The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage

https://doi.org/10.1016/j.cnsns.2008.06.024Get rights and content

Abstract

A SIS epidemic model incorporating media coverage is presented in this paper. The dynamics of this disease model under constant and pulse vaccination are analyzed. First, stability analysis of the model with constant vaccination shows that the disease free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the endemic equilibrium is globally asymptotically stable if it exists. Second, we consider the impulsive vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic infection-free solution is globally asymptotically stable under some conditions, we also show that the system is permanent. Furthermore, by bifurcation theory we obtain the existence of a positive periodic solution. In order to apply vaccination pulses frequently enough so as to eradicate the disease, the threshold for the period of pulsing, i.e., τmax is shown. Our theoretical results are confirmed by numerical simulations. The effectiveness of constant and pulse vaccination policies are compared.

Introduction

Vaccination is a commonly used method for controlling disease, the study of vaccines against infectious disease has been a boon to mankind. For example, the global eradication of smallpox was announced by the World Health Assembly in May 1980. This long-dreaded disease was defeated with a vaccination program. There are now vaccines that are effective in preventing such viral infections as rabies, yellow fever, poliovirus, hepatitis B, parotitis, and encephalitis B. Eventually, vaccines will probably prevent malaria, some forms of heart disease and cancer. Even venereal disease may someday be target of vaccination programs. Vaccines has been very important to every people.

The conventional vaccination strategies lead to epidemic eradication if the proportion of the successfully vaccinated individuals is larger than a certain critical value, which is approximately equal to 95% for measles [1], while in practice, it is both difficult and expensive to implement vaccination for such a large population coverage. We are therefore led to examine the potential of other strategies that epidemics can be more efficiently controlled, such as pulse vaccination. Theoretical results showed that pulse vaccination strategy can be distinguished from the conventional strategy in leading to disease eradication at relatively low values of vaccination [2]. Recently, pulse vaccination has gained prominent achievement as a result of its highly successful application in the control of poliomyelitis and measles throughout Central and South America [3], [4]. Another example of successful application of this strategy is the UK vaccination campaign against measles in 1994 [5]. Theories of impulsive differential equations can be found in the books [6], [7]. In recent years, their applications can be found in the domain of applied sciences [8], [9], [10], [11]. In this paper, we consider the constant vaccination to newborns and susceptible individuals and the impulsive vaccination to susceptible individuals, respectively.

In the real life, many infectious diseases transmit through both horizontal and vertical modes. These include such human diseases as rubella, herpes simplex, hepatitis B, and AIDS. For human and animal disease, horizontal transmission typically occurs through direct or indirect physical contact with infectious hosts, or through disease vectors such as mosquitos, ticks, or other biting insects. Vertical transmission can be accomplished through transplacental transfer of disease agents, literatures [12], [13], [14], [15], [16] considered this phenomenon. In our paper, we assume a fraction of the offsprings of infected hosts are infected at birth, hence the infected birth flux will enter class I, the vaccines treatment is only taken to a proportional new-borns who have not been infected at birth.

In the classical epidemic models, the incidence rate is assumed to be mass action incidence with bilinear interactions given by βSI, where β is the probability of transmission per contact, and S and I represent the susceptible and infected populations, respectively. However, some factors such as media coverage, manner of life and density of population, may affect the incidence rate directly or indirectly, nonlinear incidence rate can be approximated by a variety of forms, such as βIpSq,β(1-cI)IS(c>0),kIlS1+αIh(k,l,α,h>0) which were discussed by Liu et al. [17], [19] and Yorke and London [18], respectively. In this paper, we suggest a general nonlinear incidence rate β1-β2Im+ISI(β1>β2>0,m>0) which reflects some characters of media coverage, where β1=pc1,β2=pc2,p is the transmission probability under contacts in unit time, c1 is the usual contact rate, c2 is the maximum reduced contact rate through actual media coverage, that is, β1 is the usual valid contact rate, β2 is the maximum reduced valid contact rate through actual media coverage, m is the rate of the reflection on the disease. Again, media coverage cannot totally interrupt disease transmission, so we have β1>β2. We use β2Im+I reflect the amount of contact rate reduced through media coverage. When infectious individuals appear in a region, people reduce their contact with others to avoid being infected, and the more infectious individuals being reported the less contact with others, hence, we take the above form. Few studies have appeared on this aspect [20], [22].

The paper is arranged like this. In Section 2, a SIS model incorporating media coverage and constant vaccination to newborns and susceptible individuals strategy are considered, we calculate the basic reproduction number R0, and investigate local stability of the disease free equilibrium, we also analyze the existence of the endemic equilibrium (see Fig. 1) and discuss the global stability of the disease free equilibrium and endemic equilibrium. In Section 3, pulse vaccination strategy is considered, which is applied every τ period to the susceptible. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic infection-free solution of the impulsive vaccination system which is shown in Fig. 2. Furthermore, we prove the infection-free solution is locally asymptotically stable if R¯0<1, moreover, if λ-θ-(1-α)pb0, it is globally asymptotically stable. We also show that the system is uniformly permanent if R¯0>1 (see Fig. 3), and obtain the existence of the positive periodic solution by bifurcation theory. By comparing constant vaccination with impulsive vaccination, we know that the impulsive vaccination strategy is more natural and easier to be manipulated, and get the threshold for the period of pulsing, i.e., τmax. In the last section, we give a brief discussion of our results and further numerical simulation.

Section snippets

Model formulation

We study a population which is composed of there groups of individuals who are susceptible, infected, and vaccinated, with sizes denoted by S(t),I(t) and V(t), respectively. The sum S(t)+I(t)+V(t) is total population N(t). The natural birth and death rate are assumed to be identical and denoted by b>0. We assume that a fraction p(0p1) of the offspring from the infectious class are born into the infected I, the offspring from the susceptible and vaccinated classes are all susceptible

Impulsive vaccination

Pulse vaccination can be defined as the repeated application of vaccine across an age range. Assume the pulse scheme proposes to vaccinate a fraction q of the entire susceptible population in a single pulse, applied every τ period (τ>0). So the expression qS in system (2.1) is set to zero and is presented in pulses, we have the following impulsive differential equations:S˙=(1-α)(bN-pbI)+λI+θV-β1-β2Im+ISIN-bS,I˙=pbI+β1-β2Im+ISIN-(b+λ)I,V˙=α(bN-pbI)-(b+θ)V,tnτ,S(nτ+)=(1-q)S(nτ-),I(nτ+)=I(nτ-),V(n

Discussion

In many epidemic models, the incidence rate is assume to be mass action incidence with bilinear interactions and other incidence, but did not include an important factor, that is, when infectious individuals appear in a region, media sometimes will report this, people will reduce their contact with infectious individuals to avoid being infected, and the more infectious individuals being reported the less contact with others, this will effect behaviors of system, so in this paper we suggest a

Acknowledgement

This work is supported by the National Natural Science Foundation of China (No. 10771104).

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