Abstract
This paper investigates a three-patch Rift Valley fever virus transmission model with periodic coefficients. The basic reproduction number is calculated for the model by using the next infection operator method. The threshold conditions on the extinction and permanence in the mean of the disease with probability one. The Rift Valley fever disease will be permanent in the -th patch if and dies out in the -th patch if . The numerical simulations are given to confirm the theoretical results.
1. Introduction
Rift Valley fever (RVF) is an acute viral zoonosis caused by the Rift Valley fever virus (RVF virus). This disease is primarily transmitted among domestic animals, such as sheep, cattle, buffalo, goat, and camels, with the main route of infection being mosquito bites (see [1–4]). RVF virus infection in livestock animals can produce high rate of abortion in pregnant animals and significant morbidity and mortality rate in young ruminants. During the disease epidemic among animals, humans may be infected with Rift Valley fever virus through bites from infected mosquitoes or contacts with the blood, body fluids, and organs of infected animals (see [5, 6]).
In the 1930s, RVF virus was discovered for the first time near Naivasha Lake in the Rift Valley of Kenya. In 1977, a human RVF virus epidemic is reported in Egypt, which is the largest epidemic. Subsequently, RVF virus outbreaks have been reported in Saharan and North Africa. After that, this disease occurred in Saudi Arabia and Yemen outside the Africa in the 2000s. These disease outbreaks result in high mortality and abortion in young ruminants and case significant economic losses in Africa and Middle East. With the enlargement of scope of RVF virus infection, there is a growing concern that this disease will outbreak further in other parts of Asia and Europe. In recent years, numerous mathematical models have been developed to investigate RVF virus outbreaks (see [1, 3, 6–15]).
Research results show that, in Northeastern Africa, human activities, including those associated with the Eid al-Adha feast, along with a combination of climatic factors such as rainfall level and hydrological variations, contribute to the transmission and dispersal of the disease pathogen. Moreover, sporadic outbreaks may occur when the two events occur together: (1) abundant livestock is recruited into areas at risk from RVF due to the demand for the religious festival and (2) abundant numbers of mosquitoes emerge. In [8, 14, 15], the authors consider the religious festival and discuss three-patch model for RVF virus transmission.
On the one side, a religious festival, the Eid al-Adha feast, at which time large numbers of livestock are driven towards the site of the feast. In Africa and the Middle East, the invasion path follows the same route that people use to travel to the Nile Delta, where the important Islamic festival Greater Bairam is held. Geographically, the disease seems to be along a path from southwest to northeast of Africa. This is consistent with the way to Mecca, Saudi Arabia’s capital (see [15]).
On the other side, the number of mosquitoes is affected by temperature and humidity, the greater the humidity, the more the number of mosquitoes, the humidity at about thirty degrees is the most suitable for mosquitoes to grow and propagate, and lower than ten degrees will enter hibernation. Due to rainfall and temperature, in fact, the mosquito population densities follow seasonal fluctuations, achieving maximum numbers during the rainy season and reaching lowest level during the winter season (see [4, 7, 8]). The abundant numbers of mosquitoes corresponds to an increased number of infected ruminants. Consequently, RVF virus infections fluctuate over time and highly link to seasonal variations.
The autonomous models have been discussed in [3, 11–15]. At the same time, we can see that Rift Valley fever will change periodically with the religious festivals and mosquitoes’ periodicity. Recently, in [8], Xiao et al. proposed a three-patch periodic model for RVF virus transmission with the periodic recruitment rate of livestock in first patch, the periodic carrying capacities for mosquitoes in three patches, the periodic natural death rate for livestock in the last patch, and the periodic migration rate. In order to seek the seasonal and festival-driven impacts for RVF outbreaks, the authors in [8] investigated the dynamical behaviors of the model just by using the numerical simulation method.
The main purpose of this paper is to discuss the threshold dynamics of a periodic three-patch Rift Valley fever virus transmission model, and thereout, we will make up the deficiency for the research given in [8]. By applying the methods in given [16–21], we calculate basic reproduction number in -patch of model By using the theory of persistence for dynamical systems, we will establish that serves as a threshold value for Rift Valley fever virus dynamics in periodic case.
The organization of this paper is as follows. In the next section, we develop a periodic three-patch model to describe the transmission of Rift Valley fever virus and present some useful lemmas. In Section 3, we will propose the main results, namely, we will obtain the extinction or permanence of all positive solutions of model . In Section 4, we will give the numerical simulations to verify our theoretical results.
