Stability and bifurcation control analysis of a delayed fractional-order eco-epidemiological system.

Considering the factor of artificial intervention in biological control, a delayed fractional eco-epidemiological system with an extended feedback controller is proposed. By using the digestion delay as bifurcation parameter, the stability and Hopf bifurcation are investigated, and the branching conditions are given. The system undergoes Hopf bifurcation, when the parameter τ passes through the critical value. In addition, it can be pointed out that the negative feedback gain and the feedback delay could affect the bifurcation critical value of the system. Therefore, the Hopf bifurcation can also be induced by taking the feedback delay as a bifurcation parameter. Finally, by plotting the solution curve of the system, the significance of the controller to the stability of the eco-epidemiological system is verified.

Introduction

Biological populations thrive in nature, and various organisms are inevitably invaded by various diseases during their lives[1-3]. The COVID-19 pandemic, which started in 2020, still threatens the lives and health of people all over the world. In April 2022, two poultry farms in Hokkaido, Japan, had a highly pathogenic avian influenza outbreak. To prevent the spread of the epidemic, the local government decided to cull more than 500,000 chickens and hundreds of emus. For thousands of years, mankind’s struggle against various infectious diseases has never been interrupted. It was not until 1674, Antony van Leeuwenhoek observed the existence of microorganisms with the help of a microscope, which the foundation for human beings to truly understand diseases was laid. In 1840, Jacob Henle used bacterial theory for the first time to elucidate the pathogenesis of diseases. Later, through the work of Louis Pasteur and others, human beings realized that the origin of disease mainly comes from microorganisms, which was a key step toward the conquest of diseases.

In 1927, Kermack et al.[4] proposed the famous Kermack–McKendrick compartment model and successfully studied the law of disease transmission by using differential equation theory. Since then, mathematical models had become an important tool in the research of infectious diseases. In 1986, Anderson and May[5] first combined the infectious disease system and the Lotka–Volterra system to study the invasion, persistence, and spread of infectious diseases in plant and animal communities. Zhou et al.[6] constructed a time-delayed eco-epidemiological model of prey-infected diseases, and studied the stability of the positive equilibria and the existence conditions of the Hopf bifurcation. Saifuddin et al.[7] established a kind of eco-epidemiological model with predators having weak Allee effect and prey populations are infected. They researched the Hopf bifurcation near the equilibrium point and the chaotic dynamic behavior caused by disease. Taking S(t) represents the density of susceptible prey populations, I(t) to be the density of disease-infected prey populations, P(t) to denote the density of predator populations, Moustafa et al.[8] presented the following ecological infectious disease model of prey infectionHere, is the incidence of nonlinear saturation, and the biological significance of the coefficients in the model is shown in Table 1. Considering the prey is easy to be caught after being infected, model (1) assumed that predators only prey on the diseased prey.

Table 1

Biological significance of symbols

Symbol	Biological significance
r	The intrinsic growth rate of prey population
k	Environmental capacity of prey populations
β	Contact rate factor
η	Half saturation constant of infection
ξ	Mortality of infected prey
φ	Predator attack rate on infected prey
α	Conversion rate of predators on infected prey
δ	Predator mortality

Fractional calculus is an extension of classical calculus theory[9-11]. Since fractional-order system has the property of time memory, establishing complex system using fractional calculus theory can greatly improve the ability of identification, design, and control of dynamic system. According to the needs of modeling, long memory system or short memory system can be used[12-18]. Based on the short memory fractional differential equations, Wu et al.[19] proposed a short memory fractional-order model, and derived the global stability conditions of variable-order neural networks. Because biological populations have memory and genetic characteristics, Kumar et al.[20] used long memory fractional differential equations to establish an eco-epidemiological model, and analyzed the impact of dynamically changing spread and attack rates on system dynamics. Almeida et al.[21] adjusted the order of the fractional derivative of the model to fit the real disease spread data so as to better predict the development of the disease. Mondal et al.[22] deduced that the solution of the fractional-order system converges to several equilibrium points at a slower rate as the order of the differential equation decreases, but the qualitative properties of the solution are the same as those of the integer-order system. Chinnathambi et al.[23] demonstrated that fractional order could enhance the stability of the system and suppress the appearance of oscillations. It should be noted that in most of the previous studies, the biological systems have the same fractional order[8, 24, 25]. In fact, different biological populations have their own unique characteristics[30]. In order to better meet the actual biological background, we construct a fractional-order system with different orders for each variable on the basis of model (1).

