A discrete-time analog for coupled within-host and between-host dynamics in environmentally driven infectious disease
- Buyu Wen^{1},
- Jianpeng Wang^{1} and
- Zhidong Teng^{1}Email author
https://doi.org/10.1186/s13662-018-1522-1
© The Author(s) 2018
Received: 26 August 2017
Accepted: 9 February 2018
Published: 26 February 2018
Abstract
In this paper, we establish a discrete-time analog for coupled within-host and between-host systems for an environmentally driven infectious disease with fast and slow two time scales by using the non-standard finite difference scheme. The system is divided into a fast time system and a slow time system by using the idea of limit equations. For the fast system, the positivity and boundedness of the solutions, the basic reproduction number and the existence for infection-free and unique virus infectious equilibria are obtained, and the threshold conditions on the local stability of equilibria are established. In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established. To return to the coupling system, the local stability for the virus- and disease-free equilibrium, and virus infectious but disease-free equilibrium is established. The numerical examples show that an endemic equilibrium is locally asymptotically stable and the other one is unstable when there are two endemic equilibria.
Keywords
MSC
1 Introduction
As is well known, viruses have caused abundant types of epidemic and occur almost everywhere on Earth, infecting humans, animals, plants, and so on. There are a large number of diseases, for example: influenza, hepatitis, HIV, AIDS, SARS, Ebola, MERS, etc., which are caused by viruses. Therefore, it is important to study viral infection, which can supply theory evidence for controlling diseases breaking out.
In recent years, many authors have established and investigated the various kinds of viral infection dynamical systems which are described by differential equations and difference equations. Many important and valuable results were established and successfully applied to viral infections in practice. See, for example, [1–22] and the references therein.
In [1], for system (1) the authors introduced a slow time variable \(\tau=\varepsilon t\), where \(0<\varepsilon\ll1\). In this case, t as a fast time variable. The authors further considered the parameters associated with the dynamics at the population level to be small based on the assumption that the between-host dynamics occur on a slower time scale than that of parasite-cell dynamics within the host. Let \(\Lambda=\varepsilon \tilde{\Lambda}\), \(\beta=\varepsilon\tilde{\beta}\), \(\mu =\varepsilon\tilde{\mu}\), \(\alpha=\varepsilon\tilde{\alpha}\), \(\theta=\varepsilon\tilde{\theta}\) and \(\gamma=\varepsilon\tilde {\gamma}\). Then, under the faster time variable t and the slower time variable τ, system (1) can be written as the two singular perturbation systems, see systems (6) and (7) given in [1], respectively. Using the techniques from the singular perturbation theory in [23], the authors in [1] analyzed system (1) by analyzing the corresponding fast and slow dynamics, and the fast and slow dynamics can be analyzed using the fast and slow time subsystems, see systems (8) and (9) given in [1], respectively. Here, we see that the concepts of fast and slow time systems were introduced in [23] early.
In recent years, more and more attention was paid on the discrete-time epidemic models. The reasons are as follows. Firstly, because the statistic data about infectious disease is collected by day, week, month, or year, so it is more direct, more convenient and more accurate to describe the epidemic by using discrete-time models than continuous-time models. Secondly, it is very difficult to solve a nonlinear differential equation with a given initial condition to obtain the exact solution. Thus, for many practical requirements, such as numerical calculation, it is often necessary to discretize a continuous model into the corresponding discrete model. Therefore, we see that the discrete-time analog is also alike important for studying coupled system (1). At the present time, there are various discretization methods to discretize a continuous model, including the standard methods, such as the Euler method, the Runge–Kutta method, and some other standard finite difference schemes, and the non-standard finite difference scheme, which is originally developed by Mickens (see [24–26]).
It is clear that in order to study the dynamical properties of coupled systems (3)–(4) we can firstly analyze the fast and slow two subsystems which are determined by two time scales t and s. In other words, we can treat the within-host subsystem (4) as the fast system and the between-host subsystem (3) as the slow system.
