Effects of Coinfection on the Dynamics of Two Pathogens in a Tick-Host Infection Model

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 coinfectionmodel involving two tickborne 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.)en, 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. )e final numerical simulations mainly verify the theoretical results of coinfection.


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]. e 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.. ese 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. ese 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. is paper is organized as follows. We establish a coinfection model involving tick and host populations in Section 2. en, 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.

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 T(t) and H(t) be the total tick population and host population, respectively. e 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 T s (t), T i 1 (t), T i 2 (t), and T i 3 (t), respectively. Here, the total tick population is e 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 H s (t), H i 1 (t), H i 2 (t), and H i 3 (t), 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. en, we construct the following model:

Complexity
where all parameters involved in the model are listed in Table 1. We assume the summations of transmission probabilities β 1 + β 31 and α 1 + α 31 are greater than original β 1 and α 1 without coinfection, respectively, and which also hold for the transmission probabilities of disease 2. Susceptible ticks T s (t) recruited at a constant rate Λ T are infected and can move into T i ) with infection probability β 12 , β 13 ≥ 0 (β 21 , β 23 ≥ 0). In addition, susceptible ticks T s (t) may also be infected by taking a blood meal from H i 3 (t) directly with infection probability β 3 and move into compartment T i 3 (t). We also consider μ T 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.
For the total populations of ticks and hosts, it can be shown that It follows that the asymptotic equilibria of the two populations satisfy lim t⟶+∞ T(t) � Λ T /μ T and lim t⟶+∞ H(t) � Λ H /μ H . 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].

The Reproduction Numbers
e reproduction number R 0 , 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, R 0 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.

Basic Reproduction Number. In system (3), there always exists a disease-free equilibrium
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 Biologically, R 1 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 β 1 T s 0 /H during this host's average lifetime 1/μ H 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 α 1 H s 0 /H during the period 1/μ T . e biological interpretations for R 2 and R 3 are analogous.
It is clear that there are two boundary equilibria E 1 and E 2 , which represent the cases in which just disease 1 and just disease 2 are endemic in the population, respectively. erefore, at E 1 , we have that T i 2 , T i 3 , H i 2 , and H i 3 are all zero, and where in which with H * * � H s * * + H i * * 2 . Also, the equilibrium E 2 exists if R 2 > 1.

Invasion Reproduction Number.
e 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.  13 Probability of infection for infectious hosts with only disease 1 that are fed by T i 2 or T i 3 , respectively, and also become infected by disease 2 α 21 , α 23 Probability of infection for infectious hosts with only disease 2 that are fed by T i 1 or T i 3 , respectively, and also become infected by disease 1 4 Complexity us, 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, R 1 2 , is based on a population in which disease 1 is already endemic, we concentrate on the boundary equilibrium E 1 to reckon the matrices F 2 and V 2 as follows: Next, taking the derivative of F 2 and V 2 with respect to (T i 2 , T i 3 , H i 2 , H i 3 ) T and substituting the boundary equilibrium E 1 into the variables, we have (16) en, the next generation matrix is given by where in which T s * , T i * 1 , H s * , H i * 1 , and H * are expressed in (7). e characteristic equation is

Complexity
Similarly, the invasion reproduction number for disease 1, namely, R 2 1 , can also be derived following the above process, which is given by where A � (cf + ae + dg + bk), Note that T s * * , T i * * 2 , H s * * , H i * * 2 , and H * * are defined in (13).

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.

Stability of Disease-Free Equilibrium.
Firstly, we discuss the stability of the disease-free equilibrium E 0 � (T s 0 , 0, 0, 0,

Theorem 1.
e disease-free equilibrium E 0 for system (3) is globally asymptotically stable if R 0 < 1 but unstable if R 0 > 1.
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]. en, there exists small enough ϵ > 0 and We consider the auxiliary linear system where vector x(t) � (x 1 (t), x 2 (t), x 3 (t), x 4 (t), x 5 (t), x 6 (t)) T and 6 Complexity When R 0 < 1, it follows from eorem 2 in [14] that s(M 0 (0)) < 0, which means that M 0 (0) is stable. us, we have lim t⟶+∞ x i (t) � 0 for i � 1, 2, . . . , 6. In accordance with the comparison principle, we have and by the theory of asymptotically semiflow in [17], we get is proves that E 0 is globally asymptotically stable when And we can construct the auxiliary linear system where Complexity 7 It is shown obviously that s(M 0 ′ (0)) > 0 if R 0 > 1, which implies that y i (t) ⟶ + ∞ when t ⟶ + ∞ for i � 1, 2 , . . . , 6. en, the disease-free equilibrium E 0 is unstable. □

Stability of Boundary Equilibria.
We first analyze the two subsystems containing only one disease prior to exploring the stability of the boundary equilibria E 1 and E 2 .
e subsystem with only disease 1 is given by e feasible region of this subsystem (33) is ere is a disease-free equilibrium E 1 0 � (T s 0 , 0, H s 0 , 0) in subsystem (33), in which T s 0 � Λ T /μ T and H s 0 � Λ H /μ H . If R 1 > 1, then a unique endemic equilibrium E 1 1 � (T s * , T i * 1 , H s * , H i * 1 ), where T s * , T i * 1 , H s * , and H i * 1 are expressed as (7), exists in subsystem (33). We aim to study the global dynamics of system (33) as follows.
If R 1 < 1, there exists small enough ζ > 0 and t 3 � t(ζ) > 0, and when t ≥ t 3 , we take the limiting system for T i 1 and H i 1 . Considering the auxiliary system dw(t) dt � M 1 (ζ)w(t), with vector w(t) � (w 1 (t), w 2 (t)) and matrix we have s(M 1 (0)) < 0 if R 1 < 1 and finally obtain that E 1 0 is globally asymptotically stable when R 1 < 1 according to the comparison principle.
Similarly, statement (ii) can also be proved via taking the limiting system for T i 1 and H i 1 , a detailed proof process similar to Lemma 2.3 in Lou and Zhao [18] and eorem 3.2 in Gao et al. [19]. □ Analogously, the other subsystem containing only disease 2 can be written as follows: Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 14 Complexity