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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 236352, 15 pages
http://dx.doi.org/10.1155/2012/236352
Research Article

A Dengue Vaccination Model for Immigrants in a Two-Age-Class Population

1Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia
2Department of Mathematics, Universitas Padjadjaran, Jatinangor 45363, Indonesia
3Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia

Received 18 October 2011; Revised 2 January 2012; Accepted 17 February 2012

Academic Editor: A. Zayed

Copyright © 2012 Hengki Tasman 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.

Abstract

We develop a model of dengue transmission with some vaccination programs for immigrants. We classify the host population into child and adult classes, in regards to age structure, and into susceptible, infected and recovered compartments, in regards to disease status. Since migration plays important role in disease transmission, we include immigration and emigration factors into the model which are distributed in each compartment. Meanwhile, the vector population is divided into susceptible, exposed, and infectious compartments. In the case when there is no incoming infected immigrant, we obtain the basic reproduction ratio as a threshold parameter for existence and stability of disease-free and endemic equilibria. Meanwhile, in the case when there are some incoming infected immigrants, we obtain only endemic equilibrium. This indicates that screening for the immigrants is important to ensure the effectiveness of the disease control.

1. Introduction

Dengue fever is an endemic disease in many tropical countries, especially in the urban areas. This disease is caused by the dengue virus, which is transmitted to a human by the bite of infected female Aedes aegypti mosquitoes.

There are some epidemiological and demographical factors that contribute to the transmission of the disease. Age factor is among the important demographical factors affecting the transmission of the disease. From a theoretical point of view, age structure affects the dynamics of the disease transmission [1], and hence it should be taken into account in modeling the transmission of the disease to increase the realism of the model and to obtain a more prudent decision derived from the model. From a practical point of view, many vaccination programs are directed to a certain class of age, unexceptionally in the case of dengue in which the Pediatric Dengue Vaccine Initiative targets children in their vaccination program (http://www.pdvi.org/). A study in [2] shows that a pediatric vaccination would be economically viable and highly cost effective, once a perfect dengue vaccine is made. A similar study shows that an optimal vaccination strategy could be given to only certain classes of age [3].

In literatures, most of the age-structured population models appear in the form of integropartial differential equations [46]. Some authors included age structure in epidemic models in the form of discrete compartmental differential equations, such as in [79]. The authors in [7] have generalized the model in [10] by separating the human population into age cohorts, and then for each cohort they construct a set of SIR equations. Disease-free and endemic equilibria are found, but there is no stability analysis for these equilibria. In [8], the authors have simplified their model to a two-age-class model. They allowed different transmission rates for the adult and the child classes and found disease-free and endemic equilibria. They also provided the condition for the local stability of the disease-free equilibrium in the general case. The stability condition for the endemic equilibrium has only been found for the special case, in which no infection occurs for the adult class.

The authors in [9] showed that a two-age-class model is a special case of a more general continuous age model for a certain choice of survival function. In their paper they discussed a two-age-class dengue transmission model by dividing the human population into child and adult classes and considered vaccination in the child class only. Many scientists believe that most dengue infections are asymptomatic. For every ten cases we see in the hospital, there should be at least 50–90 cases in the community who have only fever and no complications [11]. In this regards, the authors in [9] also showed that, in some circumstances, if there is an inadvertent vaccination to asymptomatic infectious children, which worsens their condition as the time span of being infectious increases, then paradoxically, vaccination can be counterproductive; that is, vaccination makes the basic reproduction number even bigger. This suggests that, in practice, screening to identify truly susceptibles is needed before implementing a vaccination program.

Beside age factor, another factor that plays important role in disease transmission is immigration. It is easy to understand that immigration of infectious individuals could ignite the spreading of a disease in a virgin populations. Diseases like HIV, SARS, and avian influenza are believed among the examples of diseases that might be caused by the immigrants of infectious individuals [12, 13]. Many mathematical models have been devised as the means to understand and to control those kinds of diseases [6, 14, 15]. The authors in [14] showed that if there is a constant influx of infective immigrants into a population, there will be no disease-free equilibrium.

Although the immigrants are not carrying a disease at all, still they have an impact on the transmission of a disease. The buildup of immigrants (also the locals) can be viewed as the buildup of susceptibles that are ready to be infected by any disease once available or enhance the spreading of the existing disease. In this respect, it is reasonable to enforce a policy to vaccinate incoming immigrant, following a screening, to ensure that they will not contribute to the buildup of susceptible.

