The Global Dynamics of an Age-Structured Hand-Foot-Mouth Disease Model with Saturation Incidence and Time Delay

Yan, Dongxue; Zhang, Mengqi; Tang, Jiashan

doi:https://doi.org/10.1155/2023/7090661

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 7090661 | https://doi.org/10.1155/2023/7090661

The Global Dynamics of an Age-Structured Hand-Foot-Mouth Disease Model with Saturation Incidence and Time Delay

Dongxue Yan,¹Mengqi Zhang,¹and Jiashan Tang¹

Academic Editor: Binxiang Dai

Received05 Sept 2022

Revised22 Dec 2022

Accepted06 Jan 2023

Published24 Jan 2023

Abstract

The global dynamic behavior of an age-structured hand-foot-mouth disease (HFMD) model with saturation incidence and time delay is investigated in the work. The time delay occurs during the transition from latent to infectious individuals. Firstly, the model is expressed as an abstract Cauchy problem. The presence of equilibria is then pointed; meanwhile, we define the model’s basic reproduction number . The conclusions show that a threshold of can be used to evaluate whether the disease is on the verge of extinction or is still present. When , the disease-free equilibrium is globally stable. While , there exists an endemic equilibrium, and the global stability of the endemic equilibrium is also demonstrated. Finally, some numerical examples are given to demonstrate the obtained conclusions.

1. Introduction

Hand-foot-mouth disease (HFMD) is a viral infection primarily caused by enteroviruses. Coxsackie Asckievirus (Cox A16) and enterovirus (EV71) are the most prevalent HFMD virus strains [1]. HFMD mostly affects children younger than the age of ten. But older children and adults may also be infected with HFMD [2]. One of the most substantial components of death in children is HFMD, which has become a main public health worry that may have unquantifiable societal and economic consequences. Thus, researches concerns about the spread of HFMD have risen in recent years.

Because there is no anti-HFMD medical therapy or vaccination available, a huge number of scholars have built mathematical models to study the transmission dynamics and illness control of HFMD. Mathematical models of HFMD infection have advanced significantly over the last few decades. A simple SIR (susceptible-infected-recovered) model was developed to forecast the number of infectious individuals and the length of the outbreak in [3]. After that, more real factors have been included into this simple SIR model, and more complicated mathematical models were proposed in studying the dynamics of HFMD. For instance, Roy and Halder [4] established an SEIR model in 2010 by dividing the infectious group into asymptomatic (E (t)) and symptomatic (I (t)) infectious individuals. Furthermore, Liu [5] formulated an SEIQR model to investigate the effect of quarantine Q (t) on the disease development. Considering the asymptomatic individuals A and contaminated environments W, Wang et al. [6] discussed a SEIARW system and found that both factors are critical to delay and avoid epidemic outbreaks. Tan and Cao [7] developed a SEIVT epidemic model with vaccination to investigate HFMD’s transmission in 2018. Recently, Sun et al. in [8] proposed a diffusive HFMD model with a fixed latent period and nonlocal infections. Besides, these mathematical models are also used in the research of other infectious diseases, such as COVID-19 [9–11]; coffee berry disease [12]; and banana black Sigatoka disease [13].

Very recently, Samanta [14] constructed and described an HFMD infection model with incidence rates and . Standard epidemiological models almost utilize a bilinear incidence rate [15–17] on account of the law of mass action. There are three types of incidence forms used in epidemiological models when the community is saturated with infectious diseases: proportionate mixing incidence [16, 18], nonlinear incidence [19, 20], and saturation incidence [21] or [22]. In this paper, the saturation incidence rates will be considered.

In the transmission of HFMD, individuals who are latently infected have no symptoms and cannot convey the virus to others until several days after exposure (usually 3–7 days [23]). As a result, time delay (latent period) is an important factor in the research of the dynamic behaviors of HFMD systems. Many scholars have investigated delayed HFMD models, for instance [14, 24, 25]. The active HFMD individuals will disseminate the bacterium in the population once the latently infectious persons become active. Thus, infected people’s infectivity is determined by their infection age, which is a significant factor for infectious diseases, particularly the ones with a prolonged latency and fluctuating infection rate like HFMD.

