Fractional-order delayed Ross-Macdonald model for malaria transmission.

This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

Introduction

Malaria is a severe life-threatening disease prevalent near the equator, especially in Sub-Saharan Africa. Malaria is an insect-borne disease since it is transmitted from one person to another via the bites of infected female Anopheles mosquitoes, rather than directly from person to person like some other diseases. The World Health Organization (WHO) reported 229 million malaria cases in 87 malaria-endemic countries, accounting for an estimated 409,000 deaths in 2019 [1]. This year, the number of malaria deaths may increase due to the COVID-19 outbreak occupying medical care service resources. Up to now, malaria remains one of the major global public health emergencies [2].

At present, most of the extensive research on malaria transmission is underpinned by mathematical models reflecting the process, law, and trend of malaria transmission. The earliest use of a mathematical model to explain malaria transmission was Ross [3], using two nonlinear differential equations to describe the cross-infection of malaria between humans and mosquitoes. Later, Macdonald [4, 5] extended the basic model of Ross and proposed the concept of basic reproduction number, representing the beginning of malaria transmission models. However, the classical Ross–Macdonald model has not considered the incubation periods of Plasmodium within mosquitoes and humans. To some extent, this model cannot explain the natural behavior of malaria transmission adequately. Saker [3] proposed the following delayed Ross–Macdonald model to describe malaria transmission.where and represent the number of infected humans and mosquitoes at time t, respectively. The parameter a is the biting rate of female mosquitoes to people per unit time, k is the ratio of mosquitoes to humans, b is the proportion of infections after bite, c is the effective transmission rate of the humans to mosquitoes, is the recovery rate of the infective humans, is the rate of death from malaria infection, and is per capita natural death rate of mosquitoes. and denote the incubation period of the parasite within the humans and mosquitoes, respectively.

This mathematical model is regarded as the simplest, straightforward, and effective one in describing the level of malaria transmission. For this model, the author studied the boundedness, persistence, and stability of system, and proved the existence of Hopf bifurcation under two special cases. Nevertheless, this discussion has some certain limitations.

Fractional calculus, as an extension of traditional calculus, has attracted remarkable attention in recent years since it can better depict memory and hereditary properties of various materials and processes [6]. Various studies have already demonstrated that a fractional-order model better depicted the dynamic behavior of system than the integer-order counterpart [7, 8]. Most biological systems have memory and hereditary attributes. The memory property of biological systems is embodied not only in integrating more information from the past, but also in immune response in immune cells [9]. Also, the hereditary property reflects in the genetic profile along with the age and status of the immune system. Cole [10] deduced that the cell membrane of biological organisms has fractional-order electrical conductance, and the biological systems are divided into non-integer order models simultaneously. Many studies [11-13] have established that the fractional-order ordinary differential equation has more advantages in modeling some biological systems than the classical integer-order. It is therefore reasonable to use fractional differential to model biological systems. At present, Caputo fractional derivatives have become a preferred method to study and model biological systems instead of first-order derivatives, such as the fractional model of HIV transmission [14], fractional prey-predator system [15], fractional happiness model [16], fractional birhythmic biological system [17], and fractional SIR epidemic model for dengue transmission [12].

Based on the above analyses, Caputo fractional derivatives are introduced into the model (1), aiming to show more clearly the dynamic behavior of malaria transmission. Therefore, the nonlinear fractional-order malaria transmission model with delays can be expressed as:where is the order of fractional derivative, and the notation denotes Caputo fractional derivative operator . Based on biological and epidemiologic significance, the following initial conditions are consideredwhere , , denotes the Banach space of all continuous functions.

The main innovation of the paper is to study a more accurate Ross–Macdonald model of malaria transmission by using fractional derivatives than the previous model. Secondly, the condition of the stability and Hopf bifurcation under different time delays cases are discussed in more detail. The structure of this paper is organized as follows: In Sect. 2, some basic definitions and lemmas of fractional calculus are formulated. Then, in Sect. 3, we discuss the existence and uniqueness of the positive equilibrium. The local stability of the positive equilibrium and the existence of Hopf bifurcation are carried out in Sect. 4. Finally, the numerical simulations are presented in Sect. 5 and summarize our results in Sect. 6.