There is no commercially dengue vaccine available yet. However, there are some potential dengue vaccines available. A survey in four South-Eastern Asian countries in 2002 revealed that there is a high and urgent perceived need for a dengue vaccine (http://www.pdvi.org/). To simulate vaccination program in gaining some insight on how vaccination would affect the transmission of the disease, even before the vaccine itself is available in the market, is among the interests of vaccine scientists and policy makers. In this paper we develop a two-age-class model for dengue transmission by considering immigration vaccination strategy, as an anticipative study before the vaccine exists.

The introduction of immigration into the system is plausible since dengue is regarded as an urban disease [16], where the rate of immigration cannot be neglected. Different from [9] in which it is assumed that vaccination targets individuals in the child class, here we look at a scenario where vaccination is given to a portion of newborns (both immigrant and local babies) and a portion of newly arrived mature immigrants, to protect them from being infected by the local dengue disease. In practical point of view, the vaccination strategy proposed in this paper is easier to be implemented than the one in [9].

2. Model Formulation

Let us assume that the host population is classified into the child class and the adult class. Each of the classes is divided into the susceptible, infected, and recovered subclasses. We also assume that the recovered hosts have life-long immunity and there is no wanning effect of the vaccine, which means that the vaccine has a life-long permanent protection. So, the recovered hosts and the vaccinated hosts can be grouped into the recovered class.

We use variables 𝑆𝐶, 𝐼𝐶, and 𝑅𝐶 to denote the size of the susceptible, infected, and recovered of child population, respectively. Similarly, we use the subscript 𝐴 for the adult population.

We denote the susceptible, exposed, and infected vector populations by 𝑆𝑉, 𝐸𝑉, and 𝐼𝑉, respectively. We consider the latent class 𝐸𝑉, since the incubation period of the disease in mosquitoes is relatively large compared to the life span of the mosquitoes.

We use the diagram in Figure 1 for the dengue transmission in the population. The parameters 𝑃𝐶 and 𝑃𝐴 are the incoming immigration recruitment rates for child and adult classes, respectively, some positive fractions 𝑓, 𝑔, and of the incoming immigrants are susceptible, infected, and recovered or vaccinated, respectively (𝑓+𝑔+=1). In practice, it is necessary to undertake screening to identify the susceptibility status of the incoming immigrants. There is also a constant birth recruitment rate 𝐵 that increases the child population.

236352.fig.001
Figure 1: A two-age-class dengue transmission diagram.

The parameters 𝑝 and 𝑞 are the fractions of susceptible incoming children (including natural birth) and susceptible incoming adults that are vaccinated; 𝑠 is the vaccine efficacy; 𝜇𝐶, 𝜇𝐴, and 𝜇𝑉 are the child, adult, and vector natural death rates; respectively, 𝜀𝐶 and 𝜀𝐴 are the per capita emigration rates for children and adults, respectively; 𝜆𝐶, 𝜆𝐴, and 𝜆𝑉 are the successful infection rates for children, adults, and vectors; respectively, 𝛿 is the transition rate from child class to adult class; 𝛾 is the recovery rate, 𝑃𝑉 and 1/𝜏 are the recruitment rate for vector and the latent period of vectors, respectively.

