Abstract
As both ticks and hosts may carry one or more pathogens, the phenomenon of coinfection of multiple tick-borne diseases becomes highly relevant and plays a key role in tick-borne disease transmission. In this paper, we propose a coinfection model involving two tick-borne diseases in a tick-host population and calculate the basic reproduction numbers at the disease-free equilibrium and two boundary equilibria. To explore the impact of coinfection, we also derive the invasion reproduction numbers which indicate the potential of a pathogen to persist when another pathogen already exists in tick and host populations. Then, we obtain the global stability of the system at the disease-free equilibrium and the boundary equilibrium, respectively, and further demonstrate the existence conditions for uniform persistence of the two diseases. The final numerical simulations mainly verify the theoretical results of coinfection.
1. Introduction
Tick-borne diseases, which mainly comprise Lyme disease, human babesiosis, tick-borne encephalitis, and human granulocytic, are becoming an increasingly significant danger to people living in countrysides or near woodlands all over the world. Specifically, Lyme disease, which is one of the most widespread tick-born diseases in the world, is caused by the pathogen Borrelia burgdorferi, and medical reports prove that there were more than 20,000 cases already as early as 2013 in America [1] and that the number has continued to increase year after year. As for human babesiosis, the parasite Babesia microti is responsible for the occurrence of this disease, and the number of human cases is in steady increase in northeastern America [2]. Tick-borne encephalitis, instead, is an arthropod-transmitted viral infection caused by TBE virus and frequently occurs in some European countries. Human granulocytic anaplasmosis is caused by Anaplasma phagocytophilum that is a bacterium, which has done great harm to human health over the last decade [3].
The main species that transmit all the tick-borne diseases mentioned above are Ixodes ticks, so their dynamics plays a significant role in studying the transmission of tick-borne diseases. For instance, a black-legged tick named Ixodes scapularis can be infected by a large number of different kinds of pathogens including B. burgdorferi, B. microti, and A. phagocytophilum or any combination of them concurrently [4]. An early study reported occurrences of coinfection of different pathogens in tick and host populations [5], and several case studies described the coinfection among tick-borne diseases from biotic experiments and ecological perspectives.
Diuk-Wasser et al. [6] analyzed the promoting effect of coinfection of B. burgdorferi and B. microti on disease transmission in conformity with epidemiological, ecological, and clinical effects. Horowitz et al. [7] applied blood culture and serology to detect how human granulocytic anaplasmosis in Lyme disease infections played a role in the apparent rate of coinfection and in the severity to illness by studying concrete cases to explore their interaction. Welc-Faleciak et al. [8] performed a retrospective study concerning tick-exposed hosts living in southeastern Poland, aimed at probing into the risk of coinfection between Borrelia species and Anaplasma phagocytophilum or Babesia spp.. These articles have clearly shown that coinfection among tick-borne pathogens or diseases is a real phenomenon through actual case studies or experimental analysis and demonstrated that coinfection can indeed have a certain impact on the disease transmission.
Mathematical models have been applied to the coinfection of infectious diseases. Gao et al. [9] established a SIS epidemic model that includes coinfection between two infectious diseases and proved a sufficient condition for coexistence of both diseases by introducing the invasion reproduction number. Tang et al. [10] integrated the relevant transmission dynamics describing coinfection of dengue and Zika into a mathematical model and focused on how dengue vaccine affected the outbreak of Zika. Wang et al. [11] proposed a Zika-dengue model emphasizing the joint dynamics between dengue and Zika and discussed the impact of vaccination against dengue and antibody-development enhancement. These studies are all related to mosquito-borne diseases; however, to our best knowledge, reports of mathematical models regarding tick-borne coinfection are relatively scarce. Exclusively, Lou et al. [2] built a tick-borne pathogen model with coinfection and discussed the promotion effect of coinfection on two diseases transmission. Unfortunately, they just considered coinfection with two determinated diseases, Lyme disease and human babesiosis, and ignored the effects of invasion on coexistence of the two pathogens.
