
In this paper, an insect-parasite-host model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and Trypanosoma rangeli as two thresholds. The local and global stability of the vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium is investigated through the derived two thresholds. Forward bifurcation, saddle-node bifurcation and Hopf bifurcation are proved analytically and illustrated numerically. We show that the model can lose the stability of the vector-free equilibrium and exhibit a supercritical Hopf bifurcation, indicating the occurrence of a stable limit cycle. We also find it unlikely to have backward bifurcation and Bogdanov-Takens bifurcation of the parasite-positive equilibrium. However, the sustained oscillations of infected vector population suggest that Trypanosoma rangeli will persist in all the populations, posing a significant challenge for the prevention and control of Chagas disease.
Citation: Lin Chen, Xiaotian Wu, Yancong Xu, Libin Rong. Modelling the dynamics of Trypanosoma rangeli and triatomine bug with logistic growth of vector and systemic transmission[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 8452-8478. doi: 10.3934/mbe.2022393
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In this paper, an insect-parasite-host model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and Trypanosoma rangeli as two thresholds. The local and global stability of the vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium is investigated through the derived two thresholds. Forward bifurcation, saddle-node bifurcation and Hopf bifurcation are proved analytically and illustrated numerically. We show that the model can lose the stability of the vector-free equilibrium and exhibit a supercritical Hopf bifurcation, indicating the occurrence of a stable limit cycle. We also find it unlikely to have backward bifurcation and Bogdanov-Takens bifurcation of the parasite-positive equilibrium. However, the sustained oscillations of infected vector population suggest that Trypanosoma rangeli will persist in all the populations, posing a significant challenge for the prevention and control of Chagas disease.
Chagas disease, known as American trypanosomiasis, is a protozoan parasitic disease caused by Trypanosoma cruzi (T. cruzi). The disease was discovered firstly by Doctor Chagas in 1908 and it was named after that. Chagas disease is an illness that can cause serious consequences including heart disease and cardiomyopathy, and many people infected with Chagas disease may die due to these complications [1,2]. It is mainly prevalent in Central and South America, such as Argentina, Bolivia, Brazil, Chile, etc. About 13% of the Latin American population is at risk of T. cruzi infection [3]. Due to convenient transportation and the globalization, Chagas disease is spreading very widely in the world, see Figure 1(a), (b) for details. An estimated 8 million people are infected with Trypanosoma cruzi worldwide, mainly in Latin America where Chagas disease remains one of the biggest public health problems, causing incapacity in infected individuals and more than 10,000 deaths per year [4,5,6]. In particular, patients with Chagas disease may be coinfected with other epidemic diseases including HIV [7] and COVID-19. These patients are at risk of severe COVID-19 manifestations and should be a priority group to be vaccinated [8].
Trypanosoma cruzi, a protozoan parasite that parasitizes human and mammalian blood and tissue cells, can be transmitted by blood-sucking triatomine bugs to cause the symptoms of Chagas disease. It is spread mainly through the faeces of the infected blood-sucking triatomine bugs. These bugs usually live in the crevices of poorly built houses in rural or suburban areas. They hide during the day and come out at night to feed on human blood. They bite exposed areas of the skin, such as the face, and defecate near the bites. If one scratches on the bites, this leads to feces spreading to the sites of eye, mouth, or any skin break, and then the parasites enter the body, and eventually go into the heart, survive and proliferate inside [3,9]. Triatomine bugs, the vectors to transmit Chagas disease, have a relatively short life span, ranging from 4 to 14 months depending largely on species and environmental conditions. They suck the blood of vertebrates, especially mammals (such as dogs, bats, armadillos, squirrels, guinea pigs and humans), and then release feces on the skin of the bitten animal. The feeding time may take 10–30 minutes [10]. Once healthy triatomine bugs ingest with the parasites of Chagas disease, they will be infected quickly [11,12,13].
Trypanosoma rangeli (T. rangeli) is a kind of parasite which is pathogenic to some vector species including triatomine bug, it always influence the transmission dynamics of the infected bugs, so this kind of transmission behavior deserves further research. Rhodnius prolixus (R. prolixus) is one of the triatomine species and T. rangeli is one of the trypanosoma which is spread between hosts and triatomine vectors. Although T. rangeli can infect mammals through the same triatomines, it is not pathogenic to human. However, it is still important to study the transmission dynamics of T. rangeli because it shares soluble antigenic epitopes with T. cruzi and the crossed serological reactions affect the diagnosis of Chagas disease [14,15]. Both T. rangeli and T. cruzi have common hosts and triatomine vectors. The transmission dynamics of T. rangeli between hosts and triatomine bugs can affect the effectiveness of T. cruzi transmission [10]. Moreover, T. rangeli has pathogenic effect on triatomine bugs in the sense that the infection of T. rangeli can change the behavior of triatomine bugs, which alters the transmission of Chagas disease [14,16]. There are some studies showing the possible interaction between T. rangeli and T. cruzi [17]. Therefore, it is essential to study the transmission behavior of T. rangeli. T. rangeli's infection pattern is similar to that of T. cruzi. The healthy population will get infected if they bite the infected counterparts through systemic transmission, where T. rangeli parasite can enter and multiply at the common hosts' and vectors' bloodstream. In addition to this normal insect-host-insect transmission mode, there is another mode of T. rangeli transmission called insect-to-insect co-feeding transmission. Susceptible triatomine bugs can get infected if they are feeding with infected counterparts on the same hosts. This has been studied in some papers [10,18,19,20,21,22] and it differs from the transmission of T. cruzi.
There are a number of mathematical models that studied the transmission dynamics of Chagas disease, such as the different transmission routes of the interaction between hosts and vectors [2,23,24], the disease transmission in the host movement and host community composition [9,16,25,26,27], the triatomine population with temporal or spatial variations [28,29,30,31,32,33], and the optimal control of Chagas disease [34,35,36,37,38]. Recently, Wu et al. [10] formulated a new model by considering the Ricker's type growth of triatomine bugs and T. rangeli's pathogenic effect on triatomine bugs. However, the logistic growth of triatomine bugs is very common but was not investigated in the model. In this paper, we assume the generation rate of triatomine bugs follows the logistic growth instead of Ricker's type function and further study the dynamical behavior of triatomine-rangeli-host transmission. The new model is shown to have interesting dynamics, provide more insights into the interaction between triatomine bugs and T. rangeli, and may help to prevent and control the Chagas disease.
The paper is outlined as follows. In the next section, we propose the model with logistic growth and T. rangli's pathogenic effect on triamine bugs. In Section 3, the existence and stability of vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium are considered. In Section 4, the bifurcation analysis including forward bifurcation and Hopf bifurcation is studied. Numerical simulations are also performed in Section 5. Conclusion and discussion are given in Section 6.