Using the transmission diagram in Figure 1, we formulate the following 9-dimensional model:𝑑𝑆𝐶𝑓𝑑𝑡=(1𝑝𝑠)𝐶𝑃𝐶+𝐵𝜆𝐶𝑆𝐶𝑁𝐻𝐼𝑉𝛿+𝜀𝐶+𝜇𝐶𝑆𝐶,𝑑𝐼(2.1)𝐶𝑑𝑡=𝑔𝐶𝑃𝐶+𝜆𝐶𝑆𝐶𝑁𝐻𝐼𝑉𝛿+𝛾+𝜀𝐶+𝜇𝐶𝐼𝐶𝑑𝑅,(2.2)𝐶𝑑𝑡=𝐶𝑃𝐶𝑓+𝑝𝑠𝐶𝑃𝐶𝐼+𝐵+𝛾𝐶𝛿+𝜀𝐶+𝜇𝐶𝑅𝐶𝑑𝑆,(2.3)𝐴𝑑𝑡=(1𝑞𝑠)𝑓𝐴𝑃𝐴𝑆+𝛿𝐶𝜆𝐴𝑆𝐴𝑁𝐻𝐼𝑉𝜀𝐴+𝜇𝐴𝑆𝐴𝑑𝐼,(2.4)𝐴𝑑𝑡=𝑔𝐴𝑃𝐴+𝜆𝐴𝑆𝐴𝑁𝐻𝐼𝑉𝐼+𝛿𝐶𝛾+𝜀𝐴+𝜇𝐴𝐼𝐴𝑑𝑅,(2.5)𝐴=𝑑𝑡𝐴+𝑞𝑠𝑓𝐴𝑃𝐴𝑅+𝛿𝐶𝐼+𝛾𝐴𝜀𝐴+𝜇𝐴𝑅𝐴𝑑𝑆,(2.6)𝑉𝑑𝑡=𝑃𝑉𝜆𝑉𝑆𝑉𝐼𝐶+𝐼𝐴𝑁𝐻𝜇𝑉𝑆𝑉𝑑𝐸,(2.7)𝑉𝑑𝑡=𝜆𝑉𝑆𝑉𝐼𝐶+𝐼𝐴𝑁𝐻𝜏+𝜇𝑉𝐸𝑉𝑑𝐼,(2.8)𝑉𝐸𝑑𝑡=𝜏𝑉𝜇𝑉𝐼𝑉,(2.9) where 𝑁𝐻 is the total population of host. Furthermore, we use 𝑁𝐶=𝑆𝐶+𝐼𝐶+𝑅𝐶, 𝑁𝐴=𝑆𝐴+𝐼𝐴+𝑅𝐴, and 𝑁𝑉=𝑆𝑉+𝐸𝑉+𝐼𝑉 as the total populations of child, adult, and vector, respectively. These populations are governed by the following equations:𝑑𝑁𝐶𝑑𝑡=𝑃𝐶+𝐵𝛿+𝜀𝐶+𝜇𝐶𝑁𝐶,𝑑𝑁(2.10)𝐴𝑑𝑡=𝑃𝐴𝑁+𝛿𝐶𝜀𝐴+𝜇𝐴𝑁𝐴𝑑𝑁,(2.11)𝑉𝑑𝑡=𝑃𝑉𝜇𝑉𝑁𝑉.(2.12)

When 𝑡, we have that 𝑁𝐶(𝑃𝐶+𝐵)/(𝛿+𝜀𝐶+𝜇𝐶), 𝑁𝐴(𝛿(𝑃𝐶+𝐵)+(𝛿+𝜀𝐶+𝜇𝐶)𝑃𝐴)/(𝛿+𝜀𝐶+𝜇𝐶)(𝜀𝐴+𝜇𝐴), and 𝑁𝑉𝑃𝑉/𝜇𝑉.

First, we consider that the host and vector populations have reached the limiting states; these are 𝑁𝐶=(𝑃𝐶+𝐵)/(𝛿+𝜀𝐶+𝜇𝐶), 𝑁𝐴=(𝛿(𝑃𝐶+𝐵)+(𝛿+𝜀𝐶+𝜇𝐶)𝑃𝐴)/(𝛿+𝜀𝐶+𝜇𝐶)(𝜀𝐴+𝜇𝐴), 𝑁𝑉=𝑃𝑉/𝜇𝑉, and 𝑁𝐻=𝑁𝐶+𝑁𝐴. Then, we scale model (2.1)–(2.9) with following transformations 𝑆𝐶=𝑆𝐶/𝑁𝐶, 𝐼𝐶=𝐼𝐶/𝑁𝐶, 𝑅𝐶=𝑅𝐶/𝑁𝐶, 𝑆𝐴=𝑆𝐴/𝑁𝐴, 𝐼𝐴=𝐼𝐴/𝑁𝐴, 𝑅𝐴=𝑅𝐴/𝑁𝐴, 𝑆𝑉=𝑆𝑉/𝑁𝑉, 𝐸𝑉=𝐸𝑉/𝑁𝑉, and 𝐼𝑉=𝐼𝑉/𝑁𝑉. Thus, we obtain the following reduced model:𝑑𝑆𝐶𝑓𝑑𝑡=(1𝑝𝑠)𝐶𝑄𝐶+𝑇𝛽𝐶𝑆𝐶𝐼𝑉𝛿+𝜀𝐶+𝜇𝐶𝑆𝐶,(2.13)𝑑𝐼𝐶𝑑𝑡=𝑔𝐶𝑄𝐶+𝛽𝐶𝑆𝐶𝐼𝑉𝛿+𝛾+𝜀𝐶+𝜇𝐶𝐼𝐶,(2.14)𝑑𝑆𝐴𝑑𝑡=(1𝑞𝑠)𝑓𝐴𝑄𝐴+𝛿𝜎𝑆𝐶𝛽𝐴𝑆𝐴𝐼𝑉𝜀𝐴+𝜇𝐴𝑆𝐴,(2.15)𝑑𝐼𝐴𝑑𝑡=𝑔𝐴𝑄𝐴+𝛽𝐴𝑆𝐴𝐼𝑉+𝛿𝜎𝐼𝐶𝛾+𝜀𝐴+𝜇𝐴𝐼𝐴,(2.16)𝑑𝐸𝑉𝑑𝑡=𝑆𝑉𝜃𝐶𝐼𝐶+𝜃𝐴𝐼𝐴𝜏+𝜇𝑉𝐸𝑉,(2.17)𝑑𝐼𝑉𝑑𝑡=𝜏𝐸𝑉𝜇𝑉𝐼𝑉,(2.18) where 𝑁𝑇=𝐵/𝐶, 𝑄𝐶=𝑃𝐶/𝑁𝐶, 𝛽𝐶=𝜆𝐶𝑁𝑉/𝑁𝐻, 𝜃𝐶=𝜆𝑉𝑁𝐶/𝑁𝐻, 𝑁𝜎=𝐶/𝑁𝐴, 𝑄𝐴=𝑃𝐴/𝑁𝐴, 𝛽𝐴=𝜆𝐴𝑁𝑉/𝑁𝐻, 𝜃𝐴=𝜆𝑉𝑁𝐴/𝑁𝐻, and 𝑆𝑉=1𝐸𝑉𝐼𝑉. The values of 𝑅𝐶 and 𝑅𝐴 in the limiting state can be evaluated using 𝑅𝐶=1𝑆𝐶𝐼𝐶 and 𝑅𝐴=1𝑆𝐴𝐼𝐴.