Based on the previous studies mentioned above, our objective here is principally to formulate a tick-borne coinfection model comprising tick dynamics and host dynamics with two pathogens and obtain the coexistence conditions of two tick-borne diseases. This paper is organized as follows. We establish a coinfection model involving tick and host populations in Section 2. Then, we calculate basic reproduction numbers and invasion reproduction numbers and present their explicit expressions in Section 3. We further discuss the threshold dynamics at the disease-free equilibrium and two boundary equilibria, as well as the coexistence conditions of two tick-borne diseases in Section 4. Numerical simulations validate the coinfection theories and reveal the coexistence conditions of two diseases, and finally, we provide some discussions in Section 5.
2. The Model
We formulate a SIS-type model consisting of two tick-borne pathogens to describe the disease dynamics of coinfection as well as the effect on the spread of two tick-borne diseases. Note that we only consider one life stage of tick, nymph, which can account for the infection on the host to a great extent [12]. Both ticks and hosts can be infected with two or more pathogens through tick bites and blood meals.
Let and be the total tick population and host population, respectively. The tick population is divided into four subclasses: susceptible ticks to two diseases, infected ticks with only disease 1, infected ticks with only disease 2, and infected ticks with both diseases, which can be described as , , , and , respectively. Here, the total tick population is
The host population is also partitioned into four categories: susceptible hosts to two diseases, infected hosts with only disease 1, infected hosts with only disease 2, and infected hosts with both diseases, which are expressed as , , , and , respectively. In the same way,
In combination with the above statements, a transmission diagram describing coinfection of two diseases transmission among ticks and hosts is explicitly depicted in Figure 1. Then, we construct the following model:where all parameters involved in the model are listed in Table 1. We assume the summations of transmission probabilities and are greater than original and without coinfection, respectively, and which also hold for the transmission probabilities of disease 2. Susceptible ticks recruited at a constant rate are infected and can move into by blood feeding on or ( or ) with rates of disease transfer , (, ). can move to by feeding on or ( or ) with infection probability , (, ). In addition, susceptible ticks may also be infected by taking a blood meal from directly with infection probability and move into compartment . We also consider which represents the exit rate from the current compartment of the tick population. Similarly, the infection process with two diseases in the host population can be derived.
(a)
(b)
For the total populations of ticks and hosts, it can be shown that
It follows that the asymptotic equilibria of the two populations satisfy and . Here, Lemma 1 illustrates the basic properties of solutions in system (3), and the proof corresponding to this lemma can be derived in the Appendix of Wang [11].
Lemma 1. Let . All solutions of system (3) with nonnegative initial values remain nonnegative for . Specifically, the solutions satisfy under the condition that the initial value for . Furthermore, the solution set is positively invariant.
3. The Reproduction Numbers
The reproduction number , which is an essential threshold concept in epidemiological studies, denotes the average number of secondary cases caused by an infectious individual when it is introduced in an entirely susceptible population [11]. Mathematically, can measure the maximum reproductive capacity and determine whether the disease will become endemic or die out [13]. In this section, we calculate basic reproduction numbers and invasion reproduction numbers.
3.1. Basic Reproduction Number
In system (3), there always exists a disease-free equilibrium in which and . Based on the calculation of the relevant next generation matrices [14], we can easily obtain the specific form of the basic reproduction number, that is,where for . Biologically, indicates the average number of secondary cases infected by an infectious tick or host with disease 1 when this infection is drawn into a susceptible population. Concretely, it is the product of the number of new ticks infected with disease 1, which are those produced by susceptible ticks feeding on an infectious host with disease 1 at a rate during this host’s average lifetime and the number of new hosts infected with disease 1 generated by an infectious tick with disease 1 when taking blood meals of susceptible hosts at a rate during the period . The biological interpretations for and are analogous.