The model developed in the reference [10] is
S′h(t)=Λh−˜βhIv(t)Sh(t)−μ1Sh(t),I′h(t)=˜βhIv(t)Sh(t)−μ1Ih(t),S′v(t)=r(Sv(t)+θIv(t))e−σ(Sv(t)+Iv(t))−˜βvSv(t)Ih(t)−βcSv(t)Iv(t)−μ2Sv(t),I′v(t)=˜βvSv(t)Ih(t)+βcSv(t)Iv(t)−dIv(t)−μ2Iv(t), | (2.1) |
where the population is divided into four compartments: susceptible and infected competent hosts, susceptible and infected triatomine bugs, denoted by Sh,Ih,Sv,Iv in order. Λh is the constant recruitment rate of susceptible competent host per unit time. The transmission rate from infected bugs to susceptible competent hosts is denoted by ˜βh=baNc+αNq, where b is the transmission probability from infected bugs to susceptible competent hosts per bite, a is the number of bites per triatomine bug per unit time, α is the biting preference of quasi-competent hosts to competent hosts, Nc is the total number of competent hosts, and Nq is the total number of quasi-competent hosts. The transmission rate from infected competent hosts to susceptible bugs is denoted by ˜βv=caNc+αNq, where c is the transmission probability from infected hosts to susceptible triatomine bug per bite. The total infection rate through co-feeding transmission between susceptible and infected bugs is βc, which is transmitted by both the competent and quasi-competent hosts. Here βc=1δβchNc(aNc+αNqτ1ω)2+1δβcqNq(aαNc+αNqτ2ω)2, where ω is the unit time, δ is the ratio of night to unit time, βch and βcq are the transmission rates from infected bugs to susceptible bugs on an average competent host and quasi-competent host during night, respectively. τ1 and τ2 are the feeding times per bite on a competent host and a quasi-competent host, respectively. The Ricker's type function b(x)=rxe−σx was chosen to model the reproduction rate of R. prolixus. Integrating the pathogenic effect, the growth rate of triatomine bugs is modeled as r(Sv+θIv)e−σ(Sv+Iv), where r is the maximal number of offsprings that a triatomine bug can produce per unit time, θ∈[0,1] is the reproduction reduction of bugs due to the pathogenic effect of T. rangeli on bugs, σ is the density-dependency strength measuring the reproduction of bugs. μ1 and μ2 are the natural death rates of competent hosts and triatomine bugs, respectively. d is the death rate of infected vectors induced by pathogenic effect.
Model (2.1) includes the systemic and co-feeding transmission routes among vectors and hosts. Two thresholds were derived to study the dynamical behavior of this model [10]. Sustained oscillations were found numerically by changing the parameters d and θ. Furthermore, the oscillation amplitude is larger if d is larger or θ is smaller.
In this paper, we assume the generation rate of triatomine bugs follows the logistic growth instead of Ricker's type function. Here we only consider the systemic transmission, then model (2.1) can be changed to
S′h(t)=Λh−βhIv(t)Sh(t)−μ1Sh(t),I′h(t)=βhIv(t)Sh(t)−μ1Ih(t),S′v(t)=r(Sv(t)+θIv(t))(1−Sv(t)+Iv(t)K)−βvSv(t)Ih(t)−μ2Sv(t),I′v(t)=βvSv(t)Ih(t)−dIv(t)−μ2Iv(t), | (2.2) |
where βh=baNh,βv=caNh,Nh is the total number of hosts. The infection rate of susceptible competent hosts after bugs' biting at time t is βhIv(t)Sh(t), and the infection rate of susceptible vectors after bugs' biting at time t is βvSv(t)Ih(t). In the logistic growth r(Sv+θIv)(1−Sv+IvK) of triatomine bugs, K is the carrying capacity. The other parameters are the same as those in model (2.1). All the parameters are non-negative and their biological meanings and ranges are given in Table 1.
Parameter | Range | Description |
Λh | varied | Recruitment rate of susceptible competent host per unit time [10,39] |
a | [0.2, 33]/day | Number of bites per triatomine bug per unit time [10,39] |
b | [0.00271, 0.06] | Transmission probability from infected bugs to susceptible |
competent hosts per bite [2,9,10] | ||
c | [0.00026, 0.49] | Transmission probability from infected hosts to susceptible |
triatomine bug per bite [9,25] | ||
r | [0.0274, 0.7714]/day | Maximum number of offsprings that a triatomine bug can |
produce per unit time [2,25] | ||
θ | [0, 1] | Reproduction reduction of bugs due to the infection of parasites [10] |
Nh | varied | Total number of hosts [2] |
K | varied | Carrying capacity [10] |
μ1 | [0.000038, 0.0025]/day | Natural death rate of hosts [9,25] |
μ2 | [0.0045, 0.0083]/day | Natural death rate of triatomine bugs [9,25] |
d | [0.0188, 0.0347]/day | Death rate of infected vectors due to pathogenic effect [10] |
Denote the total population of competent hosts by Nh=Sh+Ih. Adding the first and second equations leads to
N′h(t)=Λh−μ1Nh(t). |
It follows that
limt→∞Nh(t)=Λhμ1≜Nh. |
Thus, in the limiting system we have Sh(t)=Nh−Ih(t).
Accordingly, system (2.2) can be reduced to the following three-dimensional limiting system:
I′h(t)=βhIv(t)(Nh−Ih(t))−μ1Ih(t),S′v(t)=r(Sv(t)+θIv(t))(1−Sv(t)+Iv(t)K)−βvSv(t)Ih(t)−μ2Sv(t),I′v(t)=βvSv(t)Ih(t)−dIv(t)−μ2Iv(t). | (2.3) |
It is easy to know that the feasible region of system (2.3) is
D={(Ih,Sv,Iv)|0≤Ih≤Nh,0≤Sv,0≤Iv,Sv+Iv≤K}. |
Let the right-hand side of the equations of system (2.3) be zero. There are one vector-free equilibrium E0(0,0,0) and one parasite-free equilibrium ES(0,K(r−μ2)r,0). We will derive two thresholds Rv and R0 to study the dynamic behavior of system (2.3) where Rv is the triatomine bug basic reproduction number and R0 is the T. rangeli basic reproduction number.
The Jacobian matrix of system (2.3) at the vector-free equilibrium E0(0,0,0) is
J(E0)=(−μ10βhNh0r−μ2θr00−(d+μ2)). |
The eigenvalues of the Jacobian matrix J(E0) at E0(0,0,0) are −μ1,−(d+μ2) and r−μ2, respectively. Let
Rv=rμ2. |
We will show that it provides a threshold to determine the persistence or extinction of the vector population.