2. Model
We investigated the festival-driven and seasonal impacts on the patterns of RVF outbreaks among livestock in Africa and Middle East.
We assume that a disease invades and subdivides the target livestock into four classes: the susceptible class , the exposed class , the infectious class , and the recovered class ; the female mosquitoes are divided into three compartments: uninfected compartment , exposed compartment , and infected compartment in each patch.
We assume that is the number of livestock imported daily at time in 1-patch, is the transmission rate of disease from mosquitoes to livestock at time in -patch, is the transmission rate of disease from livestock to mosquitoes at time in -patch, is the time-dependent natural death rate for livestock in -patch (average death rate for different livestock, i.e., cattle and sheep), is the time-dependent natural death rate for female mosquitoes in -patch, is the rate at which immunity is lost after recovery for livestock at time in -patch, is the rate of becoming infectious for livestock at time in -patch, is the rate of becoming infectious for mosquitoes at time in -patch, is the recovery rate of leaving the infectious class for livestock at time in -patch, is the time-dependent birth rate for mosquitoes in -patch, is the time-dependent carrying capacity of mosquitoes in -patch, is the disease-induced death rate for livestock at time in -patch, is moment speed of animals from -patch to -patch, and is the length of journey for animals within patch . Motivated by the above assumption, we propose the following periodic three-patch RVF virus transmission model. where .
For any continuous and bounded function defined for , we denote and .
Based on the biological background of model , the initial condition for model (for any ) has in the following form:
For model , we give assumptions as follows:
functions and are continuous and -periodic defined on . Furthermore, functions , , , , , , , and are positive, and functions , , , and are nonnegative.
and for all .
Consider a -periodic linear equation as follows: where and are -periodic continuous functions defined for . We have the following lemma.
Lemma 1 (see [17]). Assume that and for all . Then, equation (3) has a unique positive -periodic solution which is globally attractive.
Furthermore, we consider a -periodic logistic equation as follows: where and are -periodic continuous functions defined for . We have the following lemma.
Lemma 2 (see [18]). Assume that and for all . Then, equation (4) has a unique positive -periodic solution which is globally attractive.
When and , we obtain the disease-free subsystem of model :
As consequence of Lemma 1 and Lemma 2, we have the result on the existence of disease-free periodic solution of model .
Lemma 3. System (5) has a unique positive -periodic solution which is globally attractive.
From Lemma 3, model has a unique disease-free periodic solution .
Let be a continuous, cooperative, irreducible, and -periodic matrix function; we consider a linear system:
Let be the fundamental solution matrix of system (6) satisfying initial condition , where is an unit matrix. Let be the spectral radius of matrix . Since is continuous, cooperative, and irreducible, we obtain that is nonnegative for all . By Perron-Frobenius theorem, is the principal eigenvalue of in the sense that it is simple and admits a positive eigenvector . We present the following lemma.
Lemma 4 (see [19]). Let . Then, there exists a positive -periodic function such that is a solution of system (6).
Now, we calculate the basic reproduction number of model by applying the next-generation matrix approach which is given in [20]. Firstly, we easily validate that model satisfies the conditions given in [20]. Let
Let with be the evolution operator of linear -periodic system:
Then for any and and , where is the unit matrix. Let be the ordered Banach space of all periodic continuous functions . Assume that is the initial distribution of infectious individuals. Then, is the rate at which new infections are produced by infected individuals who were introduced into the population at time . When , then represents the distribution of those infected individuals who were newly infected individuals at time and remain in the infected compartments at time . Hence, the cumulative distribution of new infections at time produced by all those infected individuals introduced before is given by a linear operator as follows:
Applying the results obtained in [20], the basic reproduction number for model is defined as spectral radius of operator , that is, .
Furthermore, using Theorem 2.2 given in [20], we also can obtain the following results on basic reproduction number .
Lemma 5 (see [20]). The following conclusions hold. (a) if and only if (b) if and only if (c) if and only if (d)Disease-free periodic solution for model is locally asymptotically stable if and unstable if
Remark 1. In particular, when periodic systems (1) degenerate into autonomous systems, by the next-generation matrix method (see [22]), we can obtain where and . Then,
3. Main Results
For model , we investigate the persistence and extinction of positive solutions on the three patches.
3.1. The First Patch
In this section, we discuss model . Firstly, on the nonnegativity, positivity, and boundedness of solutions for model , we have the following result.