In this paper, for fast system (6) we will investigate the dynamical behaviors, including the positivity, boundedness, basic reproduction number, the existence of equilibria and the local stability of equilibria by using the discretization method. For slow system (7), we will investigate the dynamical properties, including the positivity, boundedness, the existence of disease-free equilibrium, only a unique endemic equilibrium, and two endemic equilibria, and the local asymptotic stability for the disease-free and endemic equilibria.
Next, we will investigate the dynamical behaviors for the coupled systems (3)–(4) basing on the research results obtained for the fast and slow subsystems. We will establish some criteria on the local asymptotic stability for the infection- and disease-free equilibrium, virus infectious but disease-free equilibrium and the endemic equilibrium. Furthermore, for the special cases which there is a unique endemic equilibrium, and two endemic equilibria in coupled systems (3)–(4), by means of the numerical examples, we will indicate that the unique endemic equilibrium may be locally asymptotically stable, and an endemic equilibrium may be locally asymptotically stable but the other one may be unstable.
This paper is organized as follows. In Sects. 2 and 3, fast system (6) and slow system (7) are discussed. Some criteria on the positivity, boundedness, existence of equilibria and local asymptotic stability are stated and proved. In Sect. 4, coupled systems (3)–(4) is discussed. Some criteria on the existence of equilibria and local asymptotic stability are stated and proved. In Sect. 5, the numerical examples are given. Lastly, a discussion is presented in Sect. 6.
2 The analysis of fast system
We firstly introduce the following lemmas on the quadratic and cubic polynomial equations which are given in [27].
Lemma 1
Lemma 2
- (H)
\(g(E)\) is defined for all \(0 \leq E\leq1\) and is continuously differentiable, which satisfies \(g(0)=0\), \(g(E)\geq0\), \(g'(E)>0\) and \(g''(E)\leq0\) for all \(0 \leq E\leq1\).
Firstly, on the positivity and boundedness of the solutions and the existence of nonnegative equilibria for system (6) we have the following results.
Lemma 3
The solution \((T(s),T^{*}(s),V(s))\) of system (6) with initial value (8) is positive for all \(s\geq0\) and ultimately bounded.
Proof
Therefore, solution \((T(s),T^{*}(s),V(s))\) with initial value (8) is ultimately bounded. This completes the proof. □
Remark 1
Lemma 4
Let \(E=0\), then system (6) always has infection-free equilibrium \(B_{0}(T_{0},0,0)\), and when \(R_{f}>1\), system (6) has a unique infectious equilibrium \(B^{*}(\breve{T},\breve{T}^{*},\breve{V})\).
Proof
Lemma 5
Next, we discuss the stability of the infection-free equilibrium and infectious equilibrium for system (6). We have the following theorems.
Theorem 1
- (a)
If \(R_{f}<1\), then infection-free equilibrium \(B_{0}\) is locally asymptotically stable.
- (b)
If \(R_{f}>1\), then \(B_{0}\) is unstable.
Proof
When \(R_{f}>1\), then \(f(1)<0\). Since \(\lim_{\lambda\to\infty }f(\lambda)=+\infty\), we see that \(f(\lambda)=0\) has a root \(\lambda _{3}\in(1,+\infty)\). This implies that equilibrium \(B_{0}\) is unstable. This completes the proof. □
Theorem 2
Let \(E=0\) in system (6). If \(R_{f}>1\), then infectious equilibrium \(B^{*}\) is locally asymptotically stable.
The proof of Theorem 2 will be given in Theorem 3 as the special case with \(E=0\). We hence omit it here.
Theorem 3
Let \(E>0\) in system (6). Then infectious equilibrium \(B_{1}(\breve {T}(E),\breve{T}^{*}(E), \breve{V}(E))\) is locally asymptotically stable.
Proof
3 The analysis of slow system
Firstly, on the positivity and boundedness of the solutions and the existence of nonnegative equilibria for slow system (7) we have the following lemmas.
Lemma 6
The solution \((S(t),I(t),E(t))\) of system (7) with initial value (21) is positive for all \(t\geq0\) and ultimately bounded. Furthermore, \(0\leq E(t)\leq1\) for all \(t\geq0\).