For the parasite-free equilibrium ES(0,S0v,0) with S0v=K(r−μ2)r to be biologically feasible, we need r−μ2>0, namely, Rv>1. Next, we calculate the T. rangeli basic reproduction number R0 of system (2.3). Using the method in ref. [40], we have
F=(00Nhβh000K(r−μ2)rβv00), |
V=(μ100K(r−μ2)rβvr−μ2r−rθ(1−r−μ2r)−μ200d+μ2). |
Thus, the T. rangeli basic reproduction number of system (2.3), given by the spectral radius of the next generation matrix, is
R0=ρ(FV−1)=√a2bcK(r−μ2)rNhμ1(d+μ2)=√βhβvNhS0v(d+μ2)μ1, | (3.1) |
where S0v=K(r−μ2)r.
For any parasite-positive equilibrium E∗=(I∗h,S∗v,I∗v) of system (2.3), its elements satisfy
S∗v=μ1(d+μ2)βhβv(Nh−I∗h), I∗v=μ1I∗hβh(Nh−I∗h), | (3.2) |
and I∗h is the positive root of the following equation:
f(I∗h)=A(I∗h)2+BI∗h+C=0, | (3.3) |
where
A=a2c2μ1Nh(abK(rθ−d−μ2)+θrNhμ1),B=acμ1(abK(ac(d+μ2−rθ)+(r−μ2)(d+μ2))+rNhμ1(d+μ2)(θ+1)),C=Nhμ1(a2bcK(μ2−r)(d+μ2)+rNhμ1(d+μ2)2)=(Nhμ1)2(d+μ2)2(1−R20). |
Let Δ=B2−4AC. We have
Δ=a2c2μ21[−4(d+μ2)(rNhμ1(d+μ2)+a2bcK(μ2−r))(θrNhμ1−abK(d+μ2−rθ))+(rNhμ1(θ+1)(d+μ2)+abK((d+μ2)(r−μ2)+ac(d+μ2−rθ)))2]. |
If Δ≥0, then the equation f(I∗h)=0 may have two roots, which are denoted by
I∗h1=−B−√Δ2A, I∗h2=−B+√Δ2A. |
From (3.2), we can see that S∗v>0,I∗v>0 as long as I∗h>0. Therefore, we study the existence of the parasite-positive equilibria in the following cases:
1) R0=1, namely, a2bcK(r−μ2)=rNhμ1(d+μ2).
In this case, we have B>0,C=0,Δ>0,A=acμ21r−μ2((d+μ2)(rθ−d−μ2)+acrθ(r−μ2)). By Vieta theorem, if A>0, then I∗h1<0,I∗h2=0. Thus, system (2.3) has no parasite-positive equilibrium. If A<0, then I∗h1=0,I∗h2>0, and system (2.3) has a unique parasite-positive equilibrium E2(I∗h2,S∗v(I∗h2),I∗v(I∗h2)). If A=0, then Eq (3.3) has a zero root, i.e., there is no parasite-positive equilibrium of system (2.3).
2) R0>1, namely, a2bcK(r−μ2)>rNhμ1(d+μ2).
In this case, we have B>0,C<0. If A>0, then Δ>0, I∗h1<0,I∗h2>0, and system (2.3) has a unique parasite-positive equilibrium E2(I∗h2,S∗v(I∗h2),I∗v(I∗h2)). If A<0,Δ>0, then I∗h1>0,I∗h2>0, and system (2.3) has two parasite-positive equilibria E1(I∗h1,S∗v(I∗h1),I∗v(I∗h1)) and E2(I∗h2,S∗v(I∗h2),I∗v(I∗h2)). If A<0,Δ=0, then E1=E2, which means that there is a parasite-positive equilibrium of multiplicity 2. If A<0,Δ<0, then there is no parasite-positive equilibrium. If A=0, there exists only one root of Eq (3.3) and the root is positive, i.e., one parasite-positive equilibrium E∗(I∗h,S∗v(I∗h),I∗v(I∗h)) of system (2.3).
3) R0<1, namely, a2bcK(r−μ2)<rNhμ1(d+μ2).
In this case, we have B>0,C>0. If A<0, then Δ>0, I∗h1<0,I∗h2>0, and system (2.3) has a unique parasite-positive equilibrium E2(I∗h2,S∗v(I∗h2),I∗v(I∗h2)). If A≥0, there is no positive root of (3.3), i.e., there is no parasite-positive equilibrium of system (2.3).
We summarize the results as follows:
Theorem 3.1 For system (2.3), we have the following results on the existence of equilibria.
1) The vector-free equilibrium E0(0,0,0) always exists. The parasite-free equilibrium ES(0,K(r−μ2)r,0) exists if and only if Rv>1.
2) When R0≤1, there is a unique parasite-positive equilibrium E2 if A<0; Otherwise, there is no parasite-parasite-positive equilibrium.
3) When R0>1, there is a unique parasite-positive equilibrium if A≥0, and there are two parasite-positive equilibria E1 and E2 if A<0,Δ>0, and the two equilibria coalesce to E if and only if A<0,Δ=0.
Theorem 3.2 The vector-free equilibrium E0(0,0,0) of system (2.3) is globally asymptotically stable if Rv<1 and unstable if Rv>1.
Proof. At the vector-free equilibrium E0(0,0,0), the eigenvalues of the Jacobian matrix of system (2.3) are −μ2,−(d+μ2) and r−μ2. If Rv=rμ2>1, namely, r>μ2, then there are two negative eigenvalues and one positive eigenvalue, i.e., E0 is unstable. If Rv<1, then all the eigenvalues are real and negative, which indicates that E0 is locally asymptotically stable for Rv<1.
Let Nv=Sv+Iv. We have
N′v=S′v+I′v=r(Sv(t)+θIv(t))(1−Sv(t)+Iv(t)K)−μ2Sv(t)−dIv−μ2Iv≤rNv(1−NvK)−μ2Nv≤(r−μ2)Nv. |
Thus, we have
lim supt→∞Nv(t)≤limt→∞Nv(0)e(r−μ2)t=0, |
with any feasible initial solution Nv(0)=Sv(0)+Iv(0) when Rv<1. That is to say, the solutions of Sv and Iv with any feasible initial conditions will tend to zeroes if Rv<1. For subsystem of Ih, it is cooperative with the positive invariance set [0,Nh]. The vector-free equilibrium E0 is unique for system (2.3) if Rv<1. From this, we know that E0 is globally asymptotically stable if Rv<1.
