Discrete-time COVID-19 epidemic model with chaos, stability and bifurcation.

In this paper, we explore local behavior at fixed points, chaos and bifurcations of a discrete COVID-19 epidemic model in the interior of R + 5 . It is explored that for all involved parametric values, COVID-19 model has boundary fixed point and also it has an interior fixed point under certain parametric condition(s). We have investigated local behavior at boundary and interior fixed points of COVID-19 model by linear stability theory. It is also explored the existence of possible bifurcations at respective fixed points, and proved that at boundary fixed point there exists no flip bifurcation but at interior fixed point it undergoes both flip and hopf bifurcations, and we have explored said bifurcations by explicit criterion. Moreover, chaos in COVID-19 model is also investigated by feedback control strategy. Finally, theoretical results are verified numerically.

Introduction

Motivation and literature survey

Never in human history has there been an epidemic bigger and wider than the COVID-19. In no time, it captured almost the whole world under its deadly wings [1].

According to the World Health Organization (WHO), a total of confirmed cases have thus far been reported out of which critically infected could not recover and hence lost their lives. Similarly, in Pakistan, out of 1.53 million total cases, 30,379 patients died due to the coronavirus [2], [3].

What makes it doubly dangerous is the fact that its symptoms are pretty common; that is to say, fever, cough, fatigue, loss of taste and smell etc. But, thankfully, the good news is, the world has successfully found a kind of cure. The vaccine doses of Pfizer, CanSino, Moderna, Sinopharm, Sputnik, AstraZeneca, CoronaVac, and PakVac are being given to the populations globally. This medication, however, only reduces the percentage of a possible viral infection. The best way to stay safe still remains the adoption of severe precautionary measures such as maintaining social distance and wearing a face mask [4], [5], [6].

Since its first occurrence in the Chinese city of Wuhan, we have witnessed divided views and opinions about the pandemic. While the western world (where the number of deaths was also the largest) took it too seriously, the countries like Pakistan made light of it once the initial shock wave was over. Even today, people in backward areas seem doubtful and resistant to the whole vaccination campaign. This air of carelessness is largely due to the lack of quality education. While having an unwavering faith in GOD is a good thing, disregard and neglect of one’s healthy existence can no less be considered suicidal. Despite the fact that the situation has now improved for the better, we cannot (and must not) be completely inattentive to or unmindful of the serious health hazards.

Meanwhile, numerous chemists, biologists, medics and mathematicians are attempting to understand coronavirus behavior. On the mathematical side, a few models like SIR, SEIQR, SEIR and SIRS have already been investigated to know the cause of this disease. Furthermore, Özköse & Yavuz [7] studied interactions among diabetes and COVID-19 using real world data from Turkey of following COVID-19 model: where , , , , , and are respectively denote susceptible, exposed, infected, recovered, exposed and susceptible, diabetes without and with complications populations. is recruitment rate, is natural death rate, and are infected people by contact with infected without and with symptoms, respectively. The rate of people become normally infected is , is recovery rate. and are respectively the rate of infected without and with symptoms, who have been quarantined. are mortality rates due to complications, natural mortality rate, and are the probabilities of a diabetic person developing a complication, are the probability of developing diabetes of susceptible individuals in quarantine and is the rate of susceptible who have been in quarantine total. Hikal et al. [8] studied the existence of time delay in implementing the quarantine strategy and the threshold values of the time delay that keeping the stability of the system of following continuous-time COVID-19 epidemic model: where and are respectively denote susceptible, exposed, infected, quarantined and recovered individuals. The parameter is new birth rate, natural death rate is , is the rate for which moved to , denotes transmission rates of and respectively, and respectively denote death rates of infected population without and with quarantined. Zhang et al. [9] proposed and studied the dynamics of following continuous-time COVID-19 epidemic model: where , , , , , , and respectively denote susceptible, exposed, infected but not yet symptomatic, infected with symptoms, hospitalized, recovered, quarantined and isolated exposed individuals. Moreover, is transmission rate, contact rate, is quarantined rate, is transition rate of to individuals, is probability of having symptoms among individuals, recovery rate of individuals is , is recovery rate of individuals, is the disease-induced death rate and is the rate of transmission of to . Liu et al. [10] discussed the dynamic behavior of the model: with positive and . Zeb et al. [11] proposed the following continuous-time mathematical model for COVID-19 epidemic containing isolation class: where , , , and respectively denote susceptible, exposed, infected, isolated and recovered individuals. Moreover , , , , and respectively denotes rates of vulnerable people become and , people become , people goes to , people together with , recovered rate of and natural and diseased death rate. More specifically, Zeb et al. [11] investigated the local and global stability at fixed points for the under consideration COVID-19 epidemic model (5). Further for the proposed model numerical solutions are obtained by Runge–Kutta fourth order method and nonstandard finite difference scheme. Zeb et al. [11] have presented the graphical results which shows that human to human contact is the main cause of spread of the COVID-19 disease. For more recent studies regarding COVID-19 disease, we refer [12], [13] and literature cited therein.