After the scaling, the region of biological interest of model (2.13)–(2.18) is𝑆Ω=𝐶,𝐼𝐶,𝑆𝐴,𝐼𝐴,𝐸𝑉,𝐼𝑉[]0,16𝑆𝐶+𝐼𝐶1,𝑆𝐴+𝐼𝐴1,𝐸𝑉+𝐼𝑉1.(2.19)

This region is positive invariant under the flow generated by the vector field of model (2.13)–(2.18), because the vector field on the boundary of Ω does not point out the exterior of Ω.

For the rest of the paper, we will analyze model (2.13)–(2.18) since this reduced model is the limiting system of model (2.1)–(2.9) and has the same asymptotic behavior as the original model [17, 18].

3. Model Analysis

Solving the equilibrium conditions of model (2.13)–(2.18), we obtain the following equations: 𝑆𝐶=𝑓(1𝑝𝑠)𝐶𝑄𝐶+𝑇𝛽𝐶𝐼𝑉+𝛿+𝜀𝐶+𝜇𝐶𝐼,(3.1)𝐶=𝑔𝐶𝑄𝐶+𝛽𝐶𝑆𝐶𝐼𝑉𝛿+𝛾+𝜀𝐶+𝜇𝐶,𝑆(3.2)𝐴=(1𝑞𝑠)𝑓𝐴𝑄𝐴+𝛿𝜎𝑆𝐶𝛽𝐴𝐼𝑉+𝜀𝐴+𝜇𝐴𝐼,(3.3)𝐴=𝑔𝐴𝑄𝐴+𝛽𝐴𝑆𝐴𝐼𝑉+𝛿𝜎𝐼𝐶𝛾+𝜀𝐴+𝜇𝐴𝐸,(3.4)𝑉=𝜇𝑉𝜏𝐼𝑉,(3.5) and the variable 𝐼𝑉 satisfies 𝑀(𝐼𝑉)+𝑁(𝐼𝑉)=0, where𝑀𝐼𝑉=𝑐2𝐼2𝑉+𝑐1𝐼𝑉+𝑐0𝐼𝑉,𝑁𝐼(3.6)𝑉=𝐼𝑉𝛽𝐴+𝜀𝐴+𝜇𝐴𝐼𝑉𝛽𝐶+𝛿+𝜀𝐶+𝜇𝐶𝐼𝑉𝜇𝑉×𝑔+𝜏𝜏𝐴𝑄𝐴𝜃𝐴𝛾+𝛿+𝜀𝐶+𝜇𝐶+𝑔𝐶𝑄𝐶𝜃𝐶𝛾+𝜀𝐴+𝜇𝐴+𝜃𝐴,𝑐𝛿𝜎(3.7)2=𝛽𝐴𝛽𝐶𝜇𝑉+𝜏(1𝑞𝑠)𝑓𝐴𝑄𝐴𝜃𝐴𝛾+𝛿+𝜀𝐶+𝜇𝐶+𝜇𝑉𝛾+𝜀𝐴+𝜇𝐴𝛾+𝛿+𝜀𝐶+𝜇𝐶+𝑆𝑑𝐶𝛿+𝜀𝐶+𝜇𝐶𝜃𝐶𝛾+𝜀𝐴+𝜇𝐴+𝜃𝐴,𝑐𝛿𝜎(3.8)1=𝛽𝐶𝜀𝐴+𝜇𝐴𝛾+𝛿+𝜀𝐶+𝜇𝐶×𝜇𝑉𝛾+𝜀𝐴+𝜇𝐴𝜇𝑉+𝜏𝛽𝐴𝜃𝐴𝜏𝑆𝑑𝐴+𝛽𝐴𝛿+𝜀𝐶+𝜇𝐶𝜇𝑉𝛾+𝜀𝐴+𝜇𝐴𝛾+𝛿+𝜀𝐶+𝜇𝐶𝜇𝑉+𝜏𝛽𝐶𝜏𝑆𝑑𝐶𝜃𝐶𝛾+𝜀𝐴+𝜇𝐴+𝛿𝜎𝜃𝐴+𝜀𝐴+𝜇𝐴𝛿+𝜀𝐶+𝜇𝐶𝜇𝑉𝛽+𝜏𝐴𝜃𝐴𝑆𝑑𝐴𝛾+𝛿+𝜀𝐶+𝜇𝐶+𝛽𝐶𝑆𝑑𝐶𝜃𝐶𝛾+𝜀𝐴+𝜇𝐴+𝛿𝜎𝜃𝐴+𝛽𝐴𝛽𝐶𝜃𝐴𝛿𝜎𝜏𝑆𝑑𝐶𝛾+𝛿+𝜀𝐶+𝜇𝐶,𝑐(3.9)0=𝛿+𝛾+𝜀𝐶+𝜇𝐶𝛿+𝜀𝐶+𝜇𝐶𝜀𝐴+𝜇𝐴𝛾+𝜀𝐴+𝜇𝐴𝜇𝑉𝜇𝑉+𝜏1𝑅0,(3.10)