Theorem 3.3 The parasite-free equilibrium ES(0,K(r−μ2)r,0) of system (2.3) is
1) a saddle-node point when R0=1;
2) unstable when R0>1;
3) locally asymptotically stable when R0<1.
Proof. At the parasite-free equilibrium ES(0,S0v,0), where S0v=K(r−μ2)r, the Jacobian matrix of system (2.3) is
J(ES)=(−μ10Nhβh−S0vβvμ2−r(1+θ)μ2−rS0vβv0−d−μ2). | (3.4) |
The corresponding characteristic polynomial of (3.4) is
P(λ)=−(λ+r−μ2)[λ2+b0λ+c0], | (3.5) |
where
b0=μ1+μ2+d, c0=μ1(d+μ2)−a2bcK(r−μ2)rNh. |
The eigenvalues of system (2.3) at ES are the roots of P(λ)=0 and denoted by λ1,λ2, and λ3.
Let Δ0=b20−4c0, i.e.,
Δ0=4a2bcK(r−μ2)rNh+(d−μ1+μ2)2. |
Then the eigenvalues of (3.5) are
λ1=μ2−r,λ2=−b0−√Δ02,λ3=−b0+√Δ02. |
Since r>μ2, we have Δ0>0, which means that λ2,λ3 are real.
When R0=1, we have c0=0,b0=√Δ0. Therefore, λ1=μ2−r<0,λ2=−b0=−d−μ1−μ2<0 and λ3=0. Because of the zero eigenvalue, we need to further study the type of ES.
We let x=Ih,y=Sv−S0v,z=Iv and shift the equilibrium to the origin. System (2.3) becomes
x′=abz+x(−abzNh−μ1),y′=−rz−ry2K−acKxNh−rθz2K+μ2z+acKμ2xrNh+θμ2z+y(−r−acxNh−rzK−rθzK+μ2),z′=acxyNh+acKxNh+z(−d−μ2)−acKμ2xrNh. | (3.6) |
Then we make the following transformations
x=m1X+m2Z,y=m3X+Y+m4Z,z=X+Z, |
where
m1=abμ1,m2=−abd+μ2,m3=−d+r−θμ2r−μ2,m4=−(1+θ)μ2+μ1−rd−r+μ1+2μ2. |
This leads to the following system
X′=−ab((r−μ2)(d+μ2)+ac(d+r−θμ2))Nh(r−μ2)(d+μ1+μ2)X2+XO(|Y,Z|)+O(|Y,Z|2),Y′=(μ2−r)Y+O(|X,Y,Z|2),Z′=(−d−μ1−μ2)Z+O(|X,Y,Z|2). | (3.7) |
We know that ES is a saddle-node point when R0=1. Moreover, if R0>1, then λ1<0,λ2<0,λ3>0, i.e., ES is unstable; if R0<1, then λ1<0,λ2<0,λ3<0, i.e., ES is locally asymptotically stable.
Based on the above analysis, we conclude that the parasite-free equilibrium ES of system (2.3) is a saddle-node point when R0=1, unstable when R0>1, and locally asymptotically stable when R0<1 [41].
In this section, we will study the global stability of the unique parasite-positive equilibrium E1 by Li-Muldowney global-stability criterion [42] when R0>1, Rv>1 and A≥0.
Let |⋅| denote a vector norm in Rn and the induced matrix norm in Rn×n, the space of all n×n matrices. For matrix A in Rn×n, the Lozinsk˜i measure or the logarithmic norm of A with respect to |⋅| [43] is
μ(A)=limh→0+|I+hA|−1h. |
Let y(t) be a solution of linear differential equation
y′(t)=A(t)y(t), |
where A(t) is m×m matrix-valued continuous function. For t≥t0, we have
|y(t)|≤|y(t0)|e∫tt0μ(A(t))dt. |
Let B be an n×n matrix. The second additive compound matrix of B, denoted by B[2], is an (n2)×(n2) matrix. For instance, if B=(bij) is a 3×3 matrix, then
B[2]=(b11+b22b23−b13b32b11+b33b12−b31b21b22+b33). |
Consider the following autonomous system
x′=f(x), | (3.8) |
where f:Ω→Rn,Ω⊂Rn is an open set, simply connected, and f∈C1(Ω). Let x(t,x0) be the solution of system (3.8) such that x(0,x0)=x0. Suppose x∗ is an equilibrium of system (3.8), i.e., f(x∗)=0. A set K is said to be absorbing in Ω for system (3.8) if x(t,K1)⊂K for each compact set K1⊂Ω and sufficiently large t. Assume the following assumptions hold:
(H1) System (3.8) has a unique equilibrium point x∗ in Ω.
(H2) System (3.8) has a compact absorbing set K⊂Ω.
Let Q:Ω↦Q(x) be (n2)×(n2) matrix-valued function with its inverse Q−1(x). Let μ be a Lozinsk˜i measure on RN×N, where N=(n2). Define
¯q2=lim supt→∞supx0∈K1t∫10μ(X(x(s,x0)))ds, |
where
X=QfQ−1+QJ[2]Q−1, |
the matrix Qf is obtained by replacing each entry qij of Q by its derivative in the direction of f, (qij)f, and J[2] is the second additive compound matrix of the Jacobian matrix J of system (3.8).
Lemma 3.1 [42] Assume that Ω is simply connected and assumptions (H1) and (H2) hold. Then, the unique equilibrium x∗ of system (3.8) is globally asymptotically stable in Ω if there exist a function Q and a Lozinsk˜i measure μ such that ¯q2<0.
We have the following theorem for our model.
Theorem 3.4 Assume that R0>1, Rv>1 and A≥0. The unique parasite-positive equilibrium E1 of system (2.3) is globally asymptotically stable if σ=K2βv+˜v<0, where
˜v=max{v1,v2},v1=−r(S2v+Iv(K−Iv)θ)+SvK(Ivβh+μ1)SvK+max{|Ivr(Sv+(2Iv+Sv−K)θ)SvK|,Iv(Nh−Ih)βhSv},v2=max{−Ivβh−Svβv−μ1,r(K−2Sv)−Ivr(1+θ)−K(Ihβv+μ2)K}. | (3.9) |
Proof. When R0>1, Rv>1 and A≥0, it is easy to show the uniqueness of the parasite-positive equilibrium E1. By Theorems 3.2 and 3.3, we know that the vector-free equilibrium E0 and the parasite-free equilibrium ES are unstable when Rv>1 and R0>1. It can also be checked that the conditions (H1) and (H2) are satisfied. Next we show the global stability of the unique parasite-positive equilibrium E1(Ih,Sv,Iv).
