Advanced Nonlinear Dynamics of Population Biology and Epidemiology
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Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments
Abstract
We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction number which is defined through the spectral radius of a linear integral operator. If < 1, then the diseasefree periodic solution is globally asymptotically stable and if > 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.
1. Introduction
Plant disease is an important constraint to crop production. Due to plant diseases, more than 10 of global food production is lost and 800 million people do not have adequate food in the world [1–3]. Plant pathologists cannot ignore the juxtaposition of these figures for food shortage and the reduction of crops caused by plant disease.
Nowadays, Huanglongbing (HLB) which is a century old disease caused by the bacteria Candidatus Liberibacter spp is one of the most serious problems of citrus worldwide [4]. HLB has been responsible for the near destruction of citrus industries in Asia and Africa [4]. The main symptoms on HLBinfected citrus trees are yellow shoots, leaves with blotchy mottle, and small lopsided fruits [4, 5]. The HLB is a phloemrestricted, noncultured, Gramnegative bacterium causing crippling diseases denoting “greening" in South Africa, “mottle leaf" in the Philippines, “dieback" in India, and “vein phloem degeneration" in Indonesia. The infected citrus orchards are usually destroyed or become unproductive in 5 to 8 years [4].
Most of the known plant viruses are transmitted by insect vectors and entirely dependent on the behaviour and dispersal capacity of their vectors to spread from plant to plant. HLB, a destructive disease of citrus, can be transmitted by grafting from citrus to citrus and by dodder to periwinkle. The citrus psyllid (Diaphorina Citri Kuwayama) is natural and mainly vector [4]. In this paper, we mainly consider that HLB transmitted from tree to tree by Asian citrus psyllid insect vectors.
Mathematical models play an important role in understanding the epidemiology of vectortransmitted plant diseases. Since the introduction of HLB, a lot of researches have been conducted on the epidemiology of the disease and on the vector, but the result of these two lines of inquiry integrated is very few. Analytical models have also been developed for the spread of citrus canker [6], but models for vectortransmitted bacterial pathogens are still preliminary [7]. In [8], the authors proposed a deterministic compartmental mathematic model to analyze HLB spread between citrus plants. They assumed that all coefficients of the model are constant (autonomous systems). However, in the real world, actual data and evidence show that dynamics of disease transmission are not as simple as shown in the model. In [9], Hall and Hentz have studied seasonal activity of psyllid insect vectors which is correlated with humidity. Seasonal fluctuations in the transmission of infectious diseases imply that the corresponding mathematical models may admit periodic solutions. It is interesting and important to study the globally dynamics which are determined by threshold parameter in periodic epidemiological models.
Based on above introduction, we propose a model with periodic transmission rates to investigate the seasonal HLB epidemics [10, 11]. In this model, we consider Logistic growth term for dynamics of susceptible psyllid vector. Furthermore, we assumed that the infective citrus population is generated through susceptible citrus which was bit by infective psyllid and the susceptible psyllid bit the infective citrus which will become infective psyllid. Then, the periodic system is as follows: with initial condition Here, , , , and represent susceptible citrus host, infected citrus host, susceptible psyllid, and infected psyllid, respectively. We can easily see that and are the number of citrus population and psyllid population, respectively. is the recruitment rate of citrus at time , is the infected rate of citrus host at time , and are the nature death and disease induced death rate of citrus host at time , respectively, is the intrinsic growth rate of psyllid at time , and are the infected rate and the nature death rate of psyllid at time , respectively, and is the maximum abundance of psyllid per citrus. , , , , , , and are continuous, positive periodic functions.
The paper is organized as follows. In the next section, we give the basic reproduction number of (1). In Sections 3 and 4, the results show that the dynamical properties of the model are completely determined by . That is, if , the diseasefree periodic solution is globally asymptotically stable, and if , the model is permanence. In Section 5, we present numerical simulations which demonstrate the theoretical analysis and a brief discussion of our main results.
2. Basic Reproduction Number
In the following, we introduce some notations and lemmas which will be used for our further argument.
Let be the standard ordered dimensional Euclidean space with a norm . For ; we denote if , if , and if , respectively.
Define and , where is a continuous, positive, periodic function.
Consider the following linear ordinary differential system: where is a continuous, cooperative, irreducible, and periodic matrix function. Denote be the fundamental solution matrix of (3) and be the spectral radius of . By the PerronFrobenius Theorem, we know that is the principle eigenvalue of ; that is, it is simple and admits an eigenvector .
Lemma 1 (see [12]). Let . Then there exists a positive periodic function such that is a solution of (3).
Consider the following nonautonomous linear equation:
where and are the same as in System (1). From Zhang and Teng ([13, Lemma 2.1]) and simple calculation, we have the following lemma.
Lemma 2. System (4) has a unique positive periodic solution which is globally asymptotically stable.
Consider the following nonautonomous Logistic equation:
where and are the same as in system (1). From Teng and Li ([14, Lemma 2]) and simple calculation, we can obtain the following lemma.