Preliminaries

Definition 1

[18] The Caputo fractional-order derivative of order for a continuous differentiable function is defined as following form:where denotes the Euler’s Gamma function, and m is the smallest positive integer not less than q, i.e., . In particular, when , such that

The Laplace transform of the Caputo fractional-order derivative is given aswhere .

Definition 2

[19] The Riemann–Liouville fractional-order integral of order for a continuous function is defined as:

Property 1

For a continuously differentiable function y(t), the following equation holds

Property 2

Fractional calculus is linear, that is to say, for any constant a, b satisfies

Lemma 1

[20] (Contraction mapping theory) Let and denote a non-empty complete metric space and a contracting map, respectively. If there exists such thatthen F has a unique fixed point in M.

Given an n-dimension linear fractional-order continuous dynamical system with delaywhere , . The corresponding characteristic matrix isIf , system (3) can be written asIt is generally known that the stability of system is determined by the roots of characteristic polynomials. Then, the stability results of the above two systems are shown as follows.

Lemma 2

[6, 21] For , the zero solution of system (3) is Lyapunov asymptotically stable if and only if all the roots of have negative real parts.

Remark 1

Based on Lemma 2, we can draw conclusion that the stability boundary of fractional-order delayed systems is the imaginary axis.

Lemma 3

[21] The zero solution of system (4) is Lyapunov asymptotically globally stable if and only if all the eigenvalue satisfy with .

Lemma 4

[6, 21] For , if all the eigenvalues of satisfy for and has no purely imaginary roots for any , then the zero solution of system is Lyapunov asymptotically stable.

Remark 2

The stability analysis for the nonlinear delayed fractional-order system is more complicated. At present, there is no mature theory to provide useful methods on nonlinear delayed fractional-order dynamical systems. Thus, in general, the local stability of nonlinear delayed fractional-order systems is researched with the stability theory of fractional-order delayed linear systems.

Existence and uniqueness

Let and C(H) be the class of all continuous functions defined on H, with the norm is defined by , where . Define as a solution of system (2) in the region , where and is sufficiently large.

For any satisfy the initial value condition of system (2), the following inequalities hold.Obviously,The detailed proof process is shown in “Appendix A.”

Theorem 1

For any initial value , there exists a unique solution to system (2).

Proof

Using the properties of fractional calculus, the solution of system (2) satisfiesLet and . Constructing a mapping ,for any ,where . If we choose N such that , it can be inferredthis implies that the mapping is a contraction mapping. Based on Lemma 1, system (2) has a unique solution.

The equilibrium of system (2) satisfies the following equations:System always has a trivial equilibrium point . If the basic reproduction number , system admits a unique positive equilibrium point satisfying initial conditions, where . Below, we only consider the positive equilibrium point in accordance with the real situation.

Stability and Hopf bifurcation

Assume that , the linearized of system (2) at the origin takes the form:According to the Laplace transform of Caputo fractional-order derivative in Definition 1, it yieldswhere .

The above Eq. (6) is rewritten asin whichthus,where

Case I: the time delays

In this case, the characteristic equation (8) is rewritten aswhereIt can be seen that all characteristic roots are located in the left half-plane. Therefore, in this case, the following theorem is established.

Theorem 2

If holds, is Lyapunov asymptotically globally stable for the time delays .

Remark 3

If , all characteristic roots of Eq. (8) lie to the left of the imaginary axis. However, the time delay may affect the stability of system and even induce oscillations, i.e., the root of Eq. (8) may cross the imaginary axis and enter the right half-plane as time delays increase. Therefore, the purely imaginary root is an essential critical case, in which characteristic roots may enter the left half-plane or the right half-plane under a small perturbation.

For further analysis, the fourth-order polynomial equation with real coefficients is first introduced.Define the matriceswith the corresponding determinants

Lemma 5

For Eq. (9), the following results hold:

If , then the polynomial f(x) has at least one positive real root.

