GLOBAL DYNAMICS OF A FILIPPOV PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD OF INFECTED-SUSCEPTIBLE RATIO∗

This paper presents a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio, above which control strategies of replanting or removing are needed to be carried out. Based on the Filippov approach, we study the sliding mode dynamics and further the global dynamics. It is shown that there is a unique equilibrium, which is a diseasefree equilibrium, an endemic equilibrium or a pseudo-equilibrium. Moreover, the equilibrium is proved to be globally asymptotically stable. Our results indicate that the control goal can be achieved by taking appropriate replanting and removing rate.


Introduction
Recently plant diseases have become a threat to crop yield as well as quality, and can further lead to health problem and unstable social effects [11,21,25]. Therefore a wide array of measures have been developed to control plant diseases. The more effective strategy is the integrated disease management, which combines several control measures to minimize losses and maximize returns. Among these measures, the cultural control measure including replanting and/or removing diseased plants is widely accepted due to the little environmental influence, see [1,6,23,24,26,34].
One important approach to understand disease transmission mechanisms is mathematical modeling, see [1-5, 9, 10, 12-20, 22, 24, 26-34]. Fishman et al. [9] considered the Citrus tristeza virus temporal spread model in a closed plant population with periodic complete removal of infected plants. However, eradicating the infected completely is generally not possible, nor biologically or economically. It is more reasonable and feasible to bring down the number of infected under an economic threshold, under which economic damage is acceptable. Thus by incorporating an economic threshold, some plant disease models were established, see [1,24,26,34].
Note that the plants population might not be closed, and the number of the infected and the susceptible plants might be always changed due to removing, recruiting as well as replanting. It is more realistic and flexible to consider an economic threshold as the ratio of the infected to the susceptible plants. Based on this motivation, we extend the model in [26] with an economic threshold of infected-susceptible ratio as follows dI(t) dt = βSI − η 2 I − vIΦ(S, I) with Φ(S, I) = 0, I S < k, 1, I S > k, (1.2) where S(t) and I(t) represent the number of susceptible plants and infected plants at the time t respectively; the constant A is the recruitment rate of susceptible plants; β is the infectious rate; η 1 and η 2 are the mortality rates of susceptible and infected plants respectively; p and v denote the replanting rate of susceptible plants and the removing rate of infected plants respectively and k ≥ 0 denotes the economic threshold of infected-susceptible ratio. In this model, when the ratio of the infected to the susceptible is less than k, no control measures need to be implemented. However, once the ratio exceeds the threshold k, one should remove the infected and/or replant the susceptible to control the disease. Throughout this paper, we assume p < kv. When the infected-susceptible ratio exceeds the economic threshold k, the approach for removing the infected, as the leading role to control the plant disease, are mainly implemented to effectively bring down the ratio of the infected to the susceptible under the ratio threshold k. Furthermore, we assume p < η 1 indicating that the replanting rate is less than the death rate of susceptible plants. Then the replanting rate p meets p < min{kv, η 1 }, which not only contributes to minimizing the loss and maximizing the production but also prevents the excessive increase in infected plants from replanting appropriate number of susceptible plants.
The goal of this paper is to study the global dynamics of the model (1.1) with (1.2). We find that the plant disease is able to be controlled by choosing appropriate removing and replanting rate. The rest of the paper is organized as follows. In section 2, we provide preliminaries for planar Filippov systems and dynamics analysis of the subsystems. Sliding mode dynamics are exhibited in Section 3. Section 4 is devoted to the analysis of the global dynamics of the model (1.1). Finally, we discuss biological implications in Section 5.

Preliminaries
In this section, we give some preliminaries for planar Filippov systems and discuss the dynamics of two subsystems.
Since the system (1.1) is piecewise continuous, we consider its solutions in Filippov sense. Thus some essential definitions on Filippov systems are given as follows based on references [7,27,28].
Then the system (1.1) with (1.2) can be written as the following generic planar Filippov system:Ẋ Suppose H X (X) directs to G 2 , where H X (X) represents the gradient of H(X) and ⟨·, ·⟩ is the standard scalar product. The separating boundary Σ = {X ∈ R 2 + |H(X) = 0} can be partitioned by the following regions: Now we give definitions of some types of singular points for the system (2.1).