Problem statement and main contribution

It is important to pointed out that mathematical modeling of population dynamics has grown more appealing to theoretical ecologists and mathematical biologists because of its rich dynamics and diverse applications [14]. Continuous-time and discrete-time mathematical models are typically employed to represent population dynamics. Differential equations represent continuous-time models, while difference equations describe discrete-time models with the memory effects means that the present state evolution depends on all past states [15]. Discrete-time models are better suited to small populations or non-overlapping generations than continuous models; for instance, many insect species have no overlap between their successive generations, so their populations exhibit discrete-time behavior. In addition, compared to continuous-time mathematical model, discrete model gives richer dynamics. A discrete-time model with a single species can show chaos, and more complex dynamics, but a continuous model requires at least three species to show chaos [16]. So, goal of this study is to explore dynamical properties of the COVID-19 model, which is a discrete form of (5) via Euler Forward Formula: From (6), the simplification yields: Precisely our main contribution in this paper includes:

Investigation of equilibrium solution of COVID-19 epidemic model (7).

Study of linearized form of COVID-19 epidemic model (7).

Study of local behavior at equilibria of the model (7).

Bifurcation analysis at equilibria of COVID-19 model (7).

Study of chaos by state feedback method.

Verification of theoretical results numerically.

Paper layout

The layout of the paper is as follows: Feasible equilibria with linearized form of COVID-19 model (7) is studied in Section 2, whereas Section 3 is about the study of local dynamics at equilibrium solutions of COVID-19 model (7). The bifurcation analysis for under study model is given in Section 4. In Section 5, we study chaos control, whereas theoretical results are numerically verified in Section 6. Section 7 concludes with closing remarks and recommendations.

Feasible equilibria with linearized form of COVID-19 model (7)

The feasible equilibria of COVID-19 model (7) will be given in the section, as follows:

For existence of feasible equilibria of COVID-19 model (7) , following statements true:

Model (7) has boundary fixed point ;

COVID-19 model (7) has interior equilibrium where if .

If is equilibrium of discrete COVID-19 model (7) then Obviously system (9) satisfied identically if , and one can say that boundary equilibrium of discrete COVID-19 model (7) is . Further following algebraic system is to be solved in order to obtain interior equilibrium: First equation of (10) yields From second equation of (10), one gets From third equation of (10), one gets Using (11) in (12), one obtains Using (13) in (14), one has Using (15) in (13), one obtains Now using (15), (16) in Eq. (11), we have Using (15), (16) in forth and fifth equations of (10), one respectively obtain and Finally, from (15)–(19) we can obtain that if then where , ,   , , is interior equilibrium of COVID-19 model (7). Further since , i.e.,  then one defines as a basic reproductive number. □

Finally, at the linearized form of COVID-19 model (7) under is where, see Eq. (22) in Box I

and

COVID-19 epidemic model (7) with local dynamics

Local dynamics at and of COVID-19 model (7) will be studied in this section. At , (22) becomes with characteristic roots are By stability theory [17], [18], [19], [20], [21] and from (25) the local dynamics can be summarize in the following Lemma:

If and with then is a sink;

If (28) holds and furthermore and then is a source;

If (28) holds and furthermore and or and or and or and then is a saddle;

If or or

then is non-hyperbolic.

Hereafter following theorem is used to determine dynamic properties at of COVID-19 model (7) (Theorem 1.6 of [17]).

