Modelling the use of impulsive vaccination to control Rift Valley Fever virus transmission
 ChenXia Yang^{1} and
 LinFei Nie^{1}Email author
https://doi.org/10.1186/s1366201608351
© Yang and Nie 2016
Received: 26 November 2015
Accepted: 8 April 2016
Published: 18 May 2016
Abstract
In this paper, we propose a vectorbone dynamical model to against the transmission of Rift Valley Fever (RVF) between ruminants and mosquitoes, where impulsive vaccination for susceptible ruminants is introduced. By using the comparison principle, integral and differential inequalities, and some of analytical skills, the threshold values for the stability of the diseasefree periodic solution and uniform persistence of disease are obtained. These values characterize the evolution and extinction of the disease. Numerical simulations are carried out to illustrate the main theoretical results and the feasibility of the impulsive vaccination control strategy.
Keywords
Rift Valley Fever impulsive vaccination extinction asymptotical stability uniform persistenceMSC
34A37 34D23 92D301 Introduction
Infectious diseases have great impact on human and animals life, which are always aroused by viruses, parasites, bacteria, fungi, and some microorganisms with pathogen. These diseases spread to human (or animals) with directly or indirectly ways. There are significant data showing that many infectious diseases spread with mediums, such as Measles, Dengue fever, Plague, Hydrophobia, and so on. Rift Valley Fever virus (RVFV) is an important mosquitoborne viral zoonsis in North Africa and Kenya, which is a member of the Phlebovirus genus in the Bunyaviridae family. RVFV was first discovered in Kenya in the early 1930s [1]. RVFV is spread by either touching infected animal blood, breathing in the air around an infected animal being butchered, drinking raw milk from an infected animal, or the bite of infected mosquitoes. Animals such as cows, sheep, goats, and camels may be affected [2, 3].
Using mathematical models to investigate the transmission rules of RVFV disease is beneficial to control diseases in medical science, so the dynamic behaviors of the diseases are investigated in many literatures. We refer to some of them, [4–8] and the references therein. Particularly, Gaff et al. [9] investigated an epidemiological model for RVFV and found the stability of the diseasefree equilibrium. Saul et al. [10] investigated the dynamical behavior of RVFV with human host and computed the disease threshold \(\mathcal{R}_{0}\) for analyzing the local stability of equilibria. Xue et al. [11] presented a new compartmentalized model for RVFV and used ordinary differential equations to assess disease spread in both time and space. With the latter driven as a function of contact networks, Gao et al. [12] investigated the spatial spread of RVFV and proposed a threepatch model, introduced the basic reproduction number for each patch, and then established the threshold dynamics of the model. Of course, mathematical models also used to discuss the transmission rules of other vectorborne disease [13–15] and the references therein.
Recently, vaccines are generally used to protect animals from Rift Valley Fever in endemic regions. Right now, two types of vaccines are available for susceptible animals to reduce amplification of the virus: inactivated wholevirus and liveattenuated Smithburn vaccines [16]. Inactivated vaccines can be applied to ruminants of all ages without causing abortions, but they are expensive, and repeated doses are required. In comparison, liveattenuated vaccines are cheap and effective since they confer a lifelong immunity with a single dose. But the undesirable side effects are obvious: they may lead to fetal abnormalities and abortions in pregnant ruminants, and there is the safety concern of reversion to virulence [17]. Based on this, Farida et al. [18] used the mathematical model to investigate RVFV among ruminants and considered continuous vaccination for animals and found that vaccination was efficient to reduce the loss of animals. We know, however, that the continuous vaccination is not practical in the reality living, and the control measures are only employed in the particular moment. So for this reason, infectious models with impulsive vaccination are concerned by more and more scholars. For example, Sabin [19] controlled successfully measles and poliomyelitis throughout Central and South America, and another example of successful application of this strategy is the U.K. vaccination action against measles in [20]. Using a SIR epidemic model, Shulgin et al. [21] showed that under a planned pulse vaccination region, the sate of model converges to a stable sate with which the size of infectious population is zero. This result shows that the pulse vaccination may lead to the eradication of infectious disease, provided that the magnitude of vaccination keeps a rational proportion and the period of pulses is sustained. d’Onofrio [22] proposed a SEIR epidemic model based pulse vaccination strategy by which the local and global asymptotic stabilities of the periodic eradication solution are analyzed. More related research works on the dynamical behaviors of epidemic models with pulse vaccination also can be found in [23–27] and the references therein.
Looking at the results of existing researches on epidemic dynamical models with impulsive vaccination and using mathematical model to investigate vectorborne epidemic model, the dynamic behavior with impulsive vaccination strategy is not investigated. Therefore, we propose a novel dynamical model to control the transmission of RVFV, where impulsive vaccinate for susceptible ruminant at regular interval is proposed. The main purpose is to investigate the impulsive vaccination control strategy that governs whether the disease dies out or not, and further to examine how the control strategy affects the prevention and control of RVFV disease. The organization of this paper is as follows. We present preliminaries and formulate the control model of RVF with impulsive vaccination in the next section. In Section 3, we consider the global stability of the freedisease periodic solution and give a threshold value for the diseaseeliminating. In Section 4, we discuss the uniform persistence of the disease. Numerical simulation and discussion are carried in Section 5.
2 Model formulation and preliminaries
 \((A_{1})\) :