It is clear that there are two boundary equilibria and , which represent the cases in which just disease 1 and just disease 2 are endemic in the population, respectively.
Therefore, at , we have that , , , and are all zero, andwherein whichwith .
We substitute (7) into (8) and get
Moreover, there also exists
Combining (9) and (10), we obtain
Therefore, (11) is positive if , but when , no positive root can be found. Thus, the equilibrium exists provided .
Similarly, the boundary equilibrium takes the following form:wherein whichwith . Also, the equilibrium exists if .
3.2. Invasion Reproduction Number
The basic reproduction number works as a threshold that only includes information about diseases transmission in one population containing a single pathogen and focuses on the static state, but it cannot represent the number of secondary cases infected by an infectious individual with one disease when this infection is led into a population where another disease already exists. Thus, it is necessary to introduce a new threshold quantity: the invasion reproduction number [15].
In the following, we calculate two reproduction numbers which are the invasion reproduction number for disease 1 when disease 2 is already endemic and the invasion reproduction number for disease 2 when disease 1 is already endemic, respectively.
Considering that the derivation of the invasion reproduction number for disease 2, namely, , is based on a population in which disease 1 is already endemic, we concentrate on the boundary equilibrium to reckon the matrices and as follows:
Next, taking the derivative of and with respect to and substituting the boundary equilibrium into the variables, we have
Then, the next generation matrix is given bywherein which , , , , and are expressed in (7). The characteristic equation is
LetIt follows from that the dominant eigenvalue of equation (19) satisfies , and then we can acquire the invasion reproduction number for disease 2, that is, .
Similarly, the invasion reproduction number for disease 1, namely, , can also be derived following the above process, which is given bywhere
Note that , , , , and are defined in (13).
4. Stability and Coinfection Dynamics
To study the dynamics of two diseases transmission, it is essential to explore the stability of system (3) by studying that of the disease-free equilibrium and the one-disease equilibria for two subsystems and yield dynamic conditions for a second disease invasion and coinfection.
4.1. Stability of Disease-Free Equilibrium
Firstly, we discuss the stability of the disease-free equilibrium , where and .
Theorem 1. The disease-free equilibrium for system (3) is globally asymptotically stable if but unstable if .
Proof. As mentioned above, the tick and host populations satisfy the following two equations:which means that system (3) is globally attractive for the two populations [16]. Then, there exists small enough and such that , , , , , , , and for any . So, for , we can obtainWe consider the auxiliary linear systemwhere vector andWhen , it follows from Theorem 2 in [14] that , which means that is stable. Thus, we have for . In accordance with the comparison principle, we haveand by the theory of asymptotically semiflow in [17], we getThis proves that is globally asymptotically stable when . Similarly, there is small enough and such that , , , , , , , and for any . Thus, for , we also haveAnd we can construct the auxiliary linear systemwhereIt is shown obviously that if , which implies that when for . Then, the disease-free equilibrium is unstable.
4.2. Stability of Boundary Equilibria
We first analyze the two subsystems containing only one disease prior to exploring the stability of the boundary equilibria and .
The subsystem with only disease 1 is given by
The feasible region of this subsystem (33) is and is positively invariant.
There is a disease-free equilibrium in subsystem (33), in which and . If , then a unique endemic equilibrium , where , , , and are expressed as (7), exists in subsystem (33). We aim to study the global dynamics of system (33) as follows.
Theorem 2. The following statements are valid:(i)If , then subsystem (33) at is globally asymptotically stable in (ii)If , then there exists a unique positive solution that is globally asymptotically stable for subsystem (33) in
Proof. In subsystem (33), the tick and host populations satisfywhich imply that the subsystem is globally attractive, and there is , where and .
If , there exists small enough and , and when , we take the limiting system for and . Considering the auxiliary systemwith vector and matrixwe have if and finally obtain that is globally asymptotically stable when according to the comparison principle.