The Jacobian matrix of system (2.3) at E1 is
J=(−Ivβh−μ10(Nh−Ih)βh−SvβvJ22J23SvβvIhβv−d−μ2), |
where J22=r(1−Iv+SvK)−Ihβv−μ2−r(Sv+θIv)K,J23=rθ(1−Iv+SvK)−r(Sv+θIv)K. The second additive compound matrix J[2] is
J[2]=(−Ivβh−μ1+J22J23−(Nh−Ih)βhIhβv−Ivβh−μ1−d−μ20−Svβv−SvβvJ22−d−μ2). |
Let P(Ih,Sv,Iv)= diag(1,SvIv,SvIv). Then Pf= diag(0,˙SvIv−Sv˙IvI2v,˙SvIv−Sv˙IvI2v). We have PfP−1= diag(0,S′vSv−I′vIv,S′vSv−I′vIv), and let B=PfP−1+PJ[2]P−1.
From system (2.3), we obtain
S′vSv=r−μ2−Sv(rSv+IhKβv)+I2vrθ+Ivr(Sv+Svθ−Kθ)SvK,I′vIv=−d−μ2+IhSvβvIv. |
Straightforward calculations yield
B=(B11B12B21B22), |
where
B11=r(K−2Sv)−Iv(r+rθ+Kβh)−K(Ihβv+μ1+μ2)K,B12=(−Ivr(Sv+(2Iv+Sv−K)θ)SvK,Iv(Ih−Nh)βhSv),B21=(IhSvβvIv,−S2vβvIv)T, |
B22=(−d−Ivβh−μ1−μ2−I′vIv+S′vSv0−Svβvr(K−2Sv)−Ivr(1+θ)−K(d+Ihβv+2μ2)K−I′vIv+S′vSv). |
Take the norm |(Ih,Sv,Iv)|=max{|Ih|,|Sv|+|Iv|} in R3. μ(⋅) is the Lozinsk˜i measure with the vector norm [44]. We have
μB≤sup{g1,g2}=sup{μ1(B11)+|B12|,μ1(B22)+|B21|}, |
where |B12|,|B21| are the matrix norms with respect to l1 vector norm.
Calculations show that
μ1(B11)=r(K−2Sv)−Iv(r+rθ+Kβh)−K(Ihβv+μ1+μ2)K,|B12|=max{|Ivr(Sv+(2Iv+Sv−K)θ)SvK|,Iv(Nh−Ih)βhSv},|B21|=IhSvβvIv+S2vβvIv,μ1(B22)=−I′vIv+S′vSv−d−μ2+v2. |
Thus, we have
g1=S′vSv+v1≤S′vSv+˜v,g2=S2vβvIv+S′vSv+v2≤S′vSv+S2vβvIv+˜v,μB≤S′vSv+S2vβvIv+˜v, |
where ˜v,v1,v2 are defined by (3.9).
According to Sv+Iv≤K, we have μB≤S′vSv+K2βv+˜v, that is
μB≤S′vSv+σ, |
where σ=K2βv+˜v.
Along each solution (Ih,Sv,Iv)⊂D of system (2.3), then for t>ˉt, we have
1t∫t0μ(B)ds=1t∫ˉt0μ(B)ds+1t∫tˉtμ(B)ds≤1t∫ˉt0μ(B)ds+1tlnSv(t)Sv(ˉt)+t−ˉttσ, |
which means that ¯q2≤σ<0. This completes the proof.
T. rangeli may not be pathogenic to every triatomine species [17]. Thus, we also study the dynamics of system (2.3) in the absence of pathogenic effect on triatomine bugs, i.e., d=0 and θ=1. This allows us to compare the obtained results with and without the pathogenic effect.
Theorem 3.5 Assume Rv>1 and R0>1. In the absence of pathogenic effect on triatomine bugs, namely, θ=1 and d=0, system (2.3) admits a unique parasite-positive equilibrium E∗=(I∗h,S∗v,I∗v), which is locally asymptotically stable when d∗>0, unstable when d∗<0, where d∗ is defined by (3.13).
Proof. In the case of θ=1,d=0, system (2.3) becomes
I′h(t)=βhIv(t)(Nh−Ih(t))−μ1Ih(t),S′v(t)=r(Sv(t)+Iv(t))(1−Sv(t)+Iv(t)K)−βvSv(t)Ih(t)−μ2Sv(t),I′v(t)=βvSv(t)Ih(t)−μ2Iv(t). | (3.10) |
According to Theorem 3.1, the system (3.10) has a unique parasite-positive equilibrium E∗=(I∗h,S∗v,I∗v) when R0>1. Adding the second and third equations of system (3.10), we have
N′v=rNv(1−NvK)−μ2Nv. | (3.11) |
Letting the right-hand side of equation (3.11) be equal to zero, we obtain a unique parasite-positive equilibrium N∗v=S0v if Rv>1, and a unique zero equilibrium which is globally asymptotically stable if Rv≤1.
When Rv>1, the limiting system of system (3.10) can be reduced to
I′h(t)=βhIv(t)(Nh−Ih(t))−μ1Ih(t),I′v(t)=βvIh(t)(S0v−Iv)−μ2Iv(t). | (3.12) |
System (3.12) has an unstable parasite-free equilibrium (0,0). When R0>1, there is a unique parasite-positive equilibrium E1=(I∗h,I∗v), where S∗v is replaced by S0v−I∗v for susceptible vectors at equilibrium. The Jacobian matrix of system (3.12) at E1 is
J(E1)=(−βhI∗v−μ1βh(Nh−I∗h)βv(S0v−I∗v)−βvI∗h−μ2). |
It is easy to know that the trace of J(E1) is negative. The determinant of J(E1) is
det(J(E1))=βhβvNhI∗v−βhβvS0v(Nh−I∗h)+βvμ1I∗h+βhμ2I∗v+μ1μ2≜d∗. | (3.13) |
The eigenvalues of Jacobian matrix J(E1) have negative real parts when d∗>0. When d∗<0, one of the eigenvalues of Jacobian matrix J(E1) has a negative real part and the other has a positive real part. Therefore, the equilibrium E1(I∗h,I∗v) of system (2.3) is locally asymptotically stable if d∗>0, unstable if d∗<0. That is to say, the parasite-positive equilibrium E∗=(I∗h,S∗v,I∗v) of system (2.3) is locally asymptotically stable if d∗>0, unstable if d∗<0.