Lemma 3. System (5) has a unique positive periodic solution which is globally asymptotically stable, where .
According to Lemmas 2 and 3, it is easy to see that (1) has a unique diseasefree periodic solution .
Now, we use the generation operator approach (see [15]) to derive the basic reproduction number. Applying the symbol of the theory in Wang and Zhao [15], for system (1), we have where . Then System (1) can be written as the following form: where .
It is easy to obtain that the conditions (A1)–(A5) in [15] hold. In the following, we will check the conditions (A6) and (A7) in [15].
We know that is the diseasefree periodic solution of system (7). Denote where and are the th components of and , respectively. According to (6), we have
It is easy to see that , where is the spectral radius of . This implies that is linearly asymptotically stable in the diseasefree subspace . Thus, condition (A6) in [15] holds.
We further define where and are the th components of and , respectively. Then, from (6), we obtain that
Let be a matrix solution of the system: where is identity matrix. From (11) and (12), we have . Therefore, the condition (A7) in [15] also holds.
Let be the ordered Banach space of all periodic function from , which is equipped with maximum norm and the positive cone , for all . Define the following linear operator by Based on the assumptions above and the results of Wang and Zhang [15], we can derive the basic reproduction number of system (1) as follows: and obtain the following conclusion.
Theorem 4. For system (1), the following statements are valid:(i) if and only if ,(ii) if and only if ,(iii) if and only if ,where and are defined in (11).
It follows from Theorem 4 that the diseasefree periodic solution of system (1) is asymptotically stable if , and it is unstable if .
In order to calculate , we consider the following linear periodic system: Let , , , be the evolution operator of the System (15) on . Since is nonnegative and is cooperative, then is continuous and nonincreasing for , and . Thus, we have the following result, which will be used in our numerical calculation of the basic reproduction ratio in Section 5.
Lemma 5 (see [15]). The following statements are valid.(i)If has a positive solution, is an eigenvalue of , and hence .(ii)If , then is the unique solution of .(iii) if and only if for all .
3. Global Stability of DiseaseFree Periodic Solution
In this section we will prove the global asymptotical stability of the diseasefree periodic solution .
Let , . Denote where and . Similar to [16, 17], we firstly prove the following lemmas.
Lemma 6. is a positively invariant set for (1).
Proof. From the equations in (1), we have which implies that is a positive invariant compact set for (1). The proof is completed.
Lemma 7. Let be any solution of system (1). It holds that where , are defined in Lemmas 2 and 3, respectively.
Proof. We denote that . It follows from the first equation of (17) that , which implies that . Further, from Lemma 6, we obtain that for any , there exists a such that
Let . From the second equation of (17) and (19), we get
for all , where
Obviously, . Because is arbitrarily small, then . Hence, the proof is completed.
Theorem 8. The diseasefree periodic solution is globally asymptotically stable if , whereas it is unstable if .
Proof. From Theorem 4, we have that is unstable if , and is locally stable if . Therefore, we only need to show the global attractivity of for .
Since , by Theorem 4, we can choose sufficiently small such that
where
From Lemma 6 and (18), we have that, for above mentioned , there exists a such that , for . It follows from the second and fourth equations that for ,
Consider the following comparison system:
In view of Lemma 1, we know that there exists a positive periodic function such that , where and . It follows from (22) that and . By the comparison of theorem [18], we have and . From (18), we have
Hence, the disease free periodic solution is globally attractive. This completes the proof.
4. Permanence
In this section, we show that if , then the disease persists.
Firstly, we define , , and , and we denote as the unique solution of System (1) with the initial value .
Define Poincaré map associated with System (1) as follows: By Lemma 6, it is easy to see that both and are positively invariant and is point dissipative. Set where . We claim that Obviously, . Next we want to show . If it does not hold, then there exists a point .
Case 1. and . It is obvious that and for any . Then, from the second equation of System (1), holds. It follows that for . This is a contradiction.
Case 2. and . It is obvious that and for any . Then, from the fourth equation of System (1), holds. It follows that for . This is a contradiction.
That is to say, for any , then . Therefore we have .
Next, we present the following result of the uniform persistence of the disease.
Theorem 9. Suppose . Then there is a positive constant such that each positive solution of System (1) satisfies
Proof. By Theorem 4, we obtain . So we can choose small enough such that , where
Put . Now we proceed by contradiction to prove that
If it does not hold, then
for some . Without loss of generality, suppose that
By the continuity of the solutions with respect to the initial values, we obtain
For any , there exists a such that , where . Then we have
for all , which implies that , . Then from (1) we have
Consider the linear system
By Lemma 1 and the standard comparison principle, we have that there exists a positive periodic function such that is a solution of System (38), where and
It follows from that and as . Applying the comparison principle [18], we know that and as . This is a contradiction. Thus, we have proved that (32) holds and is weakly uniformly persistent with respect to .