Proof
When \(t=1\), by a similar argument to above, we can prove that \((S(2),I(2),E(2))\) exists uniquely and is positive. Owing to \(0< E(1)<1\), we also have \(0< E(2)<1\). Using induction, for any \(t\geq0\), we know that \((S(t),I(t),E(t))\) exists uniquely and is positive. Furthermore, we finally have \(0< E(t)<1\) for all \(t\geq0\).
Lemma 7
- (i)
System (7) always has a disease-free equilibrium \(P_{0}(\frac {A}{\mu},0,0)\).
- (ii)System (7) has a unique endemic equilibrium \(P^{*}(\bar{S},\bar {I},\bar{E})\) if and only if one of the following conditions holds:
- (a)
\(R_{s}=1\) and \(H_{M}>0\);
- (b)
\(R_{s}>1\).
- (a)
- (iii)System (7) has two endemic equilibria \(P_{1}(S_{1},I_{1},E_{1})\) and \(P_{2}(S_{2},I_{2},E_{2})\) if and only if the following condition holds:
- (c)
\(R_{s}<1\) and \(H_{M}>0\).
- (c)
- (iv)System (7) has only disease-free equilibrium \(P_{0}(\frac{A}{\mu },0,0)\) if and only if one of the following conditions holds:
- (d)
\(H_{M}<0\);
- (e)
\(R_{s}=1\) and \(H_{M}=0\).
- (d)
Proof
If condition (a) holds, then from \(H(0)=0\) and \(H_{M}>0\), we easily see that \(H(E)=0\) has a unique positive root Ē. Hence, endemic equilibrium \(P^{*}(\bar{S},\bar{I},\bar{E})\) exists and is unique.
If condition (b) holds, then from \(H(0)>0\), it follows that \(H(E)=0\) has a unique positive root Ē, and hence endemic equilibrium \(P^{*}(\bar{S},\bar{I},\bar{E})\) also exists and is unique.
Assume that condition (c) holds, then owing to \(H(0)<0\) and \(H_{M}>0\), \(H(E)=0\) has only two positive roots. Hence, system (7) has only two endemic equilibria \(P_{1}\) and \(P_{2}\).
Lastly, we prove that system (7) has only disease-free equilibrium \(P_{0}(\frac{A}{\mu},0,0)\) if one of the conditions (d) and (e) holds. In fact, when \(H_{M}<0\) we see that \(H(E)=0\) has no root. When \(R_{s}=1\), then \(H(0)=0\). Therefore, by \(H_{M}=0\) there is only disease-free equilibrium \(P_{0}(\frac{A}{\mu},0,0)\). This completes the proof. □
Remark 2
From the proof of Lemma 7, we further see that when system (7) has a unique endemic equilibrium \(P^{*}(\bar{S},\bar{I},\bar{E})\), then since \(H(E)\) is a above convex function for \(0\leq E\leq1\), we also have \(H'(\bar{E})\leq0\).
Next, we discuss the stability of the disease-free equilibrium and endemic equilibrium for system (7). We have the following theorems.
Theorem 4
- (a)
If \(R_{s}\leq1\), then disease-free equilibrium \(P_{0}\) of system (7) is locally asymptotically stable.
- (b)
If \(R_{s}>1\) then equilibrium \(P_{0}\) is unstable.
Proof
When \(R_{f}\leq1\), we have \(\hat{V}(0)=0\). Hence \(f(\lambda)=0\) has roots \(\lambda_{1}=\frac{1}{1+\mu+\alpha}\) and \(\lambda_{2}=\frac {1}{1+\gamma}\). This implies that disease-free equilibrium \(E^{0}\) is locally asymptotically stable.
- (A)
\((\varphi+1)(n+1)(w-1)M+(1+2r+\varphi)N>0\), where \(M=\varphi w+\varphi n+nw+nw\varphi+r(1+ \mu)\varphi-2\), \(N=w\varphi+n\varphi+nw+2(\varphi+n+w)+2-r\varphi\), \(\varphi=\mu+\alpha\), \(w=\gamma-\theta\bar{I}F'(\bar{E})\) and \(n=\mu+\beta\bar{E}\).