If then all roots of following fifth-degree polynomial satisfying .

If where then is a stable.

About , (22) becomes From (46), the characteristic polynomial of at is where are depicted in (45). So, by Theorem 3 if then of COVID-19 model (7) is a sink. □

Bifurcation

By bifurcation theory [22], [23], [24], we will give complete bifurcation analysis at and in this section.

Bifurcation analysis about

From (25), the computation yields but other eigenvalues or , which indicates that model (7) can undergo flip bifurcation if located in the set: But following Theorem states that if then no flip bifurcation exists for COVID-19 model (7) at .

If then no flip bifurcation exists for COVID-19 model (7) at .

It is noted here that with respect to the COVID-19 model (7) is invariant and so in order to investigate said bifurcation, under study model (7) is restricted on and then it gives the following form: From (49), one denotes the map Now if and then from (50) one gets and The above computation shows that at of COVID-19 model (7) there exists no flip bifurcation because computed parametric condition (53) violates the condition of non-degenerate for existence of flip bifurcation if . □

We will explore hopf and flip bifurcations by choosing as a bifurcation parameter about interior equilibrium of discrete COVID-19 model (7) by utilizing explicit criterion in this section.

Hopf bifurcation about

We study the hopf bifurcation for COVID-19 model (7) at by explicit criterion [25].

If then at , COVID-19 model (7) undergoes a hopf bifurcation at a critical value where are depicted in (45) and is the real root of .

By utilizing explicit criterion for existence of hopf bifurcation for , we obtain Finally,

Flip bifurcation about

We study the flip bifurcation for COVID-19 model (7) at by explicit criterion [25], [26].

If then at , COVID-19 model (7) undergoes a flip bifurcation at , where be the real solution of .

By utilizing explicit criterion for existence of flip bifurcation for , we obtain

Chaos control

Chaos control is the stabilization of one of these unstable periodic orbits by modest system perturbations. As a result, an otherwise chaotic motion becomes more steady and predictable, which is typically advantageous. To avoid major changes in the system’s natural dynamics, the perturbation must be small in comparison to the overall size of the attractor. So, we study chaos by feedback control strategy at of COVID-19 model (7). By feedback control strategy, COVID-19 model (7) becomes with control parameter is . The at of controlled COVID-19 model (58) is The characteristic polynomial of at is where Finally, by stability theory behavior of controlled COVID-19 model (58) at can be stated as following Lemma:

of model (58) is a sink if

where are shown in (61).

Numerical simulations

In this section, obtained theoretical results are numerically illustrated. In this context, the following scenarios are offered to discuss the accuracy of obtained results for the COVID-19 model (7):

Case I: Here at it is proved that COVID-19 model (7) undergoes a hopf bifurcation if and with initial values . If and then from (47) one gets: whose characteristics roots are , and with which implies eigenvalues criterion hold, and so COVID-19 model (7) may undergoes Neimark–Sacker bifurcation. But following simulation shows that it must exists for under study COVID-19 model (7). For illustration, if and then from (54) the computation yields Moreover implies . Thus from (64) all conditions of Theorem 6 hold and hence it can be concluded that hopf bifurcation exists for COVID-19 model (7). Thus hopf bifurcation diagrams and maximum Lyapunov exponent are drawn in Fig. 1. Moreover, it is cleared from Fig. 2 that of model (7) with initial values is stable focus. Finally, Fig. 3 implies that of model (7) with is unstable focus.

Fig. 1

Maximum Lyapunov Exponents along with bifurcation diagrams of COVID-19 model (7) if and with initial values .

Fig. 2

Stable focus for model (7) if and with initial values .

Fig. 3

Unstable focus for model (7) if and with initial values .

Case II: Now at , we will show that COVID-19 model (7) undergoes a flip bifurcation if and with . If and then from (47) one gets: whose one root is but rest of roots are , 0.8062431763086326, which implies that eigenvalues criterion hold for existence of P-D bifurcation, and so COVID-19 model (7) may undergo flip bifurcation. Finally, it is shown that COVID-19 model (7) necessity undergo said bifurcation. Thus, if and then from (56) the computation yields Thus from (66) it can be concluded that parametric conditions of Theorem 7 hold and so flip bifurcation must exists for under study COVID-19 model (7). Hence in this case maximum Lyapunov exponent along with flip bifurcation diagrams are plotted in Fig. 4. Finally, Fig. 5 has been plotted which showed the chaotic behavior of the COVID-19 model (7).