Similarly, statement (ii) can also be proved via taking the limiting system for and , a detailed proof process similar to Lemma 2.3 in Lou and Zhao [18] and Theorem 3.2 in Gao et al. [19].
Analogously, the other subsystem containing only disease 2 can be written as follows:
In view of the feasible region , we can find that is positively invariant. In this subsystem (37), there exists a disease-free equilibrium in which , , and a unique nontrivial equilibrium in which , , , and are expressed by (13). Then, we obtain the global behaviors at and for subsystem (37).
Theorem 3. The following statements are valid:(i)If , then subsystem (37) at is globally asymptotically stable in (ii)If , then there exists a unique positive solution that is globally asymptotically stable for subsystem (37) in
Next, we discuss the relevant dynamics of general system (3) in , which comprise not only these basic reproduction numbers describing the stability of the boundary equilibria and but also the invasion reproduction numbers explaining how a disease can invade a tick-host population where another disease is already prevalent.
Theorem 4. When , the boundary equilibrium is locally asymptotically stable for system (3) if but unstable if .
Proof. Define as the stable manifold of , andNote that and are all invariant sets for system (3) with respect to the boundary equilibrium . According to the center manifold theorem, we consider two invariant manifolds and tangent to and , respectively, which satisfy and .
Likewise, is also an invariant set for system (3), and there is an invariant manifold expressed as such that . Since is globally stable in by Theorem 2, we have . Then, we focus on the linearization of system (3) with respect to the subspace . Calculating the respective Jacobian matrix , we obtainwhere and are shown in (16). In accordance with Theorem 2 in [14],in which represents the maximum real part of the eigenvalues of .
If , that is, , the boundary equilibrium is unstable. If , the real parts of the eigenvalues of are all negative; therefore, is stable. At the same time, the manifold is also stable. Then, we can prove that ; therefore, , and the equilibrium is stable. Finally, when , the equilibrium is locally asymptotically stable for system (3) if but unstable if .
The proof method of Theorem 4 can also be applied to the following theorem, which shows how the stability of depends on the invasion reproduction number .
Theorem 5. When , the boundary equilibrium is locally asymptotically stable for system (3) if but unstable if .
4.3. Coexistence of Two Diseases
We have shown that the boundary equilibrium (or ) is unstable in the tick-host population where disease 1 (or 2) is already prevalent once the invasion reproduction number (or ) is greater than one, which means that disease 2 (or 1) is able to invade in the population. From this perspective, the instability of and may be an advantage to study the coexistence of these two diseases. Here, we present Theorem 6 involving disease persistence.
Theorem 6. If , , , and , then system (3) remains uniformly persistent, namely, there exists a constant such that every solution for system (3) with positive initial value satisfies
Proof. DefineIt can be shown that system (3) remains uniformly persistent regarding .
By Lemma 1, first of all, we know that and are positively invariant in system (3). Also, is relatively closed in , and system (3) is point dissipative. Letand setAssume . Then, we prove thatFirstly, clearly holds, so we concentrate on the proof of .
Proving by contradiction, we suppose that does not hold. As denotes the solution of system (3) with positive initial value , then for , and . Thus, is in accordance with one of these conditions:(1)There exists such that at least one of and is not zero(2)If as , then the solution satisfies the following: (i) there exists such that at least one of and is not zero; (ii) there exists such that at least one of and is not zeroWe suppose satisfies condition (1). Without loss of generality, we may as well presume that and . According to Lemma 1, the solution can be given for any . For system (3) and for , we haveThis implies that when . Likewise, we have for using condition (2), which contradicts the hypothesis . Therefore, it follows that , and then .
We know that there exist three equilibria , , and in which all are isolated invariant sets in . In addition, we can show that , , and , in which represents the stable manifold of for , namely, there is such that, for every solution of system (3) with , we obtainin which is a distance function for .