Denote
(f(Ih,Iv)g(Ih,Iv))=(βhIv(Nh−Ih)−μ1IhβvIh(S0v−Iv)−μ2Iv). |
Obviously, both f and g:R2+→R are continuously differentiable maps. We have ∂f∂Iv=βh(Nh−Ih)−μ1Ih≥0, and ∂g∂Ih=βv(S0v−Iv)−μ2Iv≥0. The system is cooperative in a domain D={(In,Iv)∈R2:Ih∈[0,Nh],Iv∈[0,S0v]}. System (3.12) has a parasite-free equilibrium (0,0) and a unique parasite-positive equilibrium E1. According to Theorem 3.2.2 in [45], E1 is globally attractive. Thus, the parasite-positive equilibrium E∗ of sub-system (3.12) is globally asymptotically stable. This shows that the pathogenic effect may cause the system to be unstable and be responsible for the occurrence of sustained oscillations.
In this section, we will analyze the existence of forward bifurcation and Hopf bifurcation of system (2.3).
Theorem 4.1 System (2.3) exhibits a forward bifurcation from ES(0,K(r−μ2)r,0) when R0=1. Furthermore, no backward bifurcation occurs.
Proof. The Jacobian matrix of system (2.3) at ES(0,K(r−μ2)r,0) is
J(ES)=(−μ10Nhβh−S0vβvμ2−r(1+θ)μ2−rS0vβv0−d−μ2). |
Its eigenvalues are
λ1=μ2−r,λ2=−b0−√Δ02,λ3=−b0+√Δ02, |
where
b0=μ1+μ2+d,Δ0=4a2bcK(r−μ2)rNh+(d−μ1+μ2)2. |
Because all parameter values are non-negative, we know that λ1 is always negative.
If R0=1, then
c∗≜rNhμ1(d+μ2)a2bK(r−μ2). |
Substituting c=c∗ into λ2 and λ3, we have λ2<0,λ3=0. Also, the parasite-free equilibrium ES is locally stable when c<c∗, and unstable when c>c∗. Therefore, c=c∗ is a bifurcation value.
We obtain a right eigenvector u and a left eigenvector ˉv associated with the zero eigenvalue, where
u=(u1,u2,u3)T=(abIh,μ1(θμ2−r−d)r−μ2Ih,μ1Ih)T,ˉv=(ˉv1,ˉv2,ˉv3)=(d+μ2,0,ab). |
By the orthogonal condition <u,ˉv>=1, we get
I∗h=1ab(d+μ1+μ2). |
By the transformation
Ih=x1,Sv=x2,Iv=x3, |
and noticing that system (2.3) has the form dxdt=f, where x=(x1,x2,x3)T and f=(f1,f2,f3)T, we have
x′1(t)=βhx3(t)(Nh−x1(t))−μ1x1(t):=f1,x′2(t)=r(x2(t)+θx3(t))(1−x2(t)+x3(t)K)−βvx2(t)x1(t)−μ2x2(t):=f2,x′3(t)=βvx2(t)x1(t)−dx3(t)−μ2x3(t):=f3. | (4.1) |
The formula of the bifurcation coefficient in system (4.1) at ES is:
ˉa=3∑i,j,k=1ˉviujuk∂2fi∂xj∂xk(ES,c∗),ˉb=3∑i,j=1ˉviuj∂2fi∂xj∂c(ES,c∗). |
Since ˉv2=0, what we need to consider are the cross derivatives of f1 and f3 in system (4.1) at the equilibrium ES. We obtain some non-zero terms
∂2f1∂x1∂x3=∂2f1∂x3∂x1=−βh, |
∂2f3∂x1∂x2=∂2f3∂x2∂x1=βv, |
∂2f3∂x1∂c=∂2f3∂c∂x1=aS0vNh. |
Now we calculate the values of ˉa and ˉb.
ˉa=3∑i,j,k=1ˉviujuk∂2fi∂xj∂xk(ES,c∗)=ˉv13∑j,k=1ujuk∂2f1∂xj∂xk(ES,c∗)+ˉv33∑j,k=1ujuk∂2f3∂xj∂xk(ES,c∗)=−2abμ1(d+μ2)(abK(r−μ2)2+rNhμ1(d+r−θμ2))NhK(r−μ2)2I∗2h<0,ˉb=3∑i,j=1ˉviuj∂2fi∂xj∂c(ES,c∗)=ˉv3u1∂2f3∂x1∂c(ES,c∗)=a3b2K(r−μ2)rNhI∗h>0. |
From [41], we know that the local dynamical behavior of system (4.1) at equilibrium ES is determined by the signs of ˉa and ˉb. From the above calculation, we have ˉa<0 and ˉb>0. Thus, there exists a locally asymptotically stable endemic equilibrium of system (4.1) showing a forward bifurcation near the equilibrium ES.
Remark 4.2 We conclude that no backward bifurcation occurs for system (4.1). The forward bifurcation of system (4.1) is shown by Figure 7 in Section 5.
For any parasite-positive equilibria E∗=(I∗h,S∗v,I∗v), the Jacobian matrix of system (2.3) at E∗ is
J(E∗)=(−I∗vβh−μ10(Nh−I∗h)βh−S∗vβvJ∗22J∗23S∗vβvI∗hβv−d−μ2), |
where
J∗22=r(1−I∗v+S∗vK)−I∗hβv−μ2−r(S∗v+θI∗v)K,J∗23=rθ(1−I∗v+S∗vK)−r(S∗v+θI∗v)K. | (4.2) |
The corresponding characteristic polynomial is
P(ξ;I∗h,S∗v,I∗v)=ξ3+a1ξ2+b1ξ+c1, | (4.3) |
where
a1=d−J∗22+I∗vβh+μ1+μ2,b1=−J∗22[d+I∗vβh+(μ1+μ2)]+dI∗vβh−J∗23I∗hβv+(I∗h−Nh)S∗vβhβv+dμ1+I∗vβhμ2+μ1μ2,c1=−J∗22[dμ1+I∗vβhμ2+μ1μ2+dI∗vβh+I∗hS∗vβhβv−NhS∗vβhβv]+I∗hNhS∗vβhβ2v−J∗23[I∗hβvμ1+I∗hI∗vβhβv]−I∗2hS∗vβhβ2v, |
J∗22,J∗23 are defined by (4.2), S∗v and I∗v are defined in (3.2) and the coordinate of I∗h is a positive root of (3.3).
From the above calculation and reference [46], we have the following theorem.
Theorem 4.3 The parasite-positive equilibrium E∗ of system (2.3) undergoes
1) a static bifurcation if c1|E∗=0 and Δ1,2|E∗>0;
2) a Hopf bifurcation if Δ2=0,dΔ2d(Bif.)≠0,Δ1|E∗>0 and c1|E∗>0.
Here a1,b1,c1 are the coefficients of the characteristic polynomial (4.3), d(Bif.) is the differentiation of the bifurcation parameter, and Δ1=a1,Δ2=a1b1−c1 are the first and the second Hurwitz arguments, respectively.