Note that as time progresses, the disease develops within the individuals with the different infectivity or many times one may easily observe vary infectiousness of an infected individual at different stages of infection. Particularly in the transmission of HFMD, latent individuals have no symptoms and cannot convey the virus to others. When asymptomatic individuals can infect others (that is to say they have infectiousness), depending on how long they have been infected, which is usually called infection age [26]. Moreover, the infectivity of the host might continuously change with time and infection age. This means that infection age may be one of the informative factors to model for the diseases like HFMD. Hence, age-structured epidemic models should be extensively examined in order to provide better knowledge and further insights into transmission mechanisms. The authors of [26–34] have widely explored the dynamic behavior of age-structured epidemic systems.

Inspired by the preceding discussion, in this work, we will investigate the HFMD model with age structure and time delay. Let be the infected individuals’ transmission rate at age . To establish the fundamental form of , we utilize the data (see Table 1) from the China Center for Disease Control (CDC) [35] in 2018 and the least square approach to match the transmission rate at varied infection age . The numerical simulation shows that the transmission rate of HFMD follows the exponential function with (see Figure 1), after the shortest period necessary during the initial infection to become infectious class. Therefore, when we discuss HFMD, the transmission rate is taken as the following exponential function:where represents the shortest time required from initial infection to become infectious class, , and are constant coefficients of (1), and . The equation (1) means that if the infection age of the infected individuals is less that the time delay , the infected individuals do not have the infectivity; otherwise, the transmission rate is .

The form of the age-structured HFMD model with saturation incidence and time delay is investigated as follows:where , and with the initial conditionswhere , , , and represent the number of susceptible persons, infectious persons, quarantined persons, and recovered persons at time , respectively. The number of latent persons with age is expressed by . As a result, system (2) is an SEIQR-type HFMD system, and the infection process flowchart is shown in Figure 2.

The parameters in model (2) are considered biologically relevant, and all of them are positive. The biological interpretations of the parameters of the system are described in Table 2.

It is worth pointing out here that, in comparison to the model in [14], system (2), which considers the infection age and infectious delay, is obviously more general and realistic. The asymptotic behavior of solutions to system (2) is the focus of this work. Firstly, we will determine the basic reproduction number , and then study the infection-free equilibrium ’s asymptotic stability for system (2) when and , respectively, through discussing the position of the corresponding characteristic equations’ eigenvalues. Meanwhile, when , the global stability of a positive equilibrium is also discussed by constructing a suitable Lyapunov function. Finally, we are particularly interested in how age-structure, time delay, and saturation incidence affect the dynamics of the system under consideration.

The following is the remainder of this work. In Section 2, we give some preliminary conclusions and prove the well-posedness of the system (2). In Section 3, the existence of the equilibria, particularly the existence and uniqueness of the endemic equilibrium, as well as the linearized system of (2) surrounding the equilibria are discussed. The stability of the equilibrium is demonstrated, in Section 4. Ultimately, in Section 5, we summarize this paper and provide some numerical simulations to demonstrate our theoretical results.

2. Preliminaries and Well-Posedness

For this second, model (2) will be converted as an abstract Cauchy problem (ACP); then we will prove the well-posedness of the system and discuss the nonnegativity and boundedness of solutions. For these intensives, we will first gather some background information about linear operators and -semigroup theory, and some notations will be utilized throughout the paper.

We start by transforming the system (2) into an abstract evolution equation. Define thatand give the linear operator’s formulation with . Then , which implies that is not dense in . A nonlinear operator given by the following:and let . Hence, based on the above notations, system (2) can be rewritten into an abstract Cauchy problem as follows:with .

