Journal of Applied Mathematics

Volume 2018, Article ID 9674138, 13 pages

https://doi.org/10.1155/2018/9674138

## Understanding Dengue Control for Short- and Long-Term Intervention with a Mathematical Model Approach

Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia

Correspondence should be addressed to D. Aldila; di.ca.iu.ics@opidalidla

Received 10 August 2017; Accepted 19 November 2017; Published 1 January 2018

Academic Editor: Lucas Jodar

Copyright © 2018 A. Bustamam 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

A mathematical model of dengue diseases transmission will be discussed in this paper. Various interventions, such as vaccination of adults and newborns, the use of insecticides or fumigation, and also the enforcement of mechanical controls, will be considered when analyzing the best intervention for controlling the spread of dengue. From model analysis, we find three types of equilibrium points which will be built upon the dengue model. In this paper, these points are the mosquito-free equilibrium, disease-free equilibrium (with and without vaccinated compartment), and endemic equilibrium. Basic reproduction number as an endemic indicator has been found analytically. Based on analytical and numerical analysis, insecticide treatment, adult vaccine, and enforcement of mechanical control are the most significant interventions in reducing the spread of dengue disease infection caused by mosquitoes rather than larvicide treatment and vaccination of newborns. From short- and long-term simulation, we find that insecticide treatment is the best strategy to control dengue. We also find that, with periodic intervention, the result is not much significantly different with constant intervention based on reduced number of the infected human population. Therefore, with budget limitations, periodic intervention of insecticide strategy is a good alternative to reduce the spread of dengue.

#### 1. Introduction

Dengue is the most rapidly growing disease in the world [1]. The disease is spread by* Aedes *mosquitoes and is therefore often referred to as a mosquito-borne viral disease. The disease has become endemic in more than 100 countries, including the Caribbean, Africa, the Americas, the Pacific, and Asia, including Indonesia [1]. As an endemic disease, dengue occurs regularly in subtropical and tropical regions of the world, and approximately 40% of people live in regions of the world where there is a risk of contracting it [2]. Dengue is a vector-borne disease transmitted from an infected human to a female* Aedes aegypti *mosquito by a bite. The mosquito, which needs regular meals of blood to mature its eggs, completes the cycle by biting a healthy human, transmitting the disease in one act [3].

Until now, the primary prevention for dengue has been control of mosquitoes, in both larval and adult forms. Larval control is carried out by larvicide treatment using long-lasting chemicals to kill larvae, which sure preferably have WHO clearance for use in drinking water [4]. Mechanical controls are also used to control larvae, with assistance from campaigns and educational programs carried out by governments. In Indonesia, such a program is known as 3M and consists of educating people about the importance of draining and shutting down and burying all tubs, buckets, or containers of water that which be used by female mosquitoes to breed and lay their eggs [5]. The larvae of* Aedes aegypti *can also grow in used goods that can hold water, and it is therefore recommended to make sure the environment around the house has no space that could allow mosquitoes to breed.

Adult mosquito control is achieved by the use of insecticide. Insecticide fumigation targets the vector* Aedes aegypti* mosquito as the main control of dengue epidemics. However, the long-term use of insecticides and larvicides poses several risks: one is resistance of the mosquito to the product, reducing its efficacy, while genetic mutation of the mosquito, making it less susceptible to the effects of the product, is another. Such products have also been linked to numerous adverse health effects including the worsening of asthma and respiratory problems [3, 6]. In Surabaya, Indonesia, larval mortality rates of under 80% indicate possible resistance of* Aedes aegypti* to the insecticide temephos [7], and outside Indonesia, resistance to insecticide has been reported in multiple countries. In recent years, the frequency of kdr mutations associated with pyrethroid resistance has increased rapidly [8]. Pyrethroids have become the most frequently used public health insecticides globally due to their low cost and low toxicity to mammals [9], and they are of considerable concern when kdr is found in wild populations of vector mosquitoes [8]. With the many cases of* Aedes aegypti* mosquito’s resistance to insecticide, it is therefore necessary to develop alternative strategies to slow its evolution.

Besides controlling dengue via control of mosquitoes population, one of the alternative strategies that is being used is dengue vaccine. In December 2015, the first vaccine against dengue by Sanofi Pasteur, Dengvaxia (CYD-TDV), was approved in three highly endemic countries: Mexico, Philippines, and Brazil [10]. This vaccine is the world’s first dengue vaccine and is already licensed for individuals aged 9–45 years for the prevention of infectious disease caused by four dengue virus serotypes (DEN 1, DEN 2, DEN 3, and DEN 4) [11]. In Indonesia, so far, the government is still conducting clinical trials to determine its effectiveness. However, assessments of the public’s acceptance of the dengue vaccine and its associated factors are widely lacking [12]. A lack of understanding about the importance of vaccination against dengue to the public will be able to reduce the success rate of vaccination interventions in various countries, especially in a country that is less intensive to educate people about the importance of dengue fever vaccination [13]. In 2017,* Dengvaxia* would have been applied if proven effective and suitable for the dengue serotypes which are pandemic in Indonesia [14].