The zeros of the polynomial 𝑀+𝑁 determine the equilibrium of model (2.13)–(2.18). We analyze the zeros of the polynomial 𝑀+𝑁 into two cases. The first case is if there is no incoming infected immigrant, so the incoming immigrants are susceptible or have permanent immunity to the dengue infection. In this case, only polynomial 𝑀 determines the equilibrium. The second case is if there are some incoming infected immigrants. In the second case, both polynomials 𝑀 and 𝑁 determine the equilibrium.

3.1. No Incoming Infected Immigrants

In this subsection, we consider the case where there is no incoming infected immigrant or mathematically 𝑔𝐶=𝑔𝐴=0. Furthermore, the condition 𝑔𝐶=𝑔𝐴=0 implies that polynomial 𝑁 becomes a zero polynomial.

In this case, model (2.13)–(2.18) has a disease-free equilibrium; that is, 𝐸𝑑=(𝑆𝑑𝐶,0,𝑆𝑑𝐴, 0,0,0), where 𝑆𝑑𝐶 and 𝑆𝑑𝐴 are exactly as in (3.11)-(3.12). This equilibrium is obtained by substituting 𝐼𝑉=0 into (3.5).

If the vaccination programme is not implemented (𝑝=𝑞=0) and all immigrants are susceptibles (𝑓𝐶=𝑓𝐴=1), then we obtain 𝑆𝑑𝐶=𝑆𝑑𝐴=1 and 𝑅𝑑𝐶=𝑅𝑑𝐴=0. In the limiting case where all susceptible immigrants and all births are vaccinated (𝑝=𝑞=1) and the vaccine efficacy is perfect (𝑠=1), we have 𝑆𝑑𝐶=𝑆𝑑𝐴=0 and 𝑅𝑑𝐶=𝑅𝑑𝐴=1.

Basic reproduction ratio is the expected number of secondary cases per primary case in a “virgin” population [19]. It is an important threshold because it determines whether an initial infection in a virgin population will end up in an endemic. This threshold parameter is given by the spectral radius of the next-generation matrix. The spectral radius of our next-generation matrix is the square root of 𝑅0, where 𝑅0 is exactly as in (3.13). This square root of 𝑅0 can be interpreted as the basic reproduction ratio under vaccination programme.

Next, we explore the existence of the endemic equilibrium of model (2.13)–(2.18) when 𝑔𝐶=𝑔𝐴=0. Here, we consider the equation 𝑐2𝐼2𝑉+𝑐1𝐼𝑉+𝑐0=0, where the coefficients 𝑐0, 𝑐1, and 𝑐2 are as in (3.8)–(3.10).

It can be seen that 𝑐2 is positive. The coefficient 𝑐0 is positive for 𝑅0<1, and it is negative for 𝑅0>1. Moreover, for 𝑅0=1, we have that 𝑐0=0 and 𝑐1,𝑐2>0. So, model (2.13)–(2.18) cannot exhibit backward bifurcation at 𝑅0=1.