Generally speaking, solving an abstract differential problem with a strong solution such as (7) is challenging. As a result, we solve (7) in integrated form as follows:

Set

In Section 3, the operator will be shown to be a Hille–Yosida operator, and thus, it produces a -semigroup on the closure of its domain, according to Theorem 3.8. in [36]. Next, the well-posedness result of the model (7) can be obtained.

Theorem 1. In , model (2) has a unique continuous solution. Furthermore, the mapping defined by for is a continuous semiflow, i.e., the mapping is continuous and meets the condition and .

From biological interpretation, , , , and represent the number of susceptible persons, infectious persons, quarantined persons, and recovered persons at time , respectively. represents the number of latent persons with age . Therefore, only nonnegative solutions of the system (2) are meaningful. The following result demonstrates that the nonnegative initial value solutions of (2) remain nonnegative and bounded eventually.

Theorem 2. For any , all the solutions of system (2) with nonnegative initial values stay nonnegative and are eventually bounded.

Proof. To begin, by integrating the second equation of (2) along the characteristic line, we can get the following result:Similar to the proof of Lemma 2.2 in [27], it is easy to check that remains nonnegative for nonnegative initial data.
Next, before proving is nonnegative for , assume that and . Thus, the third equation of model (2) can be replaced as follows:Then we have hence, for .
Through the fourth equation of system (2), we havethus, .
Similarly, from the fifth equation, we can obtainwhich indicates that
Then, we demonstrate for , is nonnegative. If we assume there such that , and for . In fact, by the first equation of system (2), there is , which obviously is inconsistent. Thus, for any .
Finally, the boundness of the solutions of model (2) will be proved. The total number of latent persons is represented by . There is a biologically defined maximum age; hence, is a plausible assumption. From system (2), we deduceTherefore,As a result, the omega limit set of model (2) is constrained in the below feasible region:which is obviously a positive invariant for model (2), implying that system (2) is ultimately bounded.

3. Equilibria and Properties of Linearization at Equilibria

The nonlinear system (2) will be linearized around the equilibrium solutions in this section. In order to do this, we first consider whether or not the equilibrium exist. System (2) always maintains a disease-free equilibrium with . For the purpose of finding the nontrivial equilibrium of system (2), the following equations are proposed:

We solve the above equation in (17), assume , , and then obtainwith .

On the other hand, by the first equation (17) and conducting some computations, we obtain that

Thus,

Note that , assume that . Because , and , , thus , hence .

We propose the definition of the basic reproduction number as follows: is defined to be the expected number of secondary cases produced in a completely susceptible population by a typical infected individual during its entire period of infection [23]. can also be calculated by the recipe in van den Driessche and Watmough [37]. Moreover, Diekmann et al. [38] have proposed that is the spectral radius of the next-generation matrix.

In fact, each phrase in has a distinct epidemiological meaning. is the likelihood of a latent person living to the age of . In addition, is the total number of infectious persons produced during a latent individual’s lifetime. is the average infection period. is the transmission rate of the infectious individual. is the overall number of susceptible individuals. is the saturation incidence. As a result, represents new cases generated by the average number of typical infected members during the course of the infection period. As determined by the preceding study, the basic reproduction number of system (2) is .

Then, from (19), we have . Thus, if , then system (2) has a single positive solution. As a result of the aforesaid study, we arrive at the following conclusion.

Theorem 3. The model (2) always has a disease-free equilibrium . Furthermore, when , a unique endemic equilibrium exists.

Let , , , , , where is any consistent condition of the model (2), and , . System (8) is identical to the following Cauchy problem

Through computations, the linearized system of (8) can be obtained as follows:in whichThus, on the compactness of the bounded linear is obviously derived.

Notice that , then the statement below can be proven.

Theorem 4. The operator is a Hille–Yosida operator.

