A discrete plant disease model with roguing and replanting
- Yueli Luo^{1},
- Shujing Gao^{1}Email author,
- Dehui Xie^{1} and
- Yanfei Dai^{1}
https://doi.org/10.1186/s13662-014-0332-3
© Luo et al.; licensee Springer 2015
Received: 23 September 2014
Accepted: 19 December 2014
Published: 30 January 2015
Abstract
In this paper, we study a discrete plant virus disease model with roguing and replanting which is derived from the continuous case by using the well-known backward Euler method. The positivity of solutions with positive initial conditions is obtained. By applying analytic techniques and constructing a discrete Lyapunov function, we obtain the result that the disease-free equilibrium is globally attractive if \(R_{0}\leq1\), and the disease is permanent if \(R_{0}>1\). Numerical simulations show that the main theoretical results are true.
Keywords
1 Introduction
Plants not only provide people with essential means of subsistence, but also they offer other creatures food and shelter. However, plants often suffer from multiple adverse factors in the process of growth, especially viruses, which causes the decline of plants yield and quality, even famine and social unrest.
It is well known that many serious diseases of crop plants are caused by viruses. For severe cases, plant diseases have caused large-scale damage to various crops, which resulted in a diminished output in whole regions; for instance, cocoa swollen shoot in Ghana [1, 2] and banana bunchy top in Australia [3–5]. When a crop is widely planted in a new area, plant disease prevention usually becomes important. In most cases, we rogue (remove) infected plants as a control strategy when disease breaks out. Since 1946, 190 million infected trees have been removed in Ghana [6]. In addition, we can also rogue not only visibly infected plants but also other neighboring plants which do not yet show symptoms [2, 7], but this measure may be unpopular with farmers since it involves removal of apparently healthy plants which may still be highly productive [8].
Recently, more and more attention has been paid to the discrete-time epidemic models. In [9], the authors pointed out that it is more direct, more convenient, and more accurate to describe a disease by using the discrete-time models than the continuous-time models since the statistic data about the disease situation is collected by day, week, month or year. Furthermore the discrete-time models have more wealthy dynamical behaviors, such as the discrete-time epidemic models, which have bifurcations, chaos, and other more complex dynamical behaviors. Many important and interesting results can be found in [10–22] and the references cited therein.
We know that there are usually two methods to construct discrete-time epidemic models: (i) by making use of the compartment model theory and the property of the epidemic disease, (ii) by using techniques (the backward Euler scheme, the forward Euler scheme, and Mickens’ nonstandard discretization) to discretize a continuous-time epidemic model. Until now, some studies have been done on discrete-time epidemic models by using the two methods mentioned above (see [9–33]). For example, by applying Mickens’ nonstandard discretization, Wang et al. [9] discussed dynamical behaviors for a class of discrete SIRS models with disease courses. Muroya et al. [22] proposed a discrete epidemic model for a disease with immunity and latency spreading in a heterogeneous host population, which was derived from the continuous case by using the well-known backward Euler method, and they obtained the result of the global stability of the disease-free equilibrium and the endemic equilibrium. According to the first method, Teng et al. [25] constructed a discrete SIS epidemic model with stage structure and standard incident rate and established sufficient conditions for the permanence and extinction of the disease of the model. Moreover, using the method of linearization, the local asymptotic stability of the endemic equilibrium was studied. Applying the forward Euler scheme, Hu et al. [30] constructed a discrete SIR epidemic model, and they studied the local stability of the disease-free equilibrium and the endemic equilibrium. In addition, numerical simulations showed plentiful and complex dynamical behaviors including bifurcations.
In this paper, we use the well-known backward Euler method to discretize a continuous-time plant virus disease model with roguing and replanting which is investigated in [8]. Our main purpose is to study dynamical behaviors of the model.
The organization of this paper is as follows. In Section 2, the model description and some preliminaries are given. Section 3 deals with the global attractivity of disease-free equilibrium of the model. In Section 4, the criterion on the permanence of the disease of the model is stated and proved. In Section 5, the numerical simulations are provided to illustrate the validity of our theoretical results. Lastly, a brief discussion is given in Section 6.
2 Model formulation and preliminaries
Lemma 2.1
Any solution \((S(n),E(n),I(n),R(n))\) of model (2.2) with initial condition (2.3), is positive for any \(n>0\) and ultimately bounded.
Proof
From (2.5)-(2.8) we see that if only \(E(1)\) is confirmed, then \(I(1)\), \(R(1)\), \(N(1)\), and \(S(1)\) will be afterwards confirmed.
Firstly, we prove that if \(E(1)>0\) and then \(I(1)>0\), \(R(1)>0\), and \(S(1)>0\). From (2.6), we directly obtain \(I(1)>0\) when \(E(1)>0\). Secondly, from \(I(1)>0\) we further obtain the result that \(R(1)>0\).