Theorem 5
Assume that (A) holds and one of conditions (a) and (b) in Lemma 7 holds. Then unique endemic equilibrium \(P^{*}(\bar{S},\bar{I},\bar{E})\) of system (7) is locally asymptotically stable.
Proof
It is difficult to discuss the local stability for the case of two positive equilibria \(P_{1}\) and \(P_{2}\) in condition (c) of Lemma 7 by using the linearization method. However, we can give the following conjecture.
Conjecture 1
Assume that \(R_{s}<1\) and \(H_{M}>0\). Let \(P_{1}(\bar{S}_{1},\bar{I}_{1},\bar {E}_{1})\) and \(P_{2}(\bar{S}_{2},\bar{I}_{2},\bar{E}_{2})\) be two positive equilibria of slow system (7) with \(\bar{E}_{1}<\bar{E}_{2}\). Then \(P_{2}\) is locally asymptotically stable, and \(P_{1}\) is unstable.
4 The analysis for coupled system
From Lemmas 4, 5 and 7, we have the following result.
Lemma 8
- (1)
Coupled system (3)–(4) always has a disease-free and infection-free equilibrium \(D_{0}(\frac{A}{\mu},0,0,T_{0},0,0)\).
- (2)
If \(R_{f}>1\), then coupled system (3)–(4) has a disease-free equilibrium \(D_{1}(\frac{A}{\mu},0,0, \breve{T},\breve {T}^{*},\breve{V})\).
- (3)
If one of the conditions (a) and (b) in Lemma 7 holds, then coupled system (3)–(4) has a unique endemic equilibrium \(D^{*}(\bar{S},\bar{I},\bar{E},\breve{T}(\bar{E}),\breve {T}^{*}(\bar{E}),\breve{V}(\bar{E}))\).
- (4)
If the condition (d) in Lemma 7 holds, then coupled system (3)–(4) has only two positive equilibria \(D_{2}(\bar{S_{1}},\bar{I_{1}},\bar{E_{1}},\breve{T}(\bar {E_{1}}),\breve{T}^{*}(\bar{E_{1}}),\breve{V}(\bar{E_{1}}))\) and \(D_{3}(\bar{S_{2}},\bar{I_{2}},\bar{E_{2}},\breve{T}(\bar{E_{2}}), \breve{T}^{*}(\bar{E_{2}}), \breve{V}(\bar{E_{2}}))\) with \(\bar {E_{1}}<\bar{E_{2}}\).
On the stability of equilibrium \(\tilde{D}(\tilde{S},\tilde {I},\tilde{E},\tilde{T},\tilde{T}^{*},\tilde{V})\) of coupled system (3) and (4), we have the following definition.
Definition 1
- (1)
D̃ is said to be stable, if for any constant \(\epsilon >0\), there is a \(\delta=\delta(\epsilon)>0\) such that, for any initial point \((S_{0},I_{0},E_{0},T_{0},T^{*}_{0},V_{0})\) at time \(s=0\) and \(t=0\) satisfying \(|S_{0}-\tilde{S}|<\delta\), \(|I_{0}-\tilde {I}|<\delta\), \(|E_{0}-\tilde{E}|<\delta\), \(|T_{0}-\tilde{T}|<\delta\), \(|T^{*}_{0}-\tilde{T^{*}}|<\delta\), and \(|V_{0}-\tilde{V}|<\delta\), one has \(|S(t)-\tilde{S}|<\delta\), \(|I(t)-\tilde{I}|<\delta\), \(|E(t)-\tilde{E}|<\delta\), \(|T(s)-\tilde{T}|<\delta\), \(|T^{*}(s)-\tilde{T^{*}}|<\delta\), and \(|V(s)-\tilde{V}|<\delta\), for all \(t\geq0\) and \(s\geq0\).