If , then the polynomial f(x) has no positive real root.

The polynomial f(x) has no positive real root if and only if the following three conditions are satisfied.

(i) (ii) or

(iii) , or , .

If , then the necessary condition for the polynomial f(x) has at least one positive real root is any one of the following conditions.

(i) (ii) (iii) (iv) (v) (vi) (vii) .

Proof

One can see that and . It is thus clear that Eq. (9) exists such that is satisfied.

If , one can see clearly and for all , which implies that f is monotonic increasing function for . Hence, Eq. (9) has no positive real root in this case.

For this case, it is only necessary to prove that f(x) is a Hurwitz polynomial. Only one of the cases is proved here, and the method of proof for other cases is similar, that is, to prove if , the polynomial f(x) has no positive real root. The detailed derivation process is as follows

Thus, the polynomial f(x) is a Hurwitz polynomial according to Hurwitz criterion, that is, f(x) has no positive real root.

The proof is derived from (III) above.

Case II: the time delays ,

In this case, the characteristic equation (8) can be rewritten aswhere . Suppose that is a purely imaginary root of Eq. (10), then taking it into the characteristic equation yieldswhereLet for simplicity. Squaring the both sides of Eq. (11) and adding them yieldswhereIt is apparent that the coefficient for . If holds, other the coefficients satisfyThis implies that Eq. (12) has no positive real root contradicting this assumption of . According to Lemma 4, this allowed us to conclude that is Lyapunov asymptotically stable for any , namely independent of the delay.

It can show that if ; therefore, Eq. (12) has at least one positive real root according to Lemma 5. Let be the eigenvalue of Eq. (10) such that . From Eq. (11), we definewhereThen, substituting into Eq. (10) and taking the derivative with respect to on two sides yieldsApplying Eq. (10), it is straightforward to obtainthus,By some calculations, it is not difficult to prove that the above expression is always positive. Hence, the transversality condition holds, i.e., system undergoes a Hopf bifurcation at when . Based on the above analyses, the following results can be obtained.

Theorem 3

For the fractional-order malaria epidemic system (2), considering , the following results hold:

If holds, is locally asymptotically stable for all .

If holds, is locally asymptotically stable for and unstable for , system (2) undergoes a Hopf bifurcation when , i.e., it produces a series of periodic solutions from as passes through the critical value .

Remark 4

For the analysis presented here, we focused only on a root passing through the positive imaginary axis for simplicity (i.e., ), since the purely imaginary roots appear in pairs.

Remark 5

When , the integer-order model (1) satisfies Theorem 3. It is verified that the fractional-order model generalizes the classic integer-order model into an arbitrary order. In addition, the research of the fractional-order model is more complex compared with the classic integer-order model.

Case III: the time delays ,

In this case, the rewritten characteristic equation (8) satisfieswhere . The analytical approach is identical to that in Case II, assuming that is the pure imaginary root of Eq. (14) and substituting s into Eq. (14) yieldswhereBy squaring and adding the two equations of Eq. (15) to eliminate variable, one can derivewhereIt is noted that for , whereas other coefficients may be positive or negative. Suppose that the condition holds, then it can be deduced that . Therefore, there is no purely imaginary root of the characteristic equation (14). If the condition holds, we obtain , then at least one root is a positive root for Eq. (16). From Eq. (15), definewhereLet be the root of Eq. (14) near , satisfying . The following transversality condition is obtained by calculation.As the analysis process is very similar to Case II, it is omitted. Summarizing the above discussion, the following stability and bifurcation results of system (2) are given.

Theorem 4

For the fractional-order malaria epidemic system (2), considering , the following results hold:

If the condition holds, is locally asymptotically stable for all .

If the condition holds, the critical value can be defined the stability boundary, that is, is locally asymptotically stable for and lose stability when . Furthermore, the fractional-order system (2) occurs a Hopf bifurcation at near .