Fig. 4

Maximum Lyapunov Exponents along with flip bifurcation diagrams for COVID-19 model (7) if and with .

Fig. 5

Chaotic attractor for COVID-19 model (7) if and with initial values .

Case III: Finally, we will fit real data obtained from published materials to our under consideration COVID-19 epidemic model (7) in order to verify our mathematical analysis. The collected real data for European countries is depicted in Table 1, whereas corresponding dynamical analysis of COVID-19 model (7) is given in Fig. 6, Fig. 7. In Fig. 6(a), the plot of population shows variable pattern of increasing and decreasing individuals at different time period and attains stable configuration after certain time, whereas Fig. 6(b) shows rate of population for infection increases with the increase of time period. Fig. 6(c) shows that number of people increases with increase in time, whereas Fig. 6(d) shows that number of people with the passage to time also increase. Finally, Fig. 6(e) shows that in the beginning recovered was less but with the passage of time recovery of infection in more. We have also plotted flip bifurcation diagrams for real date with in Fig. 7.

Table 1

Real data for European countries.

Parameter	Value	Source
A	3.43	Estimated
μ	0.165	[27], [28]
β	0.65	[27], [29]
γ	0.07	Estimated
π	0.56	[27], [30]
σ	1.22	Estimated
θ	0.05	[27], [31]

Fig. 6

Plots for fitting results of COVID-19 model (7).

Fig. 7

Flip bifurcation diagrams for fitting results of COVID-19 model (7).

Conclusion

This works is about local behavior along with topological classifications, chaos and bifurcation of COVID-19 model in . It is shown that , COVID-19 model (7) has boundary equilibrium solution , and if then it has interior equilibrium where , , , and are depicted in (8). Further local dynamical characteristics with topological classifications at equilibrium solutions and of COVID-19 model (7) are explored. It is shown that the interior fixed point of discrete COVID-19 model (7) is a sink if and with source if (28) holds and additionally and saddle if (28) holds and additionally and or or and or and non-hyperbolic if or or and moreover of discrete COVID-19 epidemic model (7) is a sink if where , and are depicted in (45). Further, to understand behavior of COVID-19 model (7) deeply, we explored possible bifurcation scenarios. By bifurcation theory, we proved that at no flip bifurcation exists if , but it undergoes both hopf and flip bifurcations at by choosing as bifurcation parameter. By utilizing explicit criterion, we have studied hopf and flip bifurcations at of COVID-19 model (7). We also studied chaos in COVID-19 model (7) by feedback control strategy. For controlled COVID-19 model (58) it is shown that is a sink if where , and are depicted in (61). Finally, various simulations are presented in order to verify theoretical results.

Recommendations

Different mathematicians have suggested mathematical models to anticipate the origin of the COVID-19 pandemic commencing from Wuhan city of China in December 2019. For instance, Refs. [7], [8], [9], [10], [11], [12], [13] are quoted here few newly developed mathematical models regarding the COVID-19 pandemic. Refs. [7], [8], [9], [10], [11], [12], [13] and more study show that various authors predict the case of this disease using mathematical models that represent systems of difference or differential equations. In addition, based on date analysis and mathematical analysis, authors have explored the effect of lockdown and medical resources on the COVID-19 transformation in Wuhan, China. So motivated by the studies stated above, in current paper we investigated local dynamical properties, bifurcation, and control in a discrete-time COVID-19 epidemic model (7) in . Our under study discrete COVID-19 model (7) asserts us to understand the nature of COVID-19 disease infection in people, and moreover this model also interprets that separation of infected people will decreases in risk of distension of COVID-19 virus. Our work predicts that vaccination of people will minimize the effect of virus and help people to survive in this pandemic fatal disease. Further, this work will asserts as a guideline for those who intend to do research in future. Our presented work is in the direction of discrete dynamical system designated by system of difference equations which open the gate of those researchers who have attention to work in this direction to predict the cause of such fatal disease in future.

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.