Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 1923479, 15 pages

https://doi.org/10.1155/2019/1923479

## Modelling the Effect of a Novel Autodissemination Trap on the Spread of Dengue in Shah Alam and Malaysia

Correspondence should be addressed to Y. Liang; moc.liamtoh@gnail.gnefnay

Received 7 October 2018; Revised 3 April 2019; Accepted 18 April 2019; Published 4 August 2019

Academic Editor: Konstantin Blyuss

Copyright © 2019 Y. Liang 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

In this paper, we will start off by introducing the classical Ross–Macdonald model for vector-borne diseases which we use to describe the transmission of dengue between humans and *Aedes* mosquitoes in Shah Alam, which is a city and the state capital of Selangor, Malaysia. We will focus on analysing the effect of using the Mosquito Home System (MHS), which is an example of an autodissemination trap, in reducing the number of dengue cases by changing the Ross–Macdonald model. By using the national dengue data from Malaysia, we are able to estimate *λ*, which represents the initial growth rate of the dengue epidemic, and this allows us to estimate the number of mosquitoes in Malaysia. A mathematical expression is also constructed which allows us to estimate the potential number of breeding sites of *Aedes* mosquitoes. By using the data available from the MHS trial carried out in Section 15 of Shah Alam, we included the potential effect of the MHS into the dengue model and thus modelled the impact MHS has on the spread of dengue within the trial area. We then extended our results to analyse the effect of the MHSs on reducing the number of dengue cases in the whole of Malaysia. A new model was constructed with a basic reproduction number, , which allows us to identify the required MHSs coverage needed to achieve extinction in Malaysia. Numerical simulations and tables of results were also produced to illustrate our results.

#### 1. Introduction

Epidemics of infectious diseases have been a constant threat towards our society. In the past, Europe suffered from 25 million deaths out of a population of 100 million due to the Black Death [1]; Russia suffered from about 25 million cases of typhus with a death rate of about 10 percent, whilst smallpox wiped out half of the population of the Aztecs of three and a half million in 1520 [2]. Although in the 21st century, many diseases such as smallpox no longer pose a threat towards mankind, but there is still a high proportion of the population that is under threat of diseases such as malaria and dengue. According to the World Health Organization, every year there are around 50–100 million dengue infections where at least 100 countries have a dengue epidemic [3]. Dengue is a vector-borne disease which is transmitted by the *Aedes* mosquitoes which are also responsible for the transmission of yellow fever and the Zika virus [3, 4].

Malaysia, a country in the Southeast of Asia, has consistently been reported to have a high number of dengue cases due to its tropical climate. Between 2014 and 2016, Malaysia had around 330,891 reported dengue cases with around 788 dengue-related deaths with a high incidence rate of 396.4 per 100,000 population in 2015 causing it to suffer from serious economic and health burdens. In the study of Packierisamy et al. [5], it is estimated that, in 2010, it had cost Malaysia around USD $73.45 million in dengue-related vector control which was around USD $2.63 per capita population. The standard and traditional way of battling against dengue is by using space spraying (chemical fogging); however, the effect tends to reduce over time [6]. In addition, over time, it is possible for the *Aedes* mosquitoes to survive and develop resistance to the chemical that is used in space spraying [7] which reduces the effectiveness of spraying in controlling the spread of dengue. An alternative way by which we can combat dengue is by using the autodissemination trap [6], which is a more proactive method as the trap contains a special solution which will lure the female *Aedes* mosquitoes to lay eggs inside the trap. Most importantly, the eggs that are laid will get killed off by the solution inside thus preventing them from hatching into adult *Aedes* mosquitoes to transmit the disease. As a result, the autodissemination trap will essentially reduce the *Aedes* population size.

In this paper, we will modify the classic Ross–Macdonald dengue model [8] to examine the effect of such an autodissemination trap called the Mosquito Home System (MHS) in controlling the spread of dengue. The MHS data used in this paper are collected from the site of the trial that took place in an environment consisting of shop houses in Section 15 of Shah Alam, the state capital of the highly dengue infected area, Selangor, Malaysia.

This paper is arranged as follows: In Section 2, we will introduce the classical Ross–Macdonald model and the basic reproduction number. We then modify the Ross–Macdonald model to twelve differential equation models which describe the spread of dengue between humans and *Aedes* mosquitoes both in Malaysia and in the trial site in Section 15 of Shah Alam, Selangor, Malaysia. We will also construct a list of different biting proportions corresponding to different times spent outside the trial site. In Section 3, we will perform thorough analysis on the effect of having different levels of MHSs on the number of dengue cases in the trial site in Shah Alam. In Section 4, we extend our results from the trial site in Shah Alam to the whole of Malaysia. A new improved model is constructed with a new basic reproduction number. The extinction condition is also derived. Lastly, in Section 5, we summarise our results. Numerical simulations produced using Euler’s method and tables of results are shown throughout this paper.

#### 2. The Modified Ross–Macdonald Dengue Model with the Effect of Autodissemination Trap

Let us start by introducing the delayed Ross–Macdonald SIR model for dengue used in [8, 9] which our modified dengue model will be based onwith initial conditions , and . , and represent, respectively, the susceptible, infected, and recovered humans, while , and denote the initial conditions for , and which represent, respectively, the susceptible, latent, and infected mosquitoes. Note that denotes the total population size for humans and represents the total population for *Aedes* mosquitoes, both constant. The biological meanings of the parameter values used in equation (1) are given in Table 1.