Definition 2.3.
A point X * is called a pseudo-equilibrium if it is an equilibrium of the sliding mode of the system (2.1), i.e. λF 1 (X * ) The following propositions imply that the solutions of the model (2.1) with any initial values in R 2 + are positive and bounded.
Thus for t ∈ [0, t 1 ), we have which is a contradiction. Thus

a positively invariant and attracting region for the system (1.1) with any given initial conditions in
Hence the set Ω is attracting.
The dynamics of the subsystems will play an important role in the analysis of the global dynamical behavior of the system (2.1). Next, we examine the global stability for the subsystems respectively. For the subsystem (2.3), the basic production number is R 1 = Aβ η1η2 and there are two possible equilibria, a disease-free equilibrium E 1 0 and an endemic equilibrium E 1 , where

Proposition 2.3. For the subsystem (2.3), the disease-free equilibrium
Proof. According to [26], for R 1 > 1, rewrite the subsystem (2.3) as and consider the Lyapunov function The time derivative of V 1 along the solutions of the system (2.5) is Then by utilizing LaSalle's invariance set principle, we conclude that the endemic equilibrium E 1 is globally asymptotically stable.
Similarly, for the case where R 1 < 1, taking a Lyapunov function and employing LaSalle's invariance set principle, we claim that the disease-free equilibrium E 1 0 is globally asymptotically stable. For the subsystem (2.4), the basic production number is R 2 = Aβ (η1−p)(η2+v) and the possible equilibria are E 2 0 and E 2 , where ).
Then we have the following proposition, whose proof is very similar to that of Proposition 2.3 and is omitted.

Proposition 2.4. For the subsystem (2.4), the disease-free equilibrium
At last in this section, it is remarked that for the system (2.1) the diseasefree equilibrium E 1 0 is always real and the disease-free equilibrium E 2 0 is always virtual. Furthermore when

Sliding mode dynamics
In this section, we will study the sliding mode dynamics for the system (2.1). Choose vector n = (−k, 1) as a normal vector on discontinuous boundary Σ. To determine the tangent points and further the sliding region on Σ, we need the following functions which are defined by and Solving g 1 (S) = 0 and g 2 (S) = 0 yields the horizontal ordinates S 1 T and S 2 T of tangent points respectively, where It can be checked that 0 < S 1 according to Filippov convex method [8], where Under the assumption p < kv, there is a unique zero S * of the function f (S), where Moreover, E * = (S * , kS * ) T is the only possible pseudo-equilibrium for the system (2.1) and this pseudo-equilibrium exists if S 1 T < S * < S 2 T . The following lemma can help us to verify whether E * = (S * , kS * ) T is a pseudo-equilibrium. Lemma 3.1. If p < kv, then the following assertions hold: Proof. We only give the proof of the assertion (i), since the assertion (ii) can be similarly obtained. Without loss of generality, we prove that S 1 T > S * is equivalent to Aβ − kη 2 2 − η 1 η 2 < 0. By p < kv, f (S * ) = 0 and the definition (3.5) of f (S), we derive that S 1 T > S * if and only if f (S 1 T ) < 0. Notice that g 1 (S 1 T ) = 0, i.e., Thus we can obtain . On the other hand, the definition (3.1) of g 1 (S) means that S 1 T > A η 1 + kη 2 is equivalent to The proof is completed.