To prevent the serious damage or even extinction of animal and plant population caused by infectious diseases, it is particularly important to reasonably intervene and control the ecological epidemiological system[26, 27]. For the treatment of leukemia, Islam et al.[28] designed an adaptive terminal and supertorsional sliding mode controller, and verified the stability of the system using Lyapunov’s stability theory. For a nonlinear four-state ODE malignant tumor model, Qaiser et al.[29] designed a fuzzy controller and two kinds of nonlinear controllers: synergetic and state feedback controllers for chemotherapeutic drug control. An extended feedback controller was introduced into the ecological epidemiological model in[30], Wang et al. confirmed that changing the feedback control time delay could affect the stability of the model, thus effectively restrained the occurrence of bifurcation. It should be pointed out that there are few studies on bifurcation control for ecological epidemiological model with time delays. We couple the feedback controller (where denotes the feedback gain and represents the feedback control time delay) into the system (1), and consider the fractional derivatives of different orders to obtain the following delay fractional-order ecological epidemiological model,where the initial condition is . Among them, are the orders of the fractional derivative of the system variable, and , is the fractional derivative in the sense of Caputo with initial time . represents the digestion delay of the predator. The biological significance of other parameters is the same as in Table 1.

This paper mainly investigates the stability of the positive equilibrium of the system (2) and the conditions for the existence of the Hopf bifurcation. The effect of the controller on the stability of the ecological epidemiology model is analyzed. In order to verify the correctness of the theoretical analysis, the L1 scheme (Oldham and Spanier[31]) is used for numerical simulation. The L1 formula is established by a piecewise linear interpolation approximation for the integrand function on each small interval[32-34]. At the same time, the modified Adams–Bashforth–Moulton predictor–corrector scheme is used to solve the numerical calculation problems of fractional-order delay differential equations [35, 36].

The structure of this article is as follows: Related definitions and lemma are listed in Sect. 2. In Sect. 3, the existence condition of the internal equilibrium point is given. The stability of the system with and without controller is discussed by regarding digestion delay as parameter, and the criteria of Hopf bifurcation are given. In particular, the Hopf bifurcation caused by feedback delay is analyzed. Some numerical simulations and analysis are presented in Sect. 4. Finally, a discussion is presented in Sect. 5.

Preliminaries

Definition 1

[9, 11] Let . The Riemann–Liouville integral of order is defined bywhere .

Definition 2

[9, 11] If f(t) is differentiable, then the Caputo fractional derivative of order for f(t) is defined as

Lemma 1

[10] Consider the under n-dimensional linear fractional-order time-delay system:where the initial conditions are given for , and . It is defined asIf all roots of the have negative real parts, then the zero solution of system (3) is Lyapunov globally asymptotically stable.

Corollary 1

[10] In the case of , , if all the roots of the equation satisfy , then the zero solution of system (3) is Lyapunov globally asymptotically stable.

Corollary 2

[10] Assume that all are rational numbers between 0 and 1, , m is the smallest of the common multiples of the denominators of . Denote by , if all the roots of the equation satisfy , then the zero solution of system (3) is Lyapunov globally asymptotically stable.

Main results

The main objective of this paper is to investigate the existence of periodic solutions by applying the Hopf bifurcation theory. Then, we discuss how the feedback controller affects the bifurcation of the system. Throughout the paper, we make an assumption

: , where is the positive root of the equation . Under , system (2) admits a positive equilibrium , where , .

By applying the transformation of variables, let . Accordingly, the following system can be obtained from model (2)According to system (4), the corresponding linear system can be expressed asConsequently, the characteristic matrix of system (5) iswhere

Influence of time delay on bifurcation dynamics of the uncontrolled system

Let’s consider the Hopf bifurcation conditions of the uncontrolled system with regarding time delay as a bifurcation parameter. When or , system (2) is a fractional system with no controller, the system is as follows,The characteristic Eq. of (6) iswhereFirstly, we consider the stability of equilibrium point in the case of in system (6).