Proof. For , there isThus, we knowIntegrating the second equation (26) with regard to the age variable and summating all the equations, we getThus, we havewhich implies that is a Hille ̶Yosida operator.
Based on Theorem 1.3 in [36] and Theorem 4, it shows that the operator is a Hille ̶Yosida operator. Furthermore, according to Theorem 3.8 in [36], we get two strongly continuous semigroups and generated by the part of and on , respectively.
By the proof of Theorem 4, the Hille ̶Yosida is estimated; thus, we know . Moreover, is compact for any . Sincewhich implies is a quasi-compact -semigroup. Thus, Theorem B.1 in [39] deduces that, for some , as when any eigenvalue of has a negative real part.
Now, we may derive the result below, based on the previous arguments.

Theorem 5. The solution semiflow of the system (2) as given in Theorem 1 satisfies the following characteristics.(i)The steady-state is locally asymptotically stable if any eigenvalue of has a strictly negative real part.(ii)If at least one of the eigenvalues of has a strictly positive part, then the steady state is not stable.

4. Stability of Equilibria

Depending on the preceding analysis, the local stability of disease-free equilibrium will be firstly analyzed in this section.

Theorem 6. When , the disease-free equilibrium of system (2) is locally asymptotically stable. Nevertheless, when , is unstable.

Proof. Let , , , , . Linearizing system (23) at , we get the system as follows:Substitute , , , , into the (30), giving the equationsSolving equation (31), we obtainSubstituting into the last equation (31) and noting is arbitrary, thenwhich is the characteristic equation of model (30) at .
Obviously, if , is a real function of that is decreasing, and satisfies thatTherefore, if , the characteristic equation has at least one real positive solution, and the equilibrium is not stable. However, when , then does not have complex roots with non-negative real parts. We assume is an arbitrary complex root with ; thus,which is an inconsistency. As a result, the complex root of must have a negative real portion. Thus, when , is locally asymptotically stable.

Theorem 7. When , the disease-free equilibrium of system (2) is globally asymptotically stable.

Proof. Based on the result of Theorem 6, here we only need to demonstrate that for every nonnegative solution of Model (2),Indeed, (15) implies thatAs a result, for , there is such that , for , and thus,Then, consider the following linear system:We may conclude that system (39) permits a solution of the typewhere , , and are positive and is a root of the corresponding characteristic equation of this system by utilizing the similar way of proving Theorem 6. The formulation (10) of solution for the second equation in the system (2) yields that , for ; thus, , which implies ; therefore, .
That is,Since , all the eigenvalues of the system (39) have a negative real part, due to the continuous reliance of on , there exists s.t. , indicating thatThus, the first equation in (2) is asymptotic to equationObviously, there is . Using the asymptotic autonomous semiflow theory (see Corollary 4.3 in [40]), we can find out thatThus, if , thenThe demonstration is now complete.
In the following, we primarily concern about the stability of endemic equilibrium . We linearize system (2) at endemic equilibrium to investigate its local stability. We do this by introducing the perturbation variables , , , , , which lead toIn order to analyze the stability of , we seek for solutions of the type , , , , . Then, the following eigenvalue issue is obtained:We obtain by solving the second and the third equations in (47):By the fourth and the fifth equations in (47), there areThe first equation in (47) yields thatBy the last equation in (47), and then combining with (48)–(50), we obtainThus, the characteristic equation (47) is derived as follows:where.
For with , we have , and ; thus,Thus, it is easy to demonstrate that the left side of the characteristic (52) holds for with :For the right side of (52), when , we obtain thatwhere the notations of defined at the beginning of Section 3. Therefore, for with the left-hand side of (52) is strictly greater than 1, but the right-hand side of (52) is not greater than 1. That is an inconsistency, i.e., the solutions of the characteristic (52) must have a negative real part. As a result, the endemic equilibrium is locally asymptotically stable, which allows us to derive the following conclusion.

Theorem 8. When , the endemic equilibrium of system (2) is locally asymptotically stable.