The ruminant population is recruited with rate Λ, μ is the natural death rate (including slaughter) of ruminant, with RVFV owing to infection with RVFV also causing mortality in ruminants. We assume that ruminants die due to RVFV infection at rate d (including slaughter) and do not consider infection with RVFV causing abortion in ruminant.
 \((A_{2})\) :

An infectious mosquito bites ruminant, and ruminant is infected by infectious mosquito successfully by rate \(p_{r}\); in turn, an infectious mosquito bites infectious ruminant and is infected by infectious ruminant successfully with rate \(p_{m}\), and we assume that each female mosquito bites at a constant rate a.
 \((A_{3})\) :

Assume that ξ is the growth rate of mosquitoes and \(M_{0}\) is the capacity for mosquitoes. Infection with RVFV induces lifelong immunity in ruminants at a rate of γ, and we also assume that η is the natural birth/death rate of mosquitoes. Moreover, in this paper, we do not take into account vertical transmission of RVFV in mosquitoes since vertical transmission is rare.
 \((A_{4})\) :

With the consideration of mechanism of prevention and control for the spread of RVFV, we consider impulsive vaccination, and only susceptible ruminants are vaccinated at rate ϕ. Assuming that after successful vaccination, ruminants are completely immune to the virus and move to the recovery compartment with immunity.

\(t\neq nT\), \(n=1,2,\ldots\)

\(t=nT\), \(n=1,2,\ldots\)
On the nonnegative of solutions for model (3) we have the following lemma.
Lemma 1
Each component of any solution of model (3) is nonnegative, and Ω is a positively invariant.
The proof of Lemma 1 is obvious, and hence we omit it here.
The following Lemma 2 is on the existent and stability of positive periodic solution for equation (4). Though its proof is straightforward, but useful.
Lemma 2
Finally, for the convenience of further statements, we introduce the definition on the uniform persistence of disease.
Definition 1
The disease in model (3) is said to be uniform persistent if there exists a positive constant \(m^{*}\) such that \(\liminf_{t\to\infty}I(t)\geq m^{*}\) and \(\liminf_{t\to\infty}V(t)\geq m^{*}\).
3 The existence and stability of the diseasefree periodic solution
Theorem 1
If \(\mathcal{R}_{0}<1\), then model (3) has a unique diseasefree periodic solution \((S_{p}(t),0, R_{p}(t),U_{p}(t),0)\), which is globally asymptotically stable.
Proof
4 The uniform persistence of RVFV
The following theorem is on the uniform persistence of disease.
Theorem 3
Proof
 (i)
\(F(\tilde{p},t)>0\) for all \(t\geq n_{4}T\);
 (ii)
\(F(\tilde{p},t)\) oscillates about 0 for all large t.
Using Corollary 1 given by Teng et al. in [30] on the existence of positive periodic solutions for the general impulsive ordinary differential equation, we have the following theorem on the existence of positive periodic solutions for periodic impulsive model (3).
5 Numerical simulation and discussion
In this paper, we consider an epidemic model of RVFV with impulsive vaccination strategy. The main purpose is to investigate the impulsive vaccination thats governs whether the RVFV disease dies out or not and further to examine how the impulsive vaccination control strategy affects the prevention and control of RVFV disease. By using the comparison principle, integral and differential inequalities, and analytical methods, some sufficient conditions for the existence and stability of diseasefree periodic solution and for uniform persistence of disease are obtained. Theoretical results show that RVFV disease can be controlled through changing the control parameters of model based on these conditions.
Lists of parameters for Rift Valley Fever virus transmission
Symbol  Description  Range  Source 

Λ  Recruitment rate of ruminant  25,00046,000  Estimate 
\(M_{0}\)  Capacity for mosquitoes  −  Estimate 
a  Female mosquito bites rate (year^{−1})  156256  Farida [18] 
\(p_{m}\)  Probability of successful infection in mosquitoes  (0.0021, 0.2762)  Saul [10] 
\(p_{r}\)  Probability of successful infection in ruminants  (0.0021, 0.2429)  Saul [10] 
d  Death rate of ruminant due to RVFV (year^{−1})  9.12536.5  Saul [10] 
μ  Natural death rate of ruminant (year^{−1})  0.10141.0139  Saul [10] 
γ  Rate of recovery in ruminant (year^{−1})  73365  Saul [10] 
η  Death rate of mosquitoes (year^{−1})  6.08121.6  Saul [10] 
ξ  Development time of mosquitoes (year^{−1})  24.373  Saul [10] 
ϕ  Vaccinated rate for susceptible ruminant (year^{−1})  01  Estimate 
Declarations
Acknowledgements
This research has been partially supported by the National Natural Science Foundation of China (Grant Nos. 11461067, 11402223, and 11271312). The authors are very grateful to the editor and the anonymous referees for their valuable comments and suggestions, which greatly improved the presentation of this work.
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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