Theorem 1. Let be the solution of model with initial condition (2). Then, is nonnegative for all and ultimately bounded, and when and , then is also positive for all .
Proof 1. In fact, by the continuous dependence of solutions with respect to initial values, we only need to prove that when and , then is positive for all . From the fifth equation of model , we have for any . Since , we get
Define . Obviously, . We only need to prove for all . Suppose that there exists a such that and for all . Then, we only need to discuss the following six cases: (1) , (2) , (3) , (4) , (5) , and (6) .
Now, we only give the proof for case (2). The remaining cases can be proved in a similar manner. Let . Since for all , then , , and hence,
Integrating from to , we get which leads to a contradiction.
Next, we show the boundedness of nonnegative solutions of model . Let , then we get
Hence, we have which implies that and are ultimately bounded. Since for any , using Lemma 3; we further have Hence, for any , there is a such that and for all , where . From the sixth equation of model , we have for all , where . Hence, we can obtain From this, for any , there is a such that for all . Hence, from the last equation of model , we further have for all , where . This shows that . From above discussions, we finally obtain that all solutions of model with initial condition (2) are ultimately bounded. This completes the proof.
Now, we show that serves as a threshold parameter. We prove that when , then disease-free periodic solution is globally asymptotically stable, that is to say, the RVF virus is extinct in the first patch, and when , then model is permanent, that is to say, the RVF virus is existent in the population in the first patch.
Theorem 2. If , then disease-free periodic solution of model is globally asymptotically stable, and if , then is unstable.
Proof 2. From conclusion of Lemma 5, it follows that if , then is unstable, and if , then is locally asymptotically stable.
We now prove that if , then is globally attractive. Let , from conclusion of Lemma 5, then . We can choose constant small enough such that , where
From Lemma 3, for above given , there exists a such that and for any . Hence, from model , we obtain that for all
Consider the following auxiliary system:
It follows from Lemma 4 that there is a positive -periodic function such that is a solution of system (23), where .
Denote We can choose constant such that According to the comparison theorem of the vector form (see [23]), we obtain for any . By , we know Then, we conclude that , that is to say, . By the equations of , , and of model , we also obtain . Hence, disease-free periodic solution is globally attractive. This shows that disease-free periodic solution is globally asymptotically stable. This completes the proof.
Theorem 3. If , then the disease in model is permanent.
Proof 3. From conclusion of Lemma 5, we know that if and only if . We can choose a small enough constant such that , where is given in (20). From and , we can choose small enough such that for all . Consider the following auxiliary equations: and from Lemma 1 and Lemma 2 we can get equations (24) and (25) admit globally uniformly attractive positive -periodic solution and , respectively. By the continuity of solutions with respect to parameter , for above , there exists a such that for all Define the sets as follows:
Define further the Poincaré map as follows: where is the solution of model satisfying initial condition . It follows from Theorem 1 that all solutions of model are ultimately bounded. Therefore, map is point dissipative and compact on .
We define where and for . Now, we claim In fact, from model , we directly obtain that the solution of model with initial value has the form . Hence, for all . This shows . Therefore, .
Next, we prove . Suppose that there is a such that .
We firstly have . Let be the solution of model with initial value at . From Theorem 1, we firstly have that is nonnegative for all . Furthermore, by and , from the first and fifth equations of model , we also have that and for all , respectively.
If , then by the second equation of model , we obtain
Integrating this inequality from to , we have
Further, from the third equation of model we can obtain
From the sixth equation of model , we get
Hence,
Lastly, from the equation of for model , we also have for all . Therefore, we finally obtain for all .
Similarly, if or or , we also can obtain for all . This shows that , which leads to a contradiction. This implies that . Hence, .
We easily prove that system (36) has a globally asymptotically stable -periodic solution , where is given in Lemma 3. It is clear that map has a unique globally attractive fixed point restricted in , which is .
Now, we define , which is said to be stable set of . We show that
According to the continuity of solutions with respect to the initial values, for above given constant , there exists such that for any with , it follows that for all where is the solution of model with initial condition and . Since is -periodic, we have for any with , where is an integer. We claim that
Suppose that (38) does not hold. Then, we have
Without loss of generality, we assume that for all . Thus,
For any , let , where and are the greatest integers less than or equal to . Then, we have
It follows that , , , and for all . Thus, from model , we have the following: for any . Using the comparison principle, we obtain for any , where and are the solutions of equations (20) and (23) with respect to parameter satisfying initial conditions and , respectively.