As the system (2) is a high-dimensional system, it is hard to estimate the global stability of the endemic equilibrium . However, we can verify the global stability of for the situation . In this case, the function and the Lyapunov functional are defined as follows:where

Using (17), we take the derivative of each portion of the Lyapunov functional described in (57) along with the solutions of system (2) separately. Firstly, we obtain

By using (10), we can rewrite

Secondly, using (18) and the fact yield

Denote that , and

Then, we have

Differentiating along the solutions of system (2) yields

Adding (58), (62), and (63) together yields

Notice that

Therefore, we havewhere

For any , with equality holding if and only if . Thus, . Hence, . Then, it can be confirmed that the singleton is the biggest invariant set of . According to [41], we know that is globally asymptotically stable, as shown by the compact global attractor .

The preceding considerations lead to the following conclusion.

Theorem 9. Suppose , and , the endemic equilibrium of system (2) is globally asymptotically stable.

Remark 1. For , and , it is difficult to construct a suitable Lyapunov function to verify the global stability of the endemic equilibrium . Here, we take and use numerical simulation to display the endemic equilibrium . Compared with the case , we find that the endemic equilibrium is still globally asymptotically stable (see Figure 3). Moreover, it can be seen that with the increasing of, the number of latent and infectious persons increase. Hence, the rate of loss of immunity has a negative impact on the control of the disease.

5. Conclusion and Discussions

In this research, an age-structured SEIQR model of HFMD has been developed and studied, with the infection age and time lag. The well-posedness of the system (2) is explored first, and following that the solutions’ positivity and boundedness for system (2) is discussed. Next, the basic reproduction number is derived, which has been shown to be the threshold for determining disease extinction or survival. That is to say, when , the disease-free equilibrium is globally asymptotically stable; otherwise, the system is unstable. Moreover, we explore the local stability of the infected equilibrium by analyzing the distribution of roots to the related characteristic equation and the global stability by constructing suitable Lyapunov functions for the epidemic model (2) when , and demonstrate the is still globally stable when by numerical simulation. Thus, by managing the basic reproduction number , we may control the spread of HFMD in the population.

In the following, we utilize numerical simulations to demonstrate the theoretical results of the system (2) by Matlab software and reveal the influences of age-structure, saturation incidence and time delay on the disease dynamics. Firstly, according to Chinese real data about HFMD, we choose the parameters in the system (2). The entire population of China in 2020 is 1412120000 and the birth rate is 8.52 per thousand [42]; thus, . And the natural mortality rate of China is 7.07 per thousand, thus . From [35], the number of infected cases in 2018 is about 2353310. We assume that , thus , . The report shows that the exposure period for HFMD is about 4–7 days; thus, the initial latent population is . We just supply an a priori estimate for the initial recovered population . The ratio of the entire youthful population aged from 1 to 14 is 17.9; therefore, . We may derive the death rate of infection . The decubation of infection was about 2 weeks [43, 44], that is, the recovery rate of infectious individuals . Besides, in all the following figures, we set represents the total number of latent individuals.(1)The maximum infection age we chose is 100, , , , , , , reference from [23]. Accordingly, we obtain . The numbers of the susceptible persons, the latent persons, the infectious persons, the quarantined persons, and the recovered persons with three different initial values are displayed in Figure 4, implies that the disease-free equilibrium of system (2) is globally asymptotically stable when . It is compatible with Theorem 6, which indicates that if we control the fundamental regeneration number to be kept below 1, the illness will be eradicated.(2)When , the impact of delay “” on the solutions of system (2) are displayed in Figure 5. We take , respectively, and keep the other parameters the same with Figure 4. Figure 5 states that as the delay becomes longer, the time for achieving disease-free equilibrium increases.(3)Under , the stability of positive equilibrium will be shown next. When we take the parameters , and others keep the same with Figure 4, then we get . Figure 6 illustrates that the solutions of system (2) with three distinct initial values will trend to as approaches infinity, revealing that the positive solution probably be globally asymptotically stable when .(4)Fourthly, Figure 7 shows the influence of delay “” on the solutions of system (2) when . The delay does not influence the stability of , but from Figure 7 we can see that as the delay (in the incidence of latent infections) increases, the number of HFMD infections, quarantines, and recoveries decreases.(5)Accordingly, the distribution of latent individuals with respect to infection age at the endemic equilibrium , is shown in Figure 8(a), which corresponding to the second solution line in Figure 6, and the distributions with both infection age and time, is shown in Figure 8(b).(6)Finally, we examine that the effect of the saturation incidence rate parameters of the system (2) on the dynamics of HFMD when . The impact of the saturation incidence rate on the solutions of the system (2) are displayed in Figure 9. When , respectively, and the other parameters are the same as in Figure 6, then . From Figure 9, we know that as the saturation incidence rate’s parameters increases, the number of the susceptible persons raises, as well as the number of the latent persons , infectious persons , quarantined persons , and recovered persons all decrease. In summary, the higher the saturation incidence rates’s parameter is, the faster the HFMD can be controlled.