For 𝑅01, we have following inequalities:𝛽𝐶𝜏𝑆𝑑𝐶𝜃𝐶𝛾+𝜀𝐴+𝜇𝐴+𝛿𝜎𝜃𝐴<𝜇𝑉𝛾+𝜀𝐴+𝜇𝐴𝛿+𝛾+𝜀𝐶+𝜇𝐶𝜇𝑉𝛽+𝜏𝐴𝜃𝐴𝜏𝑆𝑑𝐴<𝜇𝑉𝛾+𝜀𝐴+𝜇𝐴𝜇𝑉.+𝜏(3.14)

These inequalities imply 𝑐1>0 for 𝑅01. Thus, there is no positive root 𝐼𝑉 of the equation 𝑐2𝐼2𝑉+𝑐1𝐼𝑉+𝑐0=0 for 𝑅01. And there is a unique positive root 𝐼𝑒𝑉 of the equation 𝑐2𝐼2𝑉+𝑐1𝐼𝑉+𝑐0=0 which is always less than one for 𝑅0>1. Figure 2 gives three qualitative graphs of 𝑀 with respect to the three conditions of 𝑅0.

fig2
Figure 2: Three typical graphs of the polynomial 𝑀 in (3.6) with respect to three conditions of 𝑅0.

It can be verified that the equilibrium 𝐸𝑒=(𝑆𝑒𝐶,𝐼𝑒𝐶,𝑆𝑒𝐴,𝐼𝑒𝐴,𝐸𝑒𝑉,𝐼𝑒𝑉) whose coordinates satisfy equations (3.5) is in int(Ω) if and only if 𝑅0>1. We summarized these results in the following proposition.

Proposition 3.1. Let 𝑔𝐶=𝑔𝐴=0. Model (2.13)–(2.18) always has a unique disease-free equilibrium 𝐸𝑑 in Ω. For 𝑅0>1, model (2.13)–(2.18) also has a unique positive endemic equilibrium 𝐸𝑒 in int(Ω) whose components satisfy (3.5), and 𝐼𝑉 satisfies 𝑐2𝐼2𝑉+𝑐1𝐼𝑉+𝑐0=0, where the coefficients 𝑐0,𝑐1,𝑐2 are as in (3.8)–(3.10).

The next proposition gives the stability of equilibrium 𝐸𝑑.

Proposition 3.2. Let 𝑔𝐶=𝑔𝐴=0. The disease-free equilibrium 𝐸𝑑 is locally asymptotically stable if 𝑅0<1 and it is unstable if 𝑅0>1.

Proof. The linearization of model (2.13)–(2.18) at point 𝐸𝑑 gives the Jacobian matrix: 𝐴𝐴=1𝐴30𝐴2,(3.15) where 𝐴1=𝛿𝜀𝐶𝜇𝐶0𝛿𝜎𝜀𝐴𝜇𝐴,𝐴2=𝛿𝛾𝜀𝐶𝜇𝐶00𝛽𝐶𝑆𝑑𝐶𝛿𝜎𝛾𝜀𝐴𝜇𝐴0𝛽𝐴𝑆𝑑𝐴𝜃𝐶𝜃𝐴𝜇𝑉𝜏000𝜏𝜇𝑉.(3.16) Moreover, the eigenvalues of matrices 𝐴1 and 𝐴2 determine the local stability of 𝐸𝑑.
The eigenvalues of matrix𝐴1 are (𝛿+𝜀𝐶+𝜇𝐶) and (𝜀𝐴+𝜇𝐴). The matrix 𝐴2 is an M-matrix. The real parts of all eigenvalues of matrix 𝐴2 are positive if and only if det(𝐴2)>0 (see [20]). Furthermore, all eigenvalues of 𝐴2 have negative real parts if and only if det(𝐴2)>0. The determinant of matrix 𝐴2 is given by 𝐴det2=𝜇𝑉𝛾+𝜀𝐴+𝜇𝐴𝛿+𝛾+𝜀𝐶+𝜇𝐶𝜇𝑉𝑅+𝜏01.(3.17) Thus, if 𝑅0<1, then the equilibrium 𝐸𝑑 is locally asymptotically stable and it is unstable if 𝑅0>1.