Remark 6

Most studies only focused the incubation period of Plasmodium in mosquitoes, but ignored the incubation period in humans. Many authors perceived that the incubation period of Plasmodium in humans is about –100 days (see [22]), accounting for a small part of human life. In this study, the effect of the incubation period of Plasmodium in the humans for the dynamic behavior of the disease is considered. Furthermore, the analysis of Case II and Case III facilitates the comparison of the influence of the incubation period of Plasmodium in humans and mosquitoes on the dynamic behavior of malaria.

Case IV: the time delays , and

For the characteristic equation (8), the time delay is fixed in its stable interval, and is taken as a parameter. Assume that is a purely imaginary root of characteristic equation (8) with , if and only if w satisfieswhereTaking square on the both sides of (17) and adding them, we havewhereSuppose that , then Eq. (18) has at least one positive root . As long as all the parameters are given, it is easy to calculate the value of by means of numerical software Maple. From Eq. (17), we havewhereBy the implicit function theorem, the derivative with respect to for Eq. (8) yieldsConsequently,in whichSuppose thatThus, by the Hopf bifurcation theorem for fractional-order dynamical systems in [23], we can draw the following theorem.

Theorem 5

Assume that (H1) holds, and are satisfied, then is locally asymptotically stable for all , and the fractional-order system (2) undergoes a Hopf bifurcation at when .

Remark 7

The above four Cases can be contrasted with each other to observe the influence of different time delays on the disease transmission steady-state by the real data simulation.

Case V: the time delays

In addition, the incubation period of Plasmodium in humans and mosquitoes may be equal. Consequently, it appears necessary to research the influence on disease spread for . In this case, the characteristic equation (8) becomes the following the formwhere . Let be the root of Eq. (20) and substitute it into the above equation, we havewhere . It is obvious thatEliminating by , we havewhereLet f(w) be defined as followsIn the above definition, we can deduce . Thus, if the parameter holds, Eq. (23) has at least one positive real root, denoted by .

From Eq. (22), one can getLet be the characteristic root of Eq. (20) near , satisfying , then substituting s into Eq. (20) and taking the derivative with respect to on two sides, then it is easy derive thatFurther,whereHence, let us formulate the hypothesisCombining literature [23] and the above analysis results, the following theorem is obtained.

Theorem 6

For time delays , assume that the conditions (H2) and are satisfied. As increases from zero to a critical value , is locally asymptotically stable and loses its stability as continued to increase. The fractional-order system (2) undergoes a Hopf bifurcation at when .

Remark 8

Although there have been many studies on the dynamical behavior of the malaria model, most authors only considered the integer-order models. However, the fractional-order delayed models for malaria transmission have been less researched. This study combines time delays and fractional-order, enriching the malaria model dynamics to a certain extent. Besides this, this paper’s analysis method is also suitable for the general fractional-order delayed model.

Remark 9

However, in literature [3], the author has addressed two special cases of and , which have its limitations. In this work, the local stability and bifurcation of the fractional-order malaria epidemic model for different time delays are researched.

Numerical simulations

In this section, numerical simulations using realistic parameter values (see Table 1) demonstrate and support the effectiveness of the analytical results. According to the feasible range of parameters in Table 1, the specific values of the parameters for model (2) are givenIt is known through calculation that system (2) has a unique positive equilibrium point , and the basic reproduction number implies that malaria persists in a certain range.

Table 1

Parameters values of system (2)

Parameters	Values	References
a	0.2-0.55 day^-1	[22, 24]
b	0.1–0.5	[22, 25]
c	0.3–0.5	[26, 27]
k	2	[22]
μ	0.05day^-1	[28]
γ	0.01–0.05 day^-1	[22]
δ	0.05day^-1	[22, 25]
τ	10–100 day^-1	[22, 29]
σ	5–15 day^-1	[22, 30]

Case I When , the corresponding waveform plots and phase portraits show in Fig. 1. It follows from Fig. 1 that the state trajectories of system converge to the equilibrium point for different values of the fractional-order q. Furthermore, we observe that the fractional-order q has an impact on the peak of the waveform plots, which is reduced as the q value increased. From the epidemiological perspective, the number of infections will rapidly increase and reach the peak during the initial stages of the disease. After a period of time, the number of infections will decrease and remain within a certain range. Thus, models described by the fractional derivative are more accurate and efficient than integer derivatives since it is consistent with the development trend of the infectious disease.