(a)

(b)

Comprehensive to is it can know the growth of time delay adverse to the formation of disease-free equilibrium when , and causes the increasing of the number of HFMD infections, quarantines and recoveries decrease when . Therefore, the increase in the saturation incidence rate is benefit for the control of HFMD.

Data Availability

The HFMD data used to support the findings of this study are deposited in the repository: https://www.phsciencedata.cn/Share/ky_sjml.jsp?id=b9c93769-3e0f-413a-93c1-027d2009d8bc.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Dongxue Yan conceptualized the study, performed formal analysis, and wrote the original draft. Mengqi Zhang conceptualized the study, developed the methodology, and wrote the original draft. Jiashan Tang wrote the original draft, edited the manuscript, and visualized the study. All authors have read and approved the final manuscript.

Acknowledgments

The authors also would like to thank Professor Hui Cao for her suggestions, which also helped us to improve this paper. This work was supported by National Natural Science Foundation of China, (grant no. 18dz2271000, grant no. 12101323), Natural Science Foundation of Jiangsu Province of China (grant no. BK20200749), Nanjing University of Posts and Telecommunications Science Foundation (grant no. NY220093).

References

P. McMinn, K. Lindsay, D. Perera, H. M. Chan, K. P. Chan, and M. J. Cardosa, “Phylogenetic analysis of enterovirus71 strains isolated during linked epidemics in Malaysia, Singapore, and Western Australia,” Journal of Virology, vol. 75, pp. 7732–7738, 2001.
View at: Publisher Site | Google Scholar
World Health Organization, “A guide to clinical management and public health response for hand, foot and mouth disease (HFMD),” 2011, https://iris.wpro.who.int/handle/10665.1/5521.
View at: Google Scholar
F. C. S. Tiing and J. Labadin, “A simple deterministic model for the spread of hand, foot and mouth disease (HFMD) in sarawak,” in Proceedings of the 2008 Second Asia International Conference on Modelling and Simulation, pp. 947–952, Kuala Lumpur, Malaysia, May 2008.
View at: Publisher Site | Google Scholar
N. Roy and N. Halder, “Compartmental modeling of hand, foot and mouth infectious disease (HFMD),” Research Journal of Applied Sciences, vol. 5, no. 3, pp. 177–182, 2010.
View at: Publisher Site | Google Scholar
J. Liu, “Threshold dynamics for a HFMD epidemic model with periodic transmission rate,” Nonlinear Dynamics, vol. 64, pp. 89–95, 2011.
View at: Publisher Site | Google Scholar
J. Wang, Y. Xiao, and R. A. Cheke, “Modelling the effects of contaminated environments on HFMD infections in mainland China,” Biosystems, vol. 140, pp. 1–7, 2016.
View at: Publisher Site | Google Scholar
H. W. Tan and H. Cao, “The dynamics and optimal control of a hand-foot-mouth disease model,” Computational and Mathematical Methods in Medicine, vol. 2018, 2018.
View at: Publisher Site | Google Scholar
G. Q. Sun, H. T. Zhang, L. L. Chang, Z. Jin, H. Wang, and S. Ruan, “On the dynamics of a diffusive foot-and-mouth disease model with nonlocal infections,” SIAM Journal on Applied Mathematics, vol. 82, 2022.
View at: Publisher Site | Google Scholar
A. Asami and H. Nishiura, “Doubling time of infectious diseases,” Journal of Theoretical Biology, vol. 554, Article ID 111278, 2022.
View at: Publisher Site | Google Scholar
J. K. K. Asamoah, E. Okyere, and A. Abidemi, “Optimal control and comprehensive cost-effectiveness analysis for COVID-19,” Results in Physics, vol. 33, Article ID 105177, 2022.