In this section, we conduct numerical simulations for system (2.3) by using Matlab and Auto07P [47]. We choose the following parameter values,
a=0.6,b=0.06,c=0.49,r=0.0274,θ=0.9,K=1000,Nh=400,μ1=0.0025,μ2=0.0083,d=0.0246, | (5.1) |
which were also used in the reference [10]. It is easy to calculate that there are one vector-free equilibrium, one parasite-free equilibrium, and one parasite-positive equilibrium point with the above parameter values.
We study the effect of the number of competent hosts Nh and the carrying capacity of triatomine bugs K on the basic reproduction number of T. rangeli R0. R0 is proportional to K. Thus, R0 increases as K increases. In particular, R0 increases with an increasing number of hosts and the carrying capacity of bugs K. This is shown in Figure 2(a). R0 depends on βh, which is an important parameter and its expression is a combination of Nh and K. We find that the slopes of the curves increase when the number of competent hosts decreases. Because R0 is related to βh and K, the transmission rate from infected bugs to susceptible hosts is inversely proportional to the number of hosts, as shown in Figure 2(b). Figure 3(a) shows the occurrence of sustained oscillation as the parameter values are defined in (5.1) while Figure 3(b) illustrates the stability of the parasite-positive equilibrium in Theorem 3.4.
Next, we consider the influence of the maximal number of offsprings that a triatomine bug produces per unit time (r), the carrying capacity of bugs (K), the pathogenic effect (d), and the transmission probability from infected hosts to susceptible triatomine bug per bite (c) in model (2.3) by using one-parameter and two-parameter bifurcation analysis.
We start with the maximal number of offsprings that a triatomine bug can produce per unit time. We choose r as the primary bifurcation parameter and keep the other parameters fixed as in (5.1). The one-parameter bifurcation diagram is shown in Figure 4. There exist two transcritical bifurcation points TC1(0,0,0) and TC2(0,3.10847,0) when r=8.3×10−3 and r=8.326×10−3, one supercritical Hopf bifurcation point HB1(1.94275,3.12364,0.135572) when r=9.39283×10−3, and one supercritical Hopf bifurcation point HB2(8.18863,3.17343,0.580539) when r=1.23408×10−2. The numbers of infected competent hosts, susceptible triatomine bugs and infected triatomine bugs will increase gradually when 0<r<9.39283×10−3 and r>1.23408×10−2. There is an unstable interval in which supercritical Hopf bifurcation occurs when 9.39283×10−3≤r≤1.23408×10−2. The red solid curve represents the stable limit cycle branch bifurcating from the supercritical Hopf bifurcation point, which indicates the appearance and the disappearance of stable limit cycle with the increase of the parameter r. Thus, when the maximal number of offsprings of susceptible triatomine bugs increases, the number of infected competent hosts and the number of infected triatomine bugs will increase out of the Hopf bifurcation interval r∈[9.39283×10−3,1.23408×10−2]. Further, if we use μ1 and r as the primary bifurcation parameters, then we obtain a two-parameter Hopf bifurcation curve, where the line μ1=0.0025 corresponds to the parameter values for Hopf bifurcation shown in Figure 4(a)–(c). The limit cycle branch connects the two Hopf bifurcation points and the period of limit cycle is finite. From Figure 4(d), we find that there are one or two Hopf bifurcation points when 0≤μ1≤0.0055, and no Hopf bifurcation occurs when μ1>0.0055.
The carrying capacity of triatomine bugs is the number that the ecosystem can sustainably support. It may vary due to many factors. We also use the carrying capacity K as the primary bifurcation parameter. With θ=0.3, we obtain the one-parameter bifurcation diagram for system (2.3) shown in Figure 5. There are one transcritical bifurcation point TC(0,3.10847,0) and one supercritical Hopf bifurcation point HB(33.9734,3.39698,2.57824) when K=4.45926 and K=5.45626×102, respectively. The amplitudes and periods of limit cycles bifurcating from HB become larger as K increases. When the carrying capacity of susceptible triatomine bugs increases, there will be always a supercritical Hopf bifurcation point, i.e., all the state variables will vary periodically, and all the competent hosts, susceptible triatomine bugs and infected triatomine bugs will coexist. In addition, if we use r and d as the primary two bifurcation parameters, then we obtain the two-parameter Hopf bifurcation curves which are all closed curves when K=800,1000,1200, respectively. This also indicates that no Bogdanov-Takens bifurcation of the parasite-positive equilibrium occurs for system (2.3).
Next, we use d as the primary bifurcation parameter to study the influence of the pathogenic effect in system (2.3). The parameter θ is set to 0.2 and other parameters are fixed as in (5.1). We obtain the one-parameter bifurcation diagram, shown in Figure 6(a)–(c). There are two supercritical Hopf bifurcation points HB1(27.3612,8.51017,2.03959) and HB2(36.7308,1.90856,2.80866) when d=0.0756105 and d=0.0100453, respectively. Thus, when the death rate of infected vectors increases due to the strong pathogenic effect, the number of the competent hosts and infected triatomine bugs will decrease, and the number of susceptible triatomine bugs will increase except an unstable interval for the occurrence of Hopf bifurcation. Two-parameter Hopf bifurcation curve is also given to illustrate the occurrence of stable limit cycles (Figure 6(d)), where the red dotted line is K=1000. Hopf bifurcation will always occur when K≥252.498.
The transmission rate from infected competent hosts to susceptible bugs (βv) depends on the number of bites per triatomine bug per unit time (a) and the transmission probability from infected hosts to susceptible triatomine bugs per bite (c). Since the two parameters a and c have a similar role in βv, for simplicity, we only use c as the primary bifurcation parameter and fix θ=0.3. We obtain the one-parameter bifurcation diagram (Figure 7(a)–(c)). There is a transcritical bifurcation point TC(0,6.9708×102,0) when c=2.18504×10−2, which confirms the forward bifurcation as shown in Theorem 3.1. The numbers of infected hosts and infected vectors will increase firstly, then decrease dramatically to a low level. The appearance of the limit cycle indicates that T. rangeli parasites will persist with the increase of the transmission probability from infected hosts to susceptible triatomine bugs per bite. Also, there is a supercritical Hopf bifurcation point HB(44.6167,4.57113,348737) when c=0.375043. The amplitudes and periods of limit cycles bifurcating from HB become larger as c increases. From Figure 7, we can see that all state variables coexist once the number of bites of per triatomine bug per unit time is greater than 0.865432. Therefore, T. rangeli parasites will always persist if the infection rates of susceptible triatomine bug and infected triatomine bug increase. The two-parameter (c vs. K) Hopf bifurcation curve of system (2.3) is given in Figure 7(d), which tells the relationship of the carrying capacity and the transmission probability from infected hosts to susceptible triatomine bug per bite. There is only one supercritical Hopf bifurcation occurring for system (2.3).