If , we have the following equalitywhere , , , . Under hypothesis , we get , , . By the Routh–Hurwitz criterion, we can find all the roots of Eq.(8) have negative real parts. That is . From Corollary 1, one can deduce of system (6) is locally asymptotically stable.

If are rational numbers between 0 and 1, m is the smallest of the common multiples of the denominators of . Denote by , Eq.(7) can be written as . If hypothesis : is satisfied, then of system (6) is locally asymptotically stable by Corollary 2.

Based on the above discussion, Theorem 1 can be derived.

Theorem 1

Suppose the conditions and hold, then the positive equilibria of the fractional-order system (6) is locally asymptotically stable if .

Next, we consider the stability of equilibrium point in the case of in system (6). Let be a purely imaginary root of (7), so it follows from (7) thatwhereAs far as Eq. (9), one yieldswhere , , It is obtained from Eq. (10) thatSuppose Eq. (11) has a positive real root , we get . At the same time, we apply the notation . Then, in order to better search for the criterion of the occurrence for bifurcation, differentiating Eq.(7) with respect to , we haveSo we can getwhere , . Define , be the real and imaginary parts of individually. , be the real and imaginary parts of individually. Based on algebraic analysis, we can obtain from Eq.(12) thatwhereNow, if we have , then the transversality condition is true, so we can draw a conclusion as following.

Theorem 2

If assumptions and hold, then

If , all the roots of Eq. (7) have negative real parts, and the positive equilibria of system (6) is locally asymptotically stable.

If , the roots of Eq. (7) have at least one root with positive real part, and the positive equilibria of system (6) is unstable.

If , the roots of Eq. (7) have a purely imaginary root, and system (6) exhibits a Hopf bifurcation at the positive equilibria, which imply it has a branch of periodic solution bifurcating from near .

Remark 1

According to Hopf bifurcation theory [37], if the bifurcation parameter passes through the critical value, making the eigenvalue pass through the imaginary axis, Hopf bifurcation will occur in the system. At this point, the critical value is called the bifurcation point. Therefore, if , system (6) will change from the initial stable state to an unstable state, and Hopf bifurcation will occur.

Influence of time delay on bifurcation dynamics of controlled system (2)

In what follows, the influence of the controller will be investigated. For the given , the corresponding characteristic Eq. of (2) iswhereSuppose that is a purely imaginary root of (14), it follows from (14) thatwhereSolving Eq.(15) yieldswhere , , It is apparent from Eq.(16) thatSuppose Eq.(17) has a positive real root , we can get . Meanwhile, we denote . For the sake of searching for the criterion of the occurrence for bifurcation, differentiating Eq.(14) with respect to , we haveSo we can obtainwhere , . Note , be the real and imaginary parts of individually. , be the real and imaginary parts of individually. By mathematical manipulation, we conclude from Eq. (18) thatwhereAs a result, if the suppose : is true, then we can deduce the transversality criteria , so we can summarize what we have proved as the following results.

Theorem 3

In the case of and , we have

If , all the roots of Eq. (14) have negative real parts, and the positive equilibria of system (2) is locally asymptotically stable.

If , the roots of Eq. (14) have at least one root with positive real part, and the positive equilibria of system (2) is unstable.

If , the roots of Eq. (14) have a purely imaginary root, and system (2) exhibits a Hopf bifurcation at the positive equilibria, which imply it has a branch of periodic solution bifurcating from near .

The Hopf bifurcation of system (2) caused by feedback delay

In the first two subsections, we discussed the stability and bifurcation of the system with regarding gestation period as the parameter. In fact, when the time delay is given, feedback control delay can also affect the stability of the system and induce Hopf bifurcation of the system. Hence, we give the following results.

Theorem 4

Suppose conditions and satisfy, we have

If , the positive equilibria of system (2) is locally asymptotically stable.

If , the positive equilibria of system (2) is unstable.

If , system (2) exhibits a Hopf bifurcation at the positive equilibria, which implies it has a branch of periodic solution bifurcating from near .

For the detailed proof of Theorem 4, we can refer to Appendix.