Let the endemic equilibrium 𝐸𝑒=(𝑆𝑒𝐶,𝐼𝑒𝐶,𝑆𝑒𝐴,𝐼𝑒𝐴,𝐸𝑒𝑉,𝐼𝑒𝑉) exists. Linearization of model (2.13)–(2.18) at point 𝐸𝑒 gives following Jacobian matrix:Δ𝐽=10000𝛽𝐶𝑆𝑒𝐶𝛽𝐶𝐼𝑒𝑉Δ2000𝛽𝐶𝑆𝑒𝐶𝛿𝜎0Δ300𝛽𝐴𝑆𝑒𝐴0𝛿𝜎𝛽𝐴𝐼𝑒𝑉Δ40𝛽𝐴𝑆𝑒𝐴0𝑆𝑒𝑉𝜃𝐶0𝜃𝐴𝑆𝑒𝑉Δ5𝜃𝐴𝐼𝑒𝐴+𝜃𝐶𝐼𝑒𝐶0000𝜏𝜇𝑉,(3.18)

where Δ1=(𝛽𝐶𝐼𝑒𝑉+𝛿+𝜀𝐶+𝜇𝐶), Δ2=(𝛾+𝛿+𝜀𝐶+𝜇𝐶), Δ3=(𝛽𝐴𝐼𝑒𝑉+𝜀𝐴+𝜇𝐴), Δ4=(𝛾+𝜀𝐴+𝜇𝐴), Δ5=(𝜃𝐴𝐼𝑒𝐴+𝜃𝐶𝐼𝑒𝐶+𝜇𝑉+𝜏), and 𝑆𝑒𝑉=1𝐸𝑒𝑉𝐼𝑒𝑉.

It is not easy to prove analytically that all eigenvalues of 𝐽 have negative real parts for 𝑅0>1. However, from our numerical simulations (case 𝑅0>1) all of the eigenvalues have negative real parts. Figure 3 gives the projection of three orbits of three different initial conditions when 𝑅0>1 on the 𝐼𝐶𝐼𝐴 plane. The component (𝐼𝐶,𝐼𝐴) of the equilibrium 𝐸𝑒 is not (0,0). This simulation indicates that the endemic equilibrium 𝐸𝑒 is locally asymptotically stable when 𝑅0>1.

fig3
Figure 3: Projection of three orbits of model (2.13)–(2.18) on 𝐼𝐶𝐼𝐴 plane (a) and the projection near the equilibrium state (b). This projection indicates the local stability of the endemic equilibrium 𝐸𝑒 when 𝑅0>1.
3.2. Some Incoming Immigrants Are Infected

Here, we consider the case that there are some infected incoming immigrants; that is, 𝑔𝐶 or 𝑔𝐴 is larger than zero. In this case, we have following proposition.

Proposition 3.3. Let 𝑔𝐶 or 𝑔𝐴 be larger than zero. Model (2.13)–(2.18) always has a unique positive endemic equilibrium 𝐸𝑓in int(Ω) whose components satisfy (3.5) and IV satisfies 𝑀(𝐼𝑉)+𝑁(𝐼𝑉)=0.

We will give the outline of proof of Proposition 3.3.

Outline of proof. When 𝑔𝐶 or 𝑔𝐴 or both are larger than zero, the cubic polynomial 𝑁 in (3.7) always has two negative zeros and one positive zero which is less than one. Figure 4 gives the graph of the polynomial 𝑁.

236352.fig.004
Figure 4: Graph of the polynomial 𝑁.

The cubic polynomial 𝑀 in (3.6) always has a trivial zero. Depending on 𝑅0, the other two zeros could be negative, zero, or positive. Figure 2 illustrates three typical graphs of the polynomial 𝑀 with respects to 𝑅0.

Figure 5 gives the graph of polynomial 𝑀+𝑁. The graph always has two negative zeros and one positive zero which is less than one. This positive zero is the component 𝐼𝑉 of endemic equilibrium 𝐸𝑓.

236352.fig.005
Figure 5: Graph of the polynomial 𝑀+𝑁.

From Proposition 3.3, there is no disease-free equilibrium and there is only endemic equilibrium if there are always some infected incoming child or adult immigrants. So, it is very important to do screening for the child and adult immigrants. The infected immigrants should be quarantined as long as they are ill. Otherwise, we will lose the disease-free condition. Here, we get a similar conclusion as in [14]. In [14], the authors did not separate the child class and the adult class in their model.