- (2)D̃ is said to be locally asymptotically stable, if D̃ is stable and there is a constant \(\delta>0\) such that, for any solution \((S(t),I(t),E(t),T(s),T^{*}(s),V(s))\) with initial point \((S_{0},I_{0},E_{0},T_{0},T^{*}_{0},V_{0})\) at time \(s=0\) and \(t=0\) satisfying \(|S_{0}-\tilde{S}|<\delta\), \(|I_{0}-\tilde {I}|<\delta\), \(|E_{0}-\tilde{E}|<\delta\), \(|T_{0}-\tilde{T}|<\delta\), \(|T^{*}_{0}-\tilde{T^{*}}|<\delta\), and \(|V_{0}-\tilde{V}|<\delta\), one has$$\lim_{t\rightarrow\infty}\bigl(S(t),I(t),E(t)\bigr)=(\tilde{S},\tilde {I}, \tilde{E}),\qquad \lim_{s\rightarrow\infty}\bigl(T(s),T^{*}(s),V(s) \bigr)=\bigl(\tilde{T},\tilde {T^{*}},\tilde{V}\bigr). $$
Furthermore, by applying the theory of limit equations, from Theorems 1, 2 and 4, we have the following result.
Theorem 6
- (1)
If \(R_{f}<1\) and \(R_{s}\leq1\), then equilibrium \(D_{0}(\frac{A}{\mu },0,0,T_{0},0,0)\) is locally asymptotically stable, and if \(R_{f}>1\), then \(D_{0}\) is unstable.
- (2)
If \(R_{f}>1\) and \(R_{s}\leq1\), then equilibrium \(D_{1}(\frac{A}{\mu },0,0,\breve{T},\breve{T}^{*},\breve{V})\) is locally asymptotically stable, and if \(R_{s}>1\), then \(D_{1}\) is unstable.
Proof
When \(R_{f}>1\), then, by conclusion (b) of Theorem 1, we see that equilibrium \((0,0,0)\) of the last three equations of (25) is unstable. In addition, when \(R_{s}>1\) we also have \(R_{f}>1\). Therefore, \(D_{0}\) is unstable.
It is clear that when \(R_{s}\leq1\), by conclusion (a) of Theorem 4, from first three equation of (26), we know \((X(t),Y(t),Z(t))\rightarrow(0,0,0)\) as \(t\rightarrow\infty\). By Theorem 2 and from last three equations of (26), we further have \((U(s),V(s),W(s))\rightarrow(0,0,0)\) as \(s\rightarrow\infty\). Therefore, \(D_{1}\) is locally asymptotically stable. When \(R_{s}> 1\), by conclusion (b) of Theorem 4, we see that equilibrium \((0,0,0)\) of the first three equations of (26) is unstable. Therefore, \(D_{1}\) is unstable. This completes the proof. □
However, to establish the criteria of stability for endemic equilibrium \(D^{*}\) and two positive equilibria \(D_{2}\) and \(D_{3}\) is very difficult. We here only give the following conjectures.
Conjecture 2
Assume the condition (A) holds and one of conditions (a) and (b) of Lemma 7 holds. Then endemic equilibrium \(D^{*}\) of coupled system (3)–(4) is locally asymptotically stable.
Conjecture 3
Assume that \(R_{s}<1\) and \(H_{M}>0\). Then \(D_{3}\) is locally asymptotically stable and \(D_{2}\) is unstable.
In the following section, we will give a numerical example to show that Conjectures 2 and 3 may be right.
5 Numerical examples
List of parameters
Parameter | Definition | Value | Source |
---|---|---|---|
A | the recruitment rate of individuals | 4 | Ref. [5] |
β | the infection rate of hosts in a contamination | 0.0006 | Ref. [5] |
μ | the natural mortality rate of host | 0.0004 | Ref. [5] |
α | the induced mortality rate of host | 0.0004 | Ref. [5] |
g(E) | the rate which an average host is inoculated | g(E)=4 × 10^{5}E | |
θ | the rate of contamination | 1.5 × 10^{−10} | Ref. [5] |
γ | clearance rate | 0.015 | Ref. [5] |
Λ | the recruitment rate of cells | 6000 | Ref. [5] |
k | infections rate of cells | 1.5 × 10^{−6} | Ref. [5] |
m | the natural mortality rate of cells | 0.3 | Ref. [5] |
d | the induced mortality rate of cells | 0.15 | Ref. [5] |
c | the within-host mortality rate of parasites | 60 | Ref. [5] |
6 Discussions
In this paper we studied a discrete coupled within-host and between-host models (3)–(4) in environmentally driven infectious disease obeying Micken’s non-standard finite difference scheme. Since there are two fast and slow time scales in the model, and the fast time scale is sufficiently quicker than the slow time scale, the model is separated into a fast system (6) and a slow system (7).