Analysis of global dynamics
In this section, we concentrate on the global dynamics for the system (2.1). We start by giving two lemmas to preclude closed orbits. Denote the right-hand sides of the system (2.1) There is no closed orbit that contains a part of the closure of the sliding mode Σ s for the system (2.1).
Proof. We give the proof by the way of contradiction. Without loss of generality, suppose that E 1 is real and E 2 is virtual and there exists a closed orbit C that contains a part of Σ s . Then the closed orbit C must start from the tangent point T 1 and reach Σ s after some time, as shown in Fig. 1. This implies that any orbit outside C cannot converge to E 1 , which leads to a contradiction with the global asymptotical stability of E 1 in region G 1 for the system (2.3). Analogously, the orbit starting from the tangent T 2 will not reach Σ s either.

Lemma 4.2.
There is no closed orbit surrounding Σ s for the system (2.1).
Proof. Suppose there exists a closed orbit Γ that surrounds Σ s , as shown in Fig.2. Denote the part below Σ as Γ 1 and the part above Σ as Γ 2 . The closed orbit Γ intersects with Σ at the points N 1 and N 2 . Let D 1 be the bounded region delimited by Γ 1 and N 1 N 2 and D 2 be the region bounded by Γ 2 and N 1 N 2 . Let B(S, where dS = f 1 1 dt and dI = f 1 2 dt along Γ i (i = 1, 2). Similarly, which contradicts with (4.1). Hence there is no closed orbit surrounding Σ s .

Lemma 4.3 (See [8], §13). If a half trajectory T + is bounded for the system (2.1), then its limit set Ω(T ) contains either an equilibrium or a closed trajectory.
In the following, we consider global dynamics of all possible equilibria including the disease-free equilibrium E 1 0 , the endemic equilibrium E 1 or E 2 and the pseudoequilibrium E * . Theorem 4.1. Suppose that p < min{kv, η 1 } and R 1 < 1. Then the disease-free equilibrium E 1 0 is globally asymptotically stable for the system (2.1). Proof. By the assertion (i) of Lemma 3.1, T < S 2 T by Lemma 3.1, which reveals that there is no pseudo-equilibrium on the sliding mode Σ s . Also, the orbit on the sliding mode Σ s goes down along the Σ s as f (S) < 0 for S ∈ (S 1 T , S 2 T ) from the equation (3.5). Besides, Proposition 2.2 explains that solutions of the system (2.1) are bounded and no closed orbit exists for the system (2.1) on the basis of Lemma 4.1 and Lemma 4.2. In this case the ω-limit set of the system (2.1) is the unique real equilibrium E 1 0 by Lemma 4.3 so that any solution of system (2.1) eventually stabilizes at the disease-free equilibrium E 1 0 , as shown in Fig. 3. For R 1 > 1, by the equation (4.2) and p < kv, Then we have the following theorem. Theorem 4.2. Suppose that p < min{kv, η 1 } and R 1 > 1. Then for the system (2.1) the following assertions hold:

(iii) the pseudo-equilibrium E * is globally asymptotically stable if
Proof. We first give the proof of the assertion (i). If Aβ − kη 2 2 indicates that the endemic equilibrium E 2 is virtual or the disease-free equilibrium E 2 0 exists, while the endemic equilibrium E 1 is a real equilibrium if Aβ − η 1 η 2 − kη 2 2 < 0 and R 1 > 1. Therefore, by Lemma 4.3, the ω-limit set of the system (2.1) is the unique real equilibrium E 1 so that in this case E 1 is globally asymptotically stable for the system (2.1), as shown in Fig. 4. Analogously, we can obtain the assertion (ii), as shown in Fig. 5. Now we turn to the proof of the assertion (iii). The conditions Aβ −η 1 η 2 −kη 2 2 > 0 and Aβ−(η 1 −p)(η 2 +v)−k(η 2 +v) 2 < 0 imply that S 1 T < S * < S 2 T , which indicates that there exists a pseudo-equilibrium E * . The pseudo-equilibrium E * is stable on the sliding mode Σ s since f (S) > 0 for S ∈ (S 1 T , S * ) and f (S) < 0 for S ∈ (S * , S 2 T ). In addition, Aβ − η 1 η 2 − kη 2 2 > 0 and Aβ − k(η 2 + v) 2 − (η 1 − p)(η 2 + v) < 0 explain that solutions of both subsystems cannot converge to their own equilibria. Accordingly, in this case the pseudo-equilibrium E * is globally asymptotically stable for the system (2.1) (see Fig. 6).