Figures 6 and 7 show the values of the equilibrium infected child population 𝐼𝐶 and the equilibrium infected adult population 𝐼𝐴 as the function of the portion of infected child 𝑔𝐶 and adult immigrants 𝑔𝐴. In Figure 6, we use parameters which produce 𝑅0<1 around 𝑔𝐶=𝑔𝐴=0. Note that the lowest point (𝑔𝐶=𝑔𝐴=0) corresponds to the components 𝐼𝑑𝐶 and 𝐼𝑑𝐴 of the disease-free equilibrium 𝐸𝑑. When 𝑔𝐶,𝑔𝐴0, the points in the surface correspond to the components 𝐼𝑓𝐶 and 𝐼𝑓𝐴 of the endemic equilibrium 𝐸𝑓. However, in Figure 7, we use parameters which produce 𝑅0>1 around 𝑔𝐶=𝑔𝐴=0. Here, the lowest point (𝑔𝐶=𝑔𝐴=0) corresponds to the components 𝐼𝑒𝐶 and 𝐼𝑒𝐴 of the endemic equilibrium 𝐸𝑒. When 𝑔𝐶,𝑔𝐴0, the points in the surface correspond to the components 𝐼𝑓𝐶 and 𝐼𝑓𝐴 of endemic equilibrium 𝐸𝑓. Despite the difference in the resulting properties of the basic reproduction number, and since both 𝐼𝐴 and 𝐼𝐶 constitute the endemic equilibrium 𝐸𝑓, the figures in fact indicate the existence of this endemic equilibrium when 𝑔𝐶 and 𝑔𝐴 are not zero.

fig6
Figure 6: Component 𝐼𝐶(a) and component 𝐼𝐴(b) of endemic equilibria 𝐸𝑓. Here, 𝑅0<1 when 𝑔𝐶=𝑔𝐴=0.
fig7
Figure 7: Component 𝐼𝐶(a) and component 𝐼𝐴(b) of endemic equilibria 𝐸𝑓. Here, 𝑅0>1 when 𝑔𝐶=𝑔𝐴=0.

The stability of the endemic equilibrium 𝐸𝑓 is not easy to be obtained analytically. Numerical simulations indicate the local stability of the equilibrium 𝐸𝑓. Figure 8 gives three orbits of three different sets of parameter values. This simulation indicates that the equilibrium 𝐸𝑓 is locally asymptotically stable.

fig8
Figure 8: Projection of three orbits of model (2.13)–(2.18) on 𝐼𝐶𝐼𝐴 plane (a) and the projection near the equilibrium state (b). This projection indicates the local stability of the endemic equilibrium 𝐸𝑓.

4. Numerical Simulation

In the following numerical simulations, we use data in Table 1.

tab1
Table 1: Data for numerical simulations (y represents years).

In Figure 9, we simulate four different scenarios, relative to no vaccination scenario and low screening level, that is, 𝑔𝑎=𝑔𝑐=20%. The situation is described as follow, first if we raise the level of screening twice, that is, reduction of 𝑔𝑎 and 𝑔𝑐 from 20% to 10%, the infection will decrease from 100% to 85.7% for 𝐼𝑎 and 94% for 𝐼𝑐. If we gain the screening process up to four times, we have the infection decreasing from 100% to 82.1% for 𝐼𝑎 and 91.5% for 𝐼𝑐. Hence, increasing the level of screening will decrease the endemicity. But if we vaccine 40% of children and adult (𝑝=𝑞=40%), the decreasing level of infection is 71.4% for 𝐼𝑎 and 66.3% for 𝐼𝑐, and if we raise the coverage of vaccination to 80% (𝑝=𝑞=80%), we can reduce the infection up to 42.9% for 𝐼𝑎 and 32.5% for 𝐼𝑐. So, increasing the coverage of vaccination will also decrease the endemicity. The summary of the scenarios can be seen in Table 2.

tab2
Table 2: Percentages of endemicity. We use the first scenario as a reference scenario for the other four scenarios.
fig9
Figure 9: Dynamics of some scenarios. Here, we take (0.3, 0.1, 0.5, 0.1, 0.1, 0.1) as the initial condition.

5. Conclusion

In this paper we derive a mathematical model of dengue transmission with vaccination program. The model incorporates two-age classes and migration. We also consider a susceptibility distribution in the incoming migrants.

From the analysis of the model, we obtain a conclusion that the susceptibility distribution is an important factor for the existence of disease-free equilibrium. If there is no incoming infected immigrant, then we have a unique disease-free equilibrium and a unique endemic equilibrium which depend on the basic reproduction ratio. Moreover, the stability of the equilibria also depends on the basic reproduction ratio. However, if some of the incoming immigrants are infected, then we only have a unique endemic equilibrium. Hence, screening for the incoming immigrants must be done. The incoming infected immigrants should be quarantined until they are recovered. Otherwise, we will lose the disease-free state from the population.

From the sensitivity analysis of the level of screening and the coverage of vaccination, increasing one of these parameters will give the reduction of endemic level. Increasing both parameters will give larger reduction of endemic level. The resulting simulation could give prior information for policy maker in setting the scale of vaccination and understanding the effect of vaccination in the reduction of endemic level.

Acknowledgment

Part of the research is funded by Hibah Pasca Sarjana of the Directorate for Higher Education.

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