The basic properties for fast system (6), including the existence of infection-free equilibrium \(B_{0}\), infected equilibrium \(B^{*}\) (when \(E=0\)) and infected equilibrium \(B_{1}\) (when \(E>0\)), the positivity and ultimate boundedness of the solutions with positive initial values, are established. Under assumption (H), the local stability of equilibria for system (6) is completely determined by basic reproduction number \(R_{f}\). That is, when \(E=0\) in system (6), if \(R_{f}< 1\) then \(B_{0}\) is locally asymptotically stable, and if \(R_{f}>1\) then \(B_{0}\) is unstable and \(B^{*}\) is locally asymptotically stable. When \(E>0\) in system (6), then infectious equilibrium \(B_{1}\) exists always and also is locally asymptotically stable.
For slow system (7), the basic properties on the existence of disease-free equilibrium \(P_{0}\), unique endemic equilibrium \(P^{*}\) and two positive equilibria \(P_{1}\) and \(P_{2}\), and the positivity and ultimate boundedness of the solutions with positive initial values are established.
The sufficient conditions on the local stability of disease-free equilibria \(P_{0}\) and unique endemic equilibria \(P^{*}\) are established by virtue of basic reproduction number \(R_{s}\), the quantity \(H_{M}\) and condition (A). However, it is very difficult to discuss the local stability of two endemic equilibria \(P_{1}\) and \(P_{2}\). Here we only show the local stability of \(P_{1}\) and \(P_{2}\) by the numerical examples in Sect. 5.
We see that assumption (A) is a pure mathematical condition. It is only used in the proofs of theorems on the local stability of endemic equilibria \(P^{*}\) to obtain \(|b_{0}|+b_{2}>0\) (see the proof of Theorem 5). Generally, we expect that the local stability of equilibria of slow system (7) can be determined only by basic reproduction number \(R_{s}\). Therefore, an open problem is whether condition (A) can be thrown off in Theorem 5. Furthermore, we also do not obtain the global asymptotic stability of equilibria for system (7). The reason is that the construction of Lyapunov function is very difficult.
For whole coupled systems (3)–(4), the basic properties on the existence of infection- and disease-free equilibrium \(D_{0}\), viral infection and disease-free equilibrium \(D_{1}\), unique endemic equilibrium \(D^{*}\) and two endemic equilibria \(D_{2}\) and \(D_{3}\), and the local stability of equilibria \(D_{0}\) and \(D_{1}\) are established, respectively. However, it is difficult to discuss the local stability for unique endemic equilibrium \(D^{*}\), and two endemic equilibria \(D_{2}\) and \(D_{3}\). Here, we only show the local stability of \(D^{*}\), \(D_{2}\) and \(D_{3}\) by the numerical examples in Sect. 5.
Comparing the results established in this paper with the results obtained in [1, 3], we see that the dynamical properties of equilibria for discrete-time model (3)–(4) and continuous-time model (2) (see Theorems 1–3 in [1]) in fast time and slow time subsystems, respectively, are very oncoming. This shows that discrete-time model (3)–(4), as a discrete-time analog of continuous-time model (2), is fairly appropriate. Particularly, we can use model (3)–(4) to calculate the numerical approximative solution of model (2) in a neighborhood of equilibrium. In addition, in this paper we further investigate the dynamical properties for whole coupled systems (3)–(4), such as the existence of equilibrium and the local stability of equilibrium.
Declarations
Acknowledgements
We are very grateful to the reviewers for their very helpful comments and careful reading of our manuscript. This research was supported by the National Natural Science Foundation of China (Grant Nos. 11771373, 11401512, 11261056).
Authors’ contributions
The main idea of this paper was proposed by ZT. BW prepared the manuscript initially and performed all the steps of the proofs in this research. JW performed the numerical examples and simulations. All authors read and approved the final manuscript.
Competing interests
No potential conflict of interest was reported by the authors.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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