Biological implications
In this paper, we put forward a Filippov plant disease model incorporating a threshold policy and control strategies for cutting off the infected and replanting the susceptible. Our goal is to establish conditions for controlling the number of infected plants below a given tolerable threshold so that the farmers can evaluate and choose appropriate rouging and replanting control strategies to scientifically minimize the economic losses and maximize the production. Making use of the Filippov approach for sliding mode and global dynamics, we investigate the global dynamics of the model (2.1) and acquire the global stability of all possible equilibria including disease free equilibrium E 1 0 , pseudo-equilibrium E * , endemic equilibrium E 1 or E 2 as the value of parameters varies, which is summarized in Theorem 4.1 and Theorem 4.2.
These theorems also reveal that the choice of replanting rate p and the rouging rate v are of great importance in controlling the plant disease and reduce the losses. Next by varying the control parameters p and v under the assumption p < min{kv, η 1 }, we distinguish three cases to illustrate how to achieve our control goal: to maintain the infected-susceptible ratio below the ratio threshold k.
Case 1: In this case, the infected plants will eventually goes extinct whether the control measures are implemented or not by Proposition 2.3 and Theorem 4.1. Compared with no control measures (p = 0, v = 0), taking appropriate control measures (p ≥ 0, v > 0, p < min{kv, η 1 }) helps control the plants disease faster. As shown in Fig.7, without any control measures, the infected-susceptible plants ratio (i.e. I(t) S(t) ) stay below the threshold k after time t 1 . By contrast, with the same initial values and other parameters, it takes time t 2 for I(t) S(t) maintaining below k when control measures are carried out (p = 0.4, v = 0.4). Due to t 2 < t 1 , we conclude that taking control measures contributes to achieving control goal faster, which reduces the losses. Case 2: R 1 > 1 and Aβ − kη 2 2 − η 1 η 2 < 0. Similar to the discussion in Case 1, control goal can be achieved with or without control strategies since any solution of the system (2.1) with any given initial values will eventually stabilize at the endemic equilibrium E 1 by Proposition 2.3 and the assertion (i) of Theorem 4.2. In particular, Fig. 8 suggests that choosing appropriate control parameters p and v can help achieve the control goal faster and minimize the losses. Case 3: R 1 > 1 and Aβ − kη 2 2 − η 1 η 2 > 0. Under the assumption p < min{kv, η 1 }, it is worth noting that whether the control goal can be achieved depends on the control parameters p and v. As shown in Fig. 9, if we choose parameters p and v in the region U 1 , the susceptible-infected ratio I(t) S(t) can be ultimately controlled at the threshold k by the assertion (iii) of Theorem 4.2. Nevertheless, we fail to come to control goal if the parameters p and v are taken in the region U 2 since the ratio I(t) S(t) always exceed the threshold k by the assertion (ii) of Theorem 4.2.
Notice that the control strategy in this paper is based on the ratio of the infected population over the susceptible population. It should be pointed out that the same analysis would work for the case that the control strategy is based on the ratio of the infected population over the total population. The above analysis indicates that control measures are effective if appropriate removal rate v and replanting rate p are chosen. Note that we only discuss the dynamics of the system (2.1) provided p < kv and more abundant dynamics when p > kv remain for further investigation. In the case p > kv, Fig.10 shows that two pseudo-equilibira E * 1 , E * 2 and a real equilibrium E 2 coexist, and there are two heteroclinic orbits between which one connects the two pseudo-equilibira E * 1 , E * 2 and the other connects the pseudo-equilibrium E * 2 and the real equilibrium E 2 .