Without loss of generality, we only prove that . Suppose it does not hold; then, for any , that is, there is such that , , , , , , , and for any . In system (3) for , we haveThen, we consider the following auxiliary system:in which the vector andIt is clear that , where and are given by (16). By referring to Theorem 2 in [14], we get that if , then . As is continuous for small , there exists sufficiently small such that . Consequently, we obtain a positive eigenvalue of with a positive eigenvector. Let denote a positive solution for auxiliary system (50), which is strictly increasing, namely, as for . Using the comparison principle, we haveand this conflicts with the above assumption.
Thus, is not only an isolated invariant set for , but it also satisfies . Analogously, we can also verify that is not only an isolated invariant set for , but it also satisfies when . In addition, when and , is not only an isolated invariant set for , but it also satisfies . According to Thieme [20], system (3) is uniformly persistent if , , , and .
5. Numerical Illustrations and Discussion
In this section, we utilize specific numerical simulations to illustrate the impact of coinfection on the dynamics of two tick-borne pathogens and ensure the effectiveness of the previous theories. Besides, we also propose and verify the uniform persistence of the two diseases and the presence of a coexistence equilibrium in the case in which and or and in the following.
By referring to some early works [2, 11, 21–23], we fix the parameter values as follows: , , , , , , , and . First, we consider subsystems (33) and (37) and set the initial values of these subsystems to . According to Figure 2(a), it is clearly shown that the number of infectious individuals with disease 1 consisting of infected ticks and hosts will eventually reach the equilibrium , which means that disease 1 would remain endemic since . On the contrary, it follows from Figure 2(b) that disease 2 will not persist in the relative tick and host populations as expected since . Thus, we can conclude that disease 2 cannot survive alone in tick and host populations, but disease 1 can.
(a)
(b)
Next, we introduce disease 2, which cannot spread if taken singularly, into tick and host populations where disease 1 already exists in order to study the influence of coinfection. We choose the following as parameter values: , , and . The initial values of are set to , and we calculate the invasion reproduction number . From Figure 3, we see that infected ticks and hosts with only disease 2 approach and stabilize at 76.4 and 65.9, and ones with both pathogens stabilize at 153.7 and 302.9, respectively, which imply that disease 2 can invade tick and host populations in which disease 1 is already endemic if . In order to visualize theoretical results, we choose and as parameters and plot Figure 4 to show their effects on the invasion reproduction number .
In this paper, we have developed a coinfection model involving two tick-borne diseases to explore their interaction and spread. Based primarily on the works by Van den Driessche [14], we have calculated the basic reproduction numbers concerning the disease-free equilibrium and two boundary equilibria and . Considering the invasion of a disease, we have then proposed a new threshold, namely, the invasion reproduction number, to describe the number of secondary cases infected by a disease later introduced into a population where another disease is already endemic, and we have derived the expressions for these two reproduction numbers.
According to the comparison principle and the theory of asymptotically semiflow, we have proved the global stability at the disease-free equilibrium if and the local stability at the two boundary equilibria and if and , as well as the coexistence conditions for these two diseases, which are , , , and . We have also showed that the two diseases are uniformly persistent, and the transmission system of two diseases admits a coexistence equilibrium when (or ) and (or ), which implies that the existence of one disease can facilitate the invasion and transmission of another disease. These results have been verified by numerical simulations.
The highlight of this paper is the introduction of the invasion reproduction number, which can directly reflect whether one disease can invade a population where another disease is already persistent, in order to investigate the effect of coinfection. We conclude that coinfection has a positive effect on the transmission of two tick-borne diseases in model (3), that is, one disease may invade a tick-host population where another disease already exists, although it cannot remain endemic alone in the population.
However, theoretically, pathogens or diseases interacting in a population may not only promote each other but also compete or be independent one from the other [2]. Thus, in future research, we will focus on the negative effects of coinfection on diseases transmission. Furthermore, other factors such as temporal and spatial heterogeneity will also be worth discussing.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of China (no. 11701072).