The earliest mathematical models for dengue disease transmission are developed in [15, 16] which are closely related to the models for the transmission of malaria discussed in [17, 18]. The authors in [16] create the model for two types of viruses by allowing temporary cross immunity and increased susceptibility to the second infection due to the first infection. The intervention has not yet been used into the mathematical model [16]. In [19], the mathematical model with only insecticide campaign intervention is discussed. It has been shown that, with a steady insecticide campaign, it is possible to reduce the number of infected humans and mosquitoes and prevent an outbreak that could transform an epidemiological episode to an endemic disease [19]. A year after the research discussed in [19], it was updated [20], and the mathematical model for dengue was updated continuously with all controls included, that is, proportion of larvicide, proportion of adulticide, and proportion of mechanical control. The results have shown that, even with a low, although continuous, index of control adulticide over time, the results are surprisingly positive [20]. However, it has been stated that to rely only on adulticide is a risky decision [20]. The research in [6, 7, 21] supports this claim, citing the problem of* Aedes aegypti* mosquito’s resistance to insecticide. Under the new achievement in the field of vaccination technology with the discovery of the first vaccine against dengue by Sanofi Pasteur, the work in [22] devised two models, one assuming that unintentional vaccination increases the infectious period and another assuming that unintentional vaccination leads to the development of symptoms. This argument is also supported by [3], in which the mathematical model is created with the vaccine as the new compartment, arguing that the vaccine must divide the human population into classes, that is, the perfect pediatric vaccine for newborns and perfect adult vaccine (conferring 100% protection throughout life), and also classes of human with imperfect vaccine effect [3]. Other mathematical models with different intervention were also introduced in [23] which discuss the use of mosquito repellent to reduce probability of success of infection in human population and in [4] which discuss the use of sterile mosquito strategy.

According to above explanation, it is important to find the best strategy for controlling dengue spreads for both short-term and long-term interventions. Therefore, a mathematical model of dengue disease transmission by using adult and newborn vaccines with waning immunity, the use of insecticides and larvicides, and mechanical control will be developed in the next section. Equilibrium points will be found, which ensure the existence of local stability. Basic reproduction numbers will be obtained as the main factor in whether the disease will become epidemic in a population or not. Numerical analysis for comparing the dynamic of infected humans and mosquitoes will be used to support the model interpretation.

#### 2. Mathematical Model Construction

To construct our model, firstly we divide the human population into four compartments, that is, : susceptible (individuals who can be infected with dengue); : vaccinated (individuals who have had the vaccine injected into their bodies, making them resistant to infectious disease. However, the use of the vaccine does not provide perfect immunity. There will be a time when the vaccine does not work properly in the body or when the effect of the vaccine has begun to subside [3]); : infected (individuals who are infected with dengue. In this case, the infected human is incapable of transmitting the disease to other humans); : recovered (individuals who have recovered from dengue and have acquired temporal immunity to respective DEN virus).

On the other hand, we divide the mosquito population into three compartments, that is, : aquatic phase (the phase that includes the egg, larvae, and pupa stages, which live in water); : susceptible (mosquitoes that are able to infect with dengue); : infected (mosquitoes that have been infected with dengue by an infected human and are capable of transmitting dengue to humans).

Secondly, we have made some assumptions that we will use to describe a dynamic process in our model that we will construct: There is no migration in either human or mosquito population. Humans and mosquitoes are assumed to be born susceptible, there is no natural protection, and dengue is not passed onto the next generation (no vertical transmission) [19]. The transmission process in susceptible and vaccinated humans is simply by the bite of an infected mosquito. Infected humans cannot transmit the virus to other susceptible or vaccinated humans [19]. The death rate is considered to be a natural death rate in both populations. Vaccinated human status is considered temporary because of the ability of the vaccine to subside over time [3]. There is no recovered phase in mosquitoes due to their short lifespan [20]. There is no resistant (immune) effect in mosquitoes to the use of synthetic fumigation, such as insecticides and larvicides [6, 7]; in this article, we assume that there is no resistant (immunity) effect to mosquitoes due to use of synthetic fumigation, such as insecticide and larvicide.

With the assumptions, variables, and transmission diagram given in Figure 1, the model is represented as a seven-dimensional system of differential equations which are given by with parameters description being the following: : total of human population : human per capita birth rate : average number of bites of humans by mosquitoes : average of successful transmission in human : average of successful transmission in mosquitoes : human death rate : mosquito death rate : larval death rate : the rate of change from to because of the disappearance of the temporal natural immunity : the rate of change from to because of the disappearance of the temporal vaccine effect : reduction of because of vaccine : human recovery rate : average number of eggs at each deposit : ratio for number of larvae per human : transition rate from to : adult vaccination rate : newborn vaccine : larvicide rate : fumigation rate : enforcement of mechanical control proportion to reduce