Fig. 1

Waveform plots and phase portrait of system (2) with different values of fractional-order q for

Case II When and , we can obtain . The corresponding waveform plots and phase portraits are shown in Fig. 2. As we can see from the figure, the state trajectories of system (2) converge to the equilibrium point when , and lose its stability once . Figure 3 illustrates that the amplitude of the periodic solution gradually increases with the delay increases, which means that the instability becomes stronger as increasing the time delay.

Fig. 2

Waveform plots and phase portrait of system (2) with and . a, b is asymptotically stable for . c, d System (2) has a family of periodic solutions and undergoes a Hopf bifurcation for

Fig. 3

Periods and amplitudes of periodic solutions for and

Case III When and , in which case is obtained. Figure 4 indicates that is locally asymptotically stable with ; however, system (2) generates a Hopf bifurcation at for .

Fig. 4

a is locally asymptotically stable for . b loses stability and the Hopf bifurcation occurs for

In addition, the impacts of fractional-order q on the values of critical frequency and bifurcation point are illustrated in Fig. 5. It can be observed clearly from Fig. 5a that the values of critical frequency increase slowly with the increase in fractional-order q. Conversely, the increase in the order q goes along with a decrease in the bifurcation point or , as presented in Fig. 5b, which accounts for the stability interval of system is inversely proportional to the fractional-order q.

Fig. 5

a Critical frequency versus the order q. b Bifurcation point versus the order q

Case IV Then, by regarding as a parameter and considering the time delay is fixed in its stable interval for , we can obtain . It is clear, therefore, that the stability interval of system (2) is narrowed compared with Case II when the incubation period of Plasmodium in humans is considered. Thus, when the effect of the incubation period of Plasmodium in humans on the system is not considered, the system has lower prediction accuracy. The corresponding of phase portraits is shown in Fig. 6.

Fig. 6

Phase portraits show that is locally asymptotically stable for [see (a)] and lose its stability for [see (b)] with and

Case V Finally, we can obtain with and . As can be seen from Fig. 7, is local asymptotically stable when and becomes unstable when , at which point a Hopf bifurcation occurs.

Fig. 7

Waveform plots show that is locally asymptotically stable for [see (a)] and the Hopf bifurcation occurs for [see (b)] with

Following the above analysis, it can be seen that the numerical simulation results and the theoretical results match well with each other, validating the correctness of the theoretical analysis in Sect. 4.

Conclusions

In this paper, the nonlinear delayed Ross–Macdonald model for malaria transmission [3] was generalized to a novel fractional-order model, enriching the dynamics behavior and adding complexity. The objective of the present paper was to systematically discuss the effect of time delays and fractional-order on the stability and bifurcation of this malaria model. The results of this study indicated that the time delay could change the system from stable to unstable, and a Hopf bifurcation appears. The stability of system remains virtually unchanged as the delay increases from zero to a critical value. However, system loses its stability and undergoes a Hopf bifurcation when the delay is equal to or more than a critical value. It would indicate that the appropriate incubation periods (via control measures or drugs) are beneficial to control the disease at a stable level better and prevent the massive outbreak of malaria. Another interesting result was that the change of order influences stability interval size. The change of order clearly impacts the critical frequency, further leading to the bifurcation point change. The numerical simulations showed that stability interval can increase with the decrease in order, and the stability interval of system is inversely proportional to the order. This result suggested that the order is significant in modeling epidemic diseases since the best value of order can better predict disease risk and progression.

The discussion method presented in this paper is also suitable for the delayed fractional-order system. The global stability of the positive equilibrium point in the fractional delayed Ross–Macdonald model for malaria transmission is the matter of our future work.