View at: Publisher Site | Google Scholar
X. Ma, X. F. Luo, L. Li, Y. Li, and G. Q. Sun, “The influence of mask use on the spread of COVID-19 during pandemic in New York City,” Results in Physics, vol. 34, Article ID 105224, 2022.
View at: Publisher Site | Google Scholar
D. J. F. Mbogne, D. E. H. Danga, and D. Bekolle, “Nonlinear observer of the non-spatial dynamics of anthracnose: case study of coffee berry disease,” Applied Mathematics and Computation, vol. 433, Article ID 127376, 2022.
View at: Publisher Site | Google Scholar
F. P. Agouanet, I. T. Chedjou, R. M. Etoua, and J. J. Tewa, “Mathematical modelling of Banana Black Sigatoka disease with delay and seasonality,” Applied Mathematical Modelling, vol. 99, pp. 380–399, 2021.
View at: Publisher Site | Google Scholar
G. P. Samanta, “A delayed hand foot mouth disease model with pulse vaccination strategy,” Computational and Applied Mathematics, vol. 34, no. 3, pp. 1131–1152, 2015.
View at: Publisher Site | Google Scholar
R. M. Anderson and R. M. May, “Population biology of infectious diseases part I,” Nature, vol. 180, pp. 361–367, 1979.
View at: Publisher Site | Google Scholar
R. M. Anderson and R. M. May, Infectious Disease of Humans, Dynamical and Control, Oxford University Press, Oxford, UK, 1991.
W. Wang, “Global behavior of an SEIRS epidemic model with time delays,” Applied Mathematics Letters, vol. 15, pp. 423–428, 2002.
View at: Publisher Site | Google Scholar
K. L. Cooke and P. van Den Driessche, “Analysis of an SEIRS epidemic model with two delays,” Journal of Mathematical Biology, vol. 35, pp. 240–260, 1996.
View at: Publisher Site | Google Scholar
H. W. Hethcote and P. van Den Driessche, “Some epidemiological models with nonlinear incidence,” Journal of Mathematical Biology, vol. 29, pp. 271–287, 1991.
View at: Publisher Site | Google Scholar
J. Hui and L. Chen, “Impulsive vaccination of SIR epidemic models with nonlinear incidence rates,” Discrete and Continuous Dynamical Systems - Series B, vol. 4, Article ID 818401, 605 pages, 2004.
View at: Publisher Site | Google Scholar
R. M. May and R. M. Anderson, “Regulation and stability of host–parasite population interactions II: destabilizing process,” Journal of Animal Ecology, vol. 47, pp. 219–267, 1978.
View at: Publisher Site | Google Scholar
S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with nonlinear incidence rate,” Journal of Differential Equations, vol. 188, pp. 135–163, 2003.
View at: Publisher Site | Google Scholar
J. Y. Yang, Y. M. Chen, and F. Q. Zhang, “Stability analysis and optimal control of a hand-foot-mouth disease (HFMD) model,” Journal of Applied Mathematics and Computing, vol. 41, pp. 99–117, 2013.
View at: Publisher Site | Google Scholar
J. Guan, “Stability analysis of hand foot and mouth infectious disease model with time delay and non monotonic contact rate,” in Proceedings of the 2019 2nd International Conference on Intelligent Systems Research and Mechatronics Engineering (ISRME 2019), London, UK, 2019.
View at: Publisher Site | Google Scholar
G. P. Samanta, “Analysis of a delayed hand foot mouth disease epidemic model with pulse vaccination,” Systems Science and Control Engineering, vol. 2, no. 1, pp. 61–73, 2014.
View at: Publisher Site | Google Scholar
V. P. Bajiya, J. P. Tripathi, V. Kakkar, J. S. Wang, and G. Q. Sun, “Global dynamics of a multi-group SEIR epidemic model with infection age,” Chinese Annals of Mathematics, Series B, vol. 42, no. 6, pp. 833–860, 2021.