Remark 2

In the previous studies, many scholars ignored the importance of delay parameters in feedback controllers[38, 39]. In fact, as stated in Theorem 4, the feedback control delay parameter can also affect the stability of the system and induce Hopf bifurcation phenomenon.

Numerical results

For verify the feasibility of the theoretical analysis on system stability and bifurcation control, we consider the following system by using the same coefficients as in [8]:At the same time, we define initial conditions . By calculation, system (20) has a unique positive equilibria .

Case I. First of all, let’s talk about the controller-free system, which is or .

Fix , note that Theorem 1 is satisfied for the parameters selected when , the trajectory diagram and phase diagram of the uncontrol system solution are shown in Fig. 1. From this figure, we observe that the species density of susceptible prey, infected prey, and predator changes correspondingly over time from the initial state, but eventually converges to the equilibria . Namely, the positive equilibrium of the controller-free system is locally asymptotically stable if .

Fig. 1

Time series and phase-portrait of the uncontrolled system with and

Fix . By numerical calculation, we can infer and transversality condition is satisfied. With the help of Theorem 2, the positive equilibria is asymptotically stable for , and unstable for . Furthermore, the uncontrol system has a branch of periodic solution bifurcating from near . Fig. 2 depicts the solution curves and phase diagrams of the system at and , respectively.

Fig. 2

Time series and phase-portraits of the uncontrolled system with

In particular, we find that the fractional order also affects the stability of the system. Just for comparison purposes, we simply change in Case I(ii) to . After operation, we acquire that . Therefore, if we fix , the positive equilibria becomes stable from unstable when changes from 0.97 to 0.9(see Fig. 3).

Fig. 3

Time series and phase-portraits of the uncontrolled system with and

Remark 3

In order to describe the influence of fractional order on system stability more clearly, the bifurcation critical values corresponding to parameter from 0.9 to 1 are fitted in Fig. 4. As some researchers have found, the fractional-order system has a wider stability region, which inhibits the periodic oscillation behavior of the system[40, 41].

Fig. 4

The effect of on bifurcation point

Case II. Next, we begin to discuss fractional control system. For the sake of comparison, we take again.

We select , then can be obtained by numerical calculation, and transversality condition is hold. In terms of Theorem 3, the positive equilibria is asymptotically stable for , and unstable for . In the meantime, system (20) has a branch of periodic solution bifurcating from near . Therefore, the equilibria is asymptotically stable when . Compared with the control free system, it is obvious that the extended feedback controller can affect the stability of the system (20), which are simulated in Fig. 5. In particular, if we fix beforehand, the bifurcation critical value of system (20) can be changed by changing the value of the extended feedback gain parameter within a certain range, which is shown in Fig. 6.

Fig. 5

Time series and phase-portraits of system (20) with and

Fig. 6

The effect of on bifurcation point

For test whether Hopf bifurcation can be induced by feedback control delay, let’s choose and . After calculation, we can get and transversality condition is satisfied. Based on Theorem 4, it implies that the positive equilibria is asymptotically stable for , unstable for , and system (20) has a branch of periodic solution bifurcating from near . Time series and phase-portraits of system (20) with are depicted in Fig. 7.

Fig. 7

Time series and phase-portraits of system (20) with

Remark 4

After the detailed comparison above, it is not difficult to find that the desired behavior can be obtained by appropriately adjusting the extended feedback gain parameters and feedback delay parameters. Therefore, the control system proposed in this paper could better meet the needs of real life.

Conclusion

In this research, a delayed fractional ecological epidemiological model with extended feedback controller is presented. First, the stability and Hopf bifurcation of the system in the controller-free and controller states are discussed with regarding the digestion delay as a parameter, respectively, and the existence conditions of periodic solution are given. Secondly, this paper confirms that the Hopf bifurcation can be induced by taking the feedback delay as a bifurcation parameter. Finally, by drawing the solution curve of the system, the influence of the controller on the stability of the system is verified in detail. That is, the stability of the system could be maintained by adjusting the negative feedback gain parameters and feedback time delay reasonably, thereby suppressing the occurrence of periodic solutions. Furthermore, we interestingly find that the periodic oscillatory behavior of the system solution can be suppressed by fractional order, which implies that the fractional-order system has a wider stability region than the integer-order system.