Obviously, \(A>0\), \(B>0\), \(C>0\), \(D>0\), \(F>0\), \(a>0\).
By the characteristic of a quadratic equation, we know that if \(c<0\), then \(\Psi(y)=0\) has a unique positive solution \(\overline{y}\in(0,+\infty)\).
Now, we prove that \(c<0\). By calculating, we obtain the result that \(c<0\).
From the above discussions we see that \(\Psi(y)=0\) has a unique positive solution \(\overline{y}\in(0,+\infty)\). Let \(E(1)=\overline {y}\). We also have \(I(1)>0\), \(R(1)>0\), and \(S(1)>0\). Therefore, the positivity of \(S(1)>0\), \(I(1)>0\), and \(R(1)>0\) is finally obtained.
Define \(\phi(p,n)=pE(n)-I(n)\) and \(\varphi(p)=\frac{pk_{1}r}{r+\mu }+k_{3}- (1+\frac{1}{p} )k_{2}\), where \(p>0\) is a constant and \(n>0\) is an integer.
Lemma 2.2
If there exists a constant \(p>0\) such that \(\varphi(p)\leq0\), then there exists an integer \(N_{1}>0\) such that either \(\phi(p,n)\geq0\) for all \(n\geq N_{1}\) or \(\phi(p,n)\leq 0\) for all \(n\geq N_{1}\).
Proof
3 Global attractivity of disease-free equilibrium
Therefore, we can claim that \(R_{0}\) is the basic reproduction number of model (2.2).
Theorem 3.1
Disease-free equilibrium \(P_{0}\) of model (2.2) is globally attractive iff \(R_{0}\leq1\).
Proof
Case 1. \(pE(n)\leq I(n)\) for \(n\geq N_{1}\).
Case 2. \(pE(n)\geq I(n)\) for \(n\geq N_{1}\).
From (3.9) we directly obtain \((\frac{rk_{1}p}{\mu+r}-(\mu +k_{2}))E^{*}\geq0\). When \(R_{0}<1\), by (3.2) it follows \(E^{*}=0\). Furthermore, \(I^{*}=0\), \(R^{*}=0\) and \(N^{*}=\frac{rK}{r+\mu}\). This shows that (3.8) holds.
From the above discussions, we obtain the result that the disease-free equilibrium \(P_{0}\) is globally attractive. This completes the proof of Theorem 3.1. □
4 Permanence of disease
We have the following result.
Theorem 4.1
The disease \(I(n)\) in model (2.2) is permanent iff \(R_{0}>1\).
Proof
The necessity is obvious. In fact, if \(R_{0}\leq1\), then from Theorem 3.1 the disease-free equilibrium is globally attractive.
In order to obtain the permanence of disease \(I(n)\) of model (2.2), we discuss the following three cases.
Case 1. \(I(n)\geq\eta_{0}\) for all \(n\geq T_{1}\). For this case, obviously, \(I(n)\) is permanent.
When \(n\notin\bigcup_{k=1}^{\infty}[n_{k},m_{k}]\), we directly have \(I(n)\geq\eta_{0}>\xi\). Therefore, we finally obtain the result that \(I(n)\) is permanent. This completes the proof. □
5 Numerical simulations
In this section, we carry out numerical simulations on model (2.2) to demonstrate the results in Sections 3 and 4.
Example 5.1
Example 5.2
6 Conclusion
In this paper, we study the dynamic behaviors of a discrete plant virus disease model with roguing and replanting which is derived from the continuous case. By calculating, we obtain the basic reproduction number \(R_{0}\). We also prove that the disease-free equilibrium of model (2.2) is globally attractive when \(R_{0}<1\), in other words, the disease goes extinct. On the contrary, if \(R_{0}>1\), the endemic equilibrium of model (2.2) exists and the disease will be endemic.
In Theorem 4.1, we only discuss the permanence of the disease of model (2.2). However, the numerical simulations in Figure 1 show that the endemic equilibrium of model (2.2) may be globally attractive when \(R_{0}>1\).
In the real world, some plants at the beginning may show no infection, for example Calletotrichum musae. We usually use the time delay to describe this phenomenon in a mathematical model. However, the dynamic behaviors of discrete plant virus disease models with time delay are rarely considered. Therefore, an important and interesting open problem is whether we can obtain similar results on the permanence and extinction of the disease for the discrete plant virus disease models with time delay. We will discuss these problems in our future work.
Declarations
Acknowledgements
Special thanks to the anonymous referees who have given us very useful suggestions. The research has been supported by the Natural Science Foundation of China (11261004), the Science and Technology Plan Projects of Jiangxi Provincial Education Department (GJJ14665, GJJ14673), and the Postgraduate Innovation Fund of Gannan Normal University (YCZ13B002).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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