From the above analysis, we know that the dynamics of triatomine bugs in system (2.3) are similar to HIV dynamics in [41,48]. System (2.3) undergoes a forward bifurcation instead of a backward bifurcation. However, from the limit cycle branches in Figures 5(a)–(c) and 7(a)–(c), we conclude that the T. rangeli parasites relevant to Chagas disease will persist due to the sustained oscillations from Hopf bifurcation when the carrying capacity (K) or the transmission probability (c) increases. Thus, it is challenging to eliminate T. rangeli parasites, a sister trypanosoma to T. cruzi and commonly causing the misdiagnosis of the Chagas disease.
Comparing with the dynamics of model (2.3) and revisiting model (2.1) numerically, we find that model (2.1) with Ricker's type function in [10] also doesn't undergo Bagdanov-Takens bifurcation at the parasite-positive equilibrium. However, model (2.1) has periodic-doubling bifurcation of limit cycles for r, the maximal number of offsprings that a triatomine bug can produce per unit time. This indicates the occurrence of chaos for model (2.1), which differs from the dynamics of model (2.3).
In this paper, we have formulated a model with a logistic growth of of triatmine bugs to study the dynamics of infected competent hosts, susceptible and infected triatomine bugs. The existence and stability of the vector-free equilibrium, parasite-free equilibrium and the parasite-positive equilibrium are studied. The direction of transcritical bifurcation and Hopf bifurcation is also investigated. Numerical simulations are conducted to illustrate and expand the theoretical results.
For many infectious disease models, the disease-free equilibrium usually loses its stability when R0 increases to cross one, which results in a bifurcation where a curve of endemic equilibria emerges. The direction of this bifurcation is forward if the endemic curve occurs when R0 is slightly above 1 and there is no parasite-positive equilibrium near the disease-free equilibrium for R0<1. In contrast, the bifurcation is backward if the bifurcating equilibrium occurs when R0<1. Basically, a backward bifurcation implies the occurrence of multiple endemic equilibria and the coexistence of a stable endemic equilibrium with a stable disease-free equilibrium. Thus, a forward bifurcation indicates that the infectious disease may be cured while it is not easy to eliminate the disease when a backward bifurcation occurs. Model (2.3) only goes through the forward bifurcation. However, this doesn't mean that the disease would be eliminated. The persistence of sustained oscillations of T. rangeli makes the disease eradication challenging.
One-parameter bifurcation diagrams for the parameters r,K,d,c are showed, respectively. Oscillations always persists as K or c increases. This shows that the Trypanosoma rangeli will always exist in all the population and it is difficult to be eliminated. Bagdanov-Takens bifurcation of the parasite-positive equilibrium doesn't occur in models (2.1) and (2.3). Using numerical simulations, we also find that model (2.1) with Ricker's type function growth rather than the logistic function growth could go though periodic-doubling bifurcation of limit cycles when the maximal number of offsprings that a triatomine bug can produce per unit time (r) increases. This suggests the emergence of chaos for model (2.1).
The pathogenic effect to the triatomine bug population is also studied. This may provide some critical insights for the prevention and control of Chagas disease. Although T. rangeli is not pathogenic to human, it is pathogenic to triatomine bugs, which have a great effect on the dynamics of T. cruzi population and triatomine bug population. It can cause the model to lose stability and undergo Hopf bifurcation. Moreover, the amplitudes of oscillations become larger as the parameters K or c increases. It is worth noting that we illustrate the non-existence of Bagdanov-Takens bifurcation of the parasite-positive equilibrium numerically without providing analytical results due to too many parameters in model (2.3).
This work provides more information that improves our understanding of the complexity of host-parasite-vector interactions.
The authors are very grateful to Professor Jianhong Wu for his helpful suggestions. This work was partially supported by the National Natural Science Foundation of China (No. 11671114) and NSF of Zhejiang (LY20A010002). X. Wu is supported by NSF of China (No.12071300). L. Rong is supported by the NSF grant DMS-1950254.
The authors declare there is no conflict of interest.
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1. | Fanwei Meng, Lin Chen, Xianchao Zhang, Yancong Xu, TRANSMISSION DYNAMICS OF A CHAGAS DISEASE MODEL WITH STANDARD INCIDENCE INFECTION, 2023, 13, 2156-907X, 3422, 10.11948/20230071 |
Parameter | Range | Description |
Λh | varied | Recruitment rate of susceptible competent host per unit time [10,39] |
a | [0.2, 33]/day | Number of bites per triatomine bug per unit time [10,39] |
b | [0.00271, 0.06] | Transmission probability from infected bugs to susceptible |
competent hosts per bite [2,9,10] | ||
c | [0.00026, 0.49] | Transmission probability from infected hosts to susceptible |
triatomine bug per bite [9,25] | ||
r | [0.0274, 0.7714]/day | Maximum number of offsprings that a triatomine bug can |
produce per unit time [2,25] | ||
θ | [0, 1] | Reproduction reduction of bugs due to the infection of parasites [10] |
Nh | varied | Total number of hosts [2] |
K | varied | Carrying capacity [10] |
μ1 | [0.000038, 0.0025]/day | Natural death rate of hosts [9,25] |
μ2 | [0.0045, 0.0083]/day | Natural death rate of triatomine bugs [9,25] |
d | [0.0188, 0.0347]/day | Death rate of infected vectors due to pathogenic effect [10] |
Parameter | Range | Description |
Λh | varied | Recruitment rate of susceptible competent host per unit time [10,39] |
a | [0.2, 33]/day | Number of bites per triatomine bug per unit time [10,39] |
b | [0.00271, 0.06] | Transmission probability from infected bugs to susceptible |
competent hosts per bite [2,9,10] | ||
c | [0.00026, 0.49] | Transmission probability from infected hosts to susceptible |
triatomine bug per bite [9,25] | ||
r | [0.0274, 0.7714]/day | Maximum number of offsprings that a triatomine bug can |
produce per unit time [2,25] | ||
θ | [0, 1] | Reproduction reduction of bugs due to the infection of parasites [10] |
Nh | varied | Total number of hosts [2] |
K | varied | Carrying capacity [10] |
μ1 | [0.000038, 0.0025]/day | Natural death rate of hosts [9,25] |
μ2 | [0.0045, 0.0083]/day | Natural death rate of triatomine bugs [9,25] |
d | [0.0188, 0.0347]/day | Death rate of infected vectors due to pathogenic effect [10] |