From Lemma 1 and Lemma 2, equations (24) and (25) with respect to the parameter have globally uniformly attractive solutions and , respectively. Hence, there exists such that for all . By (26), (42), and (43), we obtain and for all . Thus, for all , it holds that
Consider the following auxiliary system:
System (45) can be rewritten in terms of the matrices and as follows:
It follows from Lemma 4 that there is a positive -periodic function such that is a solution of system (46), where .
Now, we denote . We can choose small enough constant such that . Thus, by comparison theorem of the vector form, it follows that for all . By , we obtain . Hence, , which implies that which leads to a contradiction. Hence, claim (25) holds. This shows .
It is clear that fixed point of is globally attractive restricting on . Hence, is acyclic in . From the theorems established in [19] on the uniform persistence of dynamical systems, we obtain that is uniformly persistent with respect to which implies that model is permanent. This completes the proof.
3.2. The Second Patch
In this section, we discuss model . First of all, we have the following result on the positivity and ultimate boundedness of solutions in model .
Theorem 4. Let be the solution of model with initial condition (2). Then, is nonnegative for all and ultimately bounded, and when and , then is also positive for all .
Proof 4. Using the same method given in Theorem 1, we can prove the positivity of solutions for model . Next, we show the boundedness of solutions for model . Let , then we get
By the comparison theorem and Lemma 1, we get that there exists constant such that , which implies that and are ultimately bounded. Since for any , then there exists constant such that . Hence, for any , there is a such that and for all . From the sixth equation of model , we have for all . Hence, we can obtain that there exists such that . From this, for any , there is a such that for all . Hence, from the last equation of model , we further have for all . This shows that there exists such that . Hence, for any , there is a such that for all .
From above discussions, we finally obtain that all solutions of model with initial condition (2) are ultimately bounded. This completes the proof.
Assume firstly . From above result, without loss of generality, we can assume that there are positive constants and such that for any solution of model and for any solution of model . Thus, from model , we further have
Consider auxiliary equations as follows:
Firstly, we consider the livestock equations of model (53) by Lemma 1, and we have
Next, we consider the mosquito system. From the fifth equation of model (53), we have
Therefore, for any , there is a such that for all ,
Considering the sixth equation of model (53), we have then we can obtain . Thus, for any , there is a such that . Considering the last equation of model (53), we have and then, we can obtain . Using comparison theorem, we can obtain the theorem as follows.
Theorem 5. If , the disease in model is permanent.
Next, we assume if , then each nonnegative solution of model converges to disease-free periodic solution , and we further obtain the limiting system of model as follows:
From a similar argument as in the above to model , we can obtain that model (59) has a unique disease-free periodic solution , where is the unique positive -periodic solution.
Furthermore, by the same way as in the above for model , we can obtain the basic reproduction number for model (59), where the operator can be defined similarly to .
It is clear that model (59) is very similar with model . Therefore, using the theory of limiting systems and the nearly same arguments as in the above, we can prove the following theorems.
Theorem 6. Let if , then disease-free periodic solution of model (59) is globally asymptotically stable for model , and if , then is unstable for model .
Theorem 7. If and , then the disease in model is permanent.
3.3. The Third Patch
In this section, we discuss model . For model , the discussions are nearly similar to model . We here directly state the following main results.
Theorem 8. Let be the solution of model with initial condition (2). Then, is nonnegative for all and ultimately bounded, and when and , then is also positive for all .
Theorem 9. If or , the disease in model is permanent.
If and , then any nonnegative solution of model converges to disease-free periodic solution of model (59). Thus, we obtain the limiting system of model as follows:
From a similar argument as in the above to model , we can obtain that model (61) has a unique disease-free periodic solution , where is the unique positive -periodic solution.
By the same way as in the above for model , we can obtain the basic reproduction number for model (61), where the operator can be defined similarly to . Thus, we further have the following results.
Theorem 10. Let and , if , then disease-free periodic solution of model (61) is globally asymptotically stable for model , and if , then is unstable for model .
Theorem 11. If and , then the disease in model is permanent.
4. Numerical Simulation
In this section, we will present some examples and the numerical simulations to confirm our theoretical results. The numerical values of most parameters in the following examples are adopted from Xiao et al. [8], Gao et al. [13], and Gaff et al. [14]. And the numerical simulation method of basic reproduction number is presented of Posny and Wang [24]. We also give our numerical simulation with reference to [25].