View at: Publisher Site | Google Scholar
C. J. Browne and S. S. Pilyugin, “Global analysis of age-structured within-host virus model,” Discrete and Continuous Dynamical Systems, vol. 18, pp. 1999–2017, 2013.
View at: Publisher Site | Google Scholar
X. C. Duan, S. L. Yuan, Z. P. Qiu, and J. L. Ma, “Global stability of an sveir epidemic model with ages of vaccination and latency,” Computers and Mathematics With Applications, vol. 68, no. 3, pp. 288–308, 2014.
View at: Publisher Site | Google Scholar
X. Z. Li, J. Yang, and M. Martcheva, Age Structured Epidemic Modeling, Springer Nature, Berlin, Germany, 2020.
P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer International Publishing, Manhattan, NY, USA, 2018.
P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman, and A. S. Perelson, “An age-structured model of HIV infection that allow for variations in the production rate of viral particles and the death rate of productively infected cells,” Mathematical Biosciences and Engineering, vol. 1, pp. 267–288, 2004.
View at: Publisher Site | Google Scholar
L. Rong, Z. Feng, and A. S. Perelson, “Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,” SIAM Journal on Applied Mathematics, vol. 67, pp. 731–756, 2007.
View at: Google Scholar
J. Wang, J. Lang, and X. Zou, “Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission,” Nonlinear Analysis: Real World Applications, vol. 34, pp. 75–96, 2017.
View at: Publisher Site | Google Scholar
D. Yan and X. Fu, “Analysis of an age-structured HIV infection model with logistic target-cell growth and antiretroviral therapy,” IMA Journal of Applied Mathematics, vol. 83, no. 6, pp. 1037–1065, 2018.
View at: Publisher Site | Google Scholar
Chinese Center for Disease Control and Prevention, “Statutory infectious diseases report from 2003-2018,” 2022, https://www.phsciencedata.cn/Share/ky-sjml.jspid=b9c93769-3e0f-413a-93c1-027d2009d8bc.
View at: Google Scholar
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, NY, USA, 2000.
P. Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.
View at: Publisher Site | Google Scholar
O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, pp. 365–382, 1990.
View at: Publisher Site | Google Scholar
M. Martcheva and H. R. Thieme, “Progression age enhanced backward bifurcation in an epidemic model with super-infection,” Journal of Mathematical Biology, vol. 46, pp. 385–424, 2003.
View at: Publisher Site | Google Scholar
K. Styblo, Selected Papers: Epidemiology of Tuberculosis, Royal Netherlands Tuberculosis Association, The Hague, Netherlands, 1991.
P. Magal, C. C. McCluskey, and G. Webb, “Lyapunov functional and global asymptotic stability for an infection-age model,” Applicable Analysis, vol. 89, pp. 1109–1140, 2010.
View at: Publisher Site | Google Scholar
China statistical organization, “China population statistic yearbook,” 2020, http://www.stats.gov.cn/tjsj/ndsj/.
View at: Google Scholar
Z. Du, W. Zhang, D. Zhang, S. Yu, and Y. Hao, “Estimating the basic reproduction rate of HFMD using the time series SIR model in Guangdong, China,” PLoS One, vol. 12, no. 7, Article ID e0179623, 2017.
View at: Publisher Site | Google Scholar
S. Takahashi, Q. Liao, B. T. P. Van et al., “Hand, foot, and mouth disease in China: modeling epidemic dynamics of enterovirus serotypes and implications for vaccination,” PLoS Medicine, vol. 13, no. 2, Article ID e1001958, 2016.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Dongxue Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

558

Downloads

336

Citations