According to the results of Lemma 7, we can easily obtain that has a global attractor . It is easy to obtain that is an isolated invariant set in and . We know that is acyclic in and every solution in converges to . According to Zhao [19], we have that is uniformly persistent with respect to . This implies that the solution of (1) is uniformly persistent with respect to . Thus we have that there exists a such that , .
5. Numerical Simulations and Sensitivity Analysis
In this section, we will make numerical simulations by means of Matlab 7.1 to support our theoretical results, to predict the trend of the disease, and to explore some control and prevention measures. Numerical values of parameters of system (1) are given in Table 1 (most of the data are taken from the literatures ([20–24])).

According to the periodicity of System (1) and Table 1, we set , where and . By the similar method, we can obtain the other parameter functions of model (1) (see Table 2). For the simulations that follows, we apply the parameters in Table 2 unless otherwise stated.
Choose . Then from Lemma 5, we can compute by means of Matlab 7.1. From Theorem 8 we obtain that the infected citrus population and the infected psyllid population of system (1) are extinct (see Figures 1 and 2).
(a)
(b)
(a)
(b)
Choose . Then from Lemma 5, we obtain that . From Theorem 9 we have that the infected citrus population and the infected psyllid population of System (1) are permanence (see Figures 3 and 4).
(a)
(b)
(a)
(b)
From the formulae for , we can predict the general tendency of the epidemic in a long term according to the current situation, which is presented in Figures 1, 2, 3, and 4. From the first two figures we know that the epidemic of Huanglongbing can be rising in a short time but cannot be outbreak with the current prevention and control measures. From Figures 3 and 4, we can see that the epidemic of Huanglongbing dropped heavily after 100 months, while there is still tendency to a stable periodic solution in a long time.
Next, we perform some sensitivity analysis to determine the influence on the parameters , and .
We choose function , where , denote the average and amplitude values of , respectively, and . From Figure 5, we can observe that the blue line is linear relation between and , and increases as increases. The red curve reflects the influence of the average value of on . Figure 5 shows that is more sensitive than on the basic reproduction number . Therefore, in the real world, decreasing the average recruitment rate of citrus is the valuable way to control Huanglongbing.
Now, we consider the combined influence of and on . Set , and . Moreover, we know that and . Other parameters can be seen in Table 2.
Case (I). We fix and , and let vary from 0.00001 to 0.015 and from 0.00001 to 0.02. For this case, it is interesting to examine how the average values of adequate contact rate and affect the basic reproduction number . Numerical results shown in Figure 6 imply that the basic reproduction number may be less than 1 when or is small enough. And the results also imply that increases as and increase. Further, we can observe that from Figure 6(i)the smaller the values of or are, the more sensitive is; (ii) increasing may be more sensitive for when is fixed; (iii) increasing may be more sensitive for when is fixed.
Case (II). We fix and , and let vary from 0.000001 to 0.005 and from 0.000002 to 0.006. Then we obtain the result of numerical simulation and it is shown in Figure 7. Obviously, Figure 7 shows that is linearly related to both and with the pattern that decreases to a relatively small value (less than 1) only when and are very small.
By the above graphs of the basic reproduction number on the average values of recruitment rate of citrus and adequate contact rate , , we know that the basic reproduction number is a monotonic increasing function by the average values. From the sensitivity analysis diagrams, we observe that falls to less than 1 by decreasing the values of those parameters.
6. Conclusion
In this paper, we have analyzed a HLB transmission model with Logistic growth in periodic environments. It is proved that is the threshold for distinguishing the disease extinction or permanence. The diseasefree periodic solution is globally asymptotically stable and the disease dies out when . When , the disease persists.
The numerical simulations shown in Figure 5 show that there are some parameter ranges of and such that the threshold parameter is smaller than 1. It indicates a useful way to eradicate Huanglongbing by limiting the recruitment of citrus, including the average value and amplitude of recruitment function.
The results shown in Figure 6 (Figure 7) show that if the amplitudes of infected functions , (the average infected rate , ) are fixed, we can control the infection of citrus and psyllid by limiting the average infected rates , (the amplitudes of infected functions , ).
According to the above theoretical analysis and numerical simulations, we can conclude that the recruitment of citrus and the infection of citrus and psyllid have significant effects on Huanglongbing transmission. In order to prevent the epidemic disease from generating endemic, making an appropriate reduction of the recruitment rate of citrus and decreasing the contact rate between psyllid and the citrus are effective measures to control Huanglongbing.
Disclosure
The paper is approved by all authors for publication. The authors would like to declare that the work described was original research that has not been published previously and not under consideration for publication elsewhere.
Conflict of Interests
No conflict of interests exists in the submission of this paper.
Acknowledgments
The research has been supported by the Natural Science Foundation of China (11261004), the Natural Science Foundation of Jiangxi Province (20122BAB211010), the Science and Technology Plan Projects of Jiangxi Provincial Education Department (GJJ13646), and the Postgraduate Innovation Fund of Jiangxi Province (YC2012S121).
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Copyright © 2014 Jianping Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.