In the following numerical examples, we choose the initial value , , , , , , , , , , , , , , , , , , , and in model – which is given in [8].
Example 1. We assume that in model –, only the capacity of mosquitoes is periodic. Thus, we take , , , , , , , , , , , , , , , , where , and , where , , and .
It is clear that and are satisfied. By the numerical calculations, we get the basic reproduction number Therefore, from Theorems 3, 5, and 9, we obtain that the disease in model – in three patches is permanent. The numerical simulations are given in Figure 1.
Example 2. We only consider the capacity of mosquitoes, and the parameters incorporating both seasonal and festival impacts are periodic. In model –, we take in the following form: with and . We also take , , and using the same definition as in (63) with , , and . The other parameters in model – are chosen as in Example 1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
It is clear that and are satisfied. By the numerical calculations, we get the basic reproduction number Therefore, from Theorems 3, 5, and 9, we obtain that the disease in model – in three patches is permanent. The numerical simulations are given in Figure 2.
Example 3. In model –, we take , , , and , where , and are given as in Example 1, where . We also take , , , , , and using the same definition as in (63) with , , , , , , , , , and , where . We further choose and as in Example 2 and , , and as in Example 1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
It is clear that and are satisfied. By the numerical calculations, we get the basic reproduction number Therefore, from Theorems 3, 5, and 9, we obtain that the disease in model – in three patches is permanent. The numerical simulations are given in Figure 3.
Example 4. In model –, take and for . We also take , , and using the same definition as in (63) with , , and , where . The other parameters in model – are chosen as in Example 3.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
It is clear that and are satisfied. By the numerical calculations, we get the basic reproduction numbers and . Therefore, from Theorem 9, we obtain that the disease in model – dies out in the first patch and is permanent in the second and third patches. The numerical simulations are given in Figure 4.
Example 5. In model – take . We also take and using the same definition as in (63) with , , , , , and . The other parameters in model – are chosen as in Example 4.
(a)
(b)
It is clear that and are satisfied. By the numerical calculations, basic reproduction numbers are , and Therefore, from Theorem 11, we obtain that the disease in model – dies out in first and second patches and is permanent in third patch. The numerical simulations are given in Figure 5.
Example 6. In model –, take for . We also take and using the same definition as in (63) with and . The other parameters in model – are chosen as in Example 4.
(a)
(b)
It is clear that and are satisfied. By the numerical calculations, basic reproduction numbers are , , and Therefore, from Theorem 10, we obtain that the disease in model – in three patches dies out. The numerical simulations are given in Figure 6.
(a)
(b)
5. Discussion
From the analysis and simulations of the model, we found that the basic reproduction number which was calculating by using the next infection operator approach determined whether the disease was persistent or extinct.
For the model , if , the disease is persistent in the patch; if the time-dependent birth rate or the transmission rate of disease of the disease or the rate of becoming infectious for livestock and female mosquitoes decreases, the basic reproductive number , then the disease dies out. This means that importation of livestock (especially the major festivals such as Eid al-AdhA) results in more livestock and increases the chance of disease outbreaks; the transportation of livestock from one patch to the other patch increases the chance of disease outbreaks in the other path but reduces the chance of disease outbreaks in the one path. Also, as summer temperatures rise, rain is abundant, mosquitoes are increasing, and disease carriers are increasing which increases the chance of disease outbreaks. That is abundances of hosts and vectors increase the probability of large disease outbreaks.
According to our work, diseases will occur during religious festivals which hold in summer. But this is not entirely consistent with the actual situation. In the model, human activity is reflected by periodic parameters, and human compartments are not directly considered, but real-time changes in human activity have different effects on the pattern of disease. Since we assumed well-mixed populations of livestock in each patch and identical movement rates regardless the clinical stage of livestock during transportation, the model may overestimate the real situation. Implementations of various kinds of vaccination strategies and unpredicted religious festival preparations or other stochastic events will also change the pattern of disease outbreaks. These factors will also be our future research work.
We see that some basic theoretical results are established for the Rift Valley fever virus transmission model. The deficiency presented in [8] is made up. There is a shortage that in this paper we do not establish a theoretical result on the local and global stability of the model when . It will be studied in our future works.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors have declared that no competing interests exist.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11771373) and the Opening Project of Key Laboratory of Xinjiang (Grant no. 2016D03022).