0 of the delay is fixed. When the amplitude becomes larger, the increase of the unstable nodes leads to the spot pattern ( figure 14(c)) and chaotic evolution curve ( figure 14(d)). We note that frequency modulation could suppress the genesis of the chaotic states and make their behaviours return to the controllability of the state of the epidemic (figures 14(e) and (f)). Therefore, the regulation of frequency and amplitude through policy interventions can remove the chaotic states from the system, even in the case of seasonality.

Conclusion
Understanding the dynamics and evolution of Covid-19 have many challenges as its nature is affected by driving factors, incubation period, and seasonality. Meanwhile, the epidemic spreads along with social networks and gradually expands its scope, which is in accord with the generation process of BA network [32]. We model the spread of Covid-19 with an incubation period using network-organized SIR, and comparison with data shows accurate prediction. The SIR model with driving factors shows that driving factors cause periodic waves when the infected rates are low and therefore susceptible to becoming infected, thus causing periodic waves, which represents that diffusion due to transport or any other means can cause periodic outbreaks of Covid-19, and this behaviour is consistent with the new confirmed cases of Covid-19. Then the SIR model with both the driving factors and network is investigated to illustrate the importance of the interaction between nodes. Numerical simulations reveal that Hopf bifurcation and the periodic perturbation lead to the emergence of the spread of the epidemic. In fact, stable and unstable nodes coexist, and they compete with each other, which generates the emergence in the spatiotemporal dynamics of Covid-19 spread. In other words, all the unstable nodes will pressure the state of stable nodes to change into their state and induce the chaotic state. The competition between the unstable nodes becomes weaker when the unstable nodes become few, and then the system restores to a regular pattern or bursting state. The same goes for the spread of infectious diseases. The unstable and stable nodes represent the infected and disease-free areas, respectively. The infected areas try to align the other disease-free areas with them, forming a competitive relationship between the infected and the disease-free regions. Furthermore, the emergence occurs in the spread of Covid-19, which can be proved qualitatively by the newly confirmed cases of Covid-19. We have also studied the effect of seasonality using periodic perturbations in the birth rate of susceptibles, delay, and driving factors. We have noted the oscillations, period-2 oscillations, and aperiodic oscillations. This period-2 behaviour and aperiodicity should be avoided with the policy interventions. This study does not show the mechanisms of the emergence of period-2 and aperiodic states. In short, compared with the classical SIR model, time-varying SIR model and probabilistic SIR model, the modified SIR model with the driving factors is more functional for expressing the evolution process of infectious diseases. The emerging pattern mechanism in spatiotemporal dynamics of pandemics should be discussed in the following. Meanwhile, Covid-19 will lead to more serious outcomes for patients with cancer [2,3,40], but how they work together is needed to further explore through multiscale modelling [40].

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.