Computational and Mathematical Methods in Medicine

Volume 2015 (2015), Article ID 206131, 14 pages

http://dx.doi.org/10.1155/2015/206131

## Estimation of the Basic Reproductive Ratio for Dengue Fever at the Take-Off Period of Dengue Infection

^{1}Departemen Matematika, FMIPA, Institut Teknologi Bandung, Bandung, Indonesia^{2}Jurusan Matematika, FST, Universitas Nusa Cendana, Kupang, Indonesia^{3}Jurusan Matematika, FMIPA, Universitas Padjadjaran, Bandung, Indonesia

Received 22 December 2014; Revised 6 July 2015; Accepted 7 July 2015

Academic Editor: Chung-Min Liao

Copyright © 2015 Jafaruddin 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

Estimating the basic reproductive ratio of dengue fever has continued to be an ever-increasing challenge among epidemiologists. In this paper we propose two different constructions to estimate which is derived from a dynamical system of host-vector dengue transmission model. The construction is based on the original assumption that in the early states of an epidemic the infected human compartment increases exponentially at the same rate as the infected mosquito compartment (previous work). In the first proposed construction, we modify previous works by assuming that the rates of infection for mosquito and human compartments might be different. In the second construction, we add an improvement by including more realistic conditions in which the dynamics of an infected human compartments are intervened by the dynamics of an infected mosquito compartment, and vice versa. We apply our construction to the real dengue epidemic data from SB Hospital, Bandung, Indonesia, during the period of outbreak Nov. 25, 2008–Dec. 2012. We also propose two scenarios to determine the take-off rate of infection at the beginning of a dengue epidemic for construction of the estimates of : scenario I from equation of new cases of dengue with respect to time (daily) and scenario II from equation of new cases of dengue with respect to cumulative number of new cases of dengue. The results show that our first construction of accommodates the take-off rate differences between mosquitoes and humans. Our second construction of the estimation takes into account the presence of infective mosquitoes in the early growth rate of infective humans and vice versa. We conclude that the second approach is more realistic, compared with our first approach and the previous work.

#### 1. Introduction

Dengue is a mosquito-borne viral disease found in more than 100 countries around the world which are mostly located in tropical and subtropical countries. This disease is transmitted through the bites of female* Aedes aegypti*. In recent years, dengue transmission has increased predominantly in urban and semiurban areas and has become a major international public health concern. Controlling of dengue fever has been conducted continuously, but the spread of dengue virus is still increasing in many countries. Many efforts have been conducted to control the spread of the virus, for instance, a reduction in the population of* Aedes aegypti* in the field [1]. Fumigation methods have been used to reduce mosquitoes, and the use of temephos has been utilized to reduce the larvae. Until recently, there was no vaccine against any of the four virus serotypes (DEN-1, DEN-2, DEN-3, and DEN-4). To cure patients, treatment in hospitals is usually given in the form of supportive care, which includes bed rest, antipyretics, and analgesics [2]. Mathematical models have proved to be useful tools in the understanding of dengue transmission [3–5]. So far the dynamics of the dengue transmission are still an interesting issue in epidemiological modeling.

The first model of dengue transmission and the stability analysis of equilibrium points are shown in [6] in which the basic reproductive ratio is constructed from the stability condition of the disease-free equilibrium. The general concept of can be seen in [7, 8] which was adapted to various infectious disease models [9, 10]. The difficulty in using for measuring the level of dengue endemicity in the field is that often depends on several parameters (biological, environmental, transmission, etc), which are difficult to be obtained from the field. With limited information about mosquitos such as the mosquito population size the estimate of can not be computed from the dynamical model only. With daily human incidence being the only available data, it is natural to ask how to extract relevant parameters from the data to fit in the model. A simple approach has been achieved by assuming that both a human and mosquito undergo linear growth at the early state of infection. As shown in [11–14] the estimation does not distinguish between the growth rate of an infected human and the growth rate of infected mosquitoes and does not yet consider how the interaction between infected mosquitoes and infected humans influences the early growth of a dengue epidemic. There are other problems that are treated in a recent paper [15] which examines two models vector-borne infections, namely, dengue transmission. In this paper, we construct estimation by taking into account the difference between the growth rate of an infected human and infected mosquito and the effect of the interaction between them and the characteristics of dengue incidence data.

The existing data do not specify the infection status (age of infection) of each patient. Hence we assume that people who come to the hospital should be identified as dengue patient at the end of incubation period (approximately at the 7th day after contact with infected mosquito). Also, we assume that there are no available alternative hosts as blood sources and there are no death and recovery in the early days of dengue infection. The main purpose of this paper is to construct estimation of the real conditions in the field, in which only the daily incidence data are available. Based on the dengue transmission model in [6], we build two different constructions for estimation and used them for estimating the value of for dengue incidence data between the dates Nov. 25, 2008, and 2012 from SB Hospital, Indonesia.

The organization of the paper is given as follows. In Section 2 a dynamical system of a host-vector transmission model is introduced. Based on this dynamical system we derive the basic reproductive ratio of the model. The constructions of the proposed estimation are presented in Section 3. Formulations of the take-off rate of dengue infection are presented in Section 4. We apply the formula to the real data of dengue incidence from SB Hospital, Bandung, Indonesia. All the numerical results are given in Section 5 along with the insights and interpretation from the numerical results. We provide conclusions in Section 6.

#### 2. Basic Reproductive Ratio of the Dengue Transmission Model

Generally, for a vector-borne disease, was understood as the number of persons who would be infected from a single person initially infected by a mosquito [3, 16]. In the host-vector system, the basic reproductive ratio is defined as the expected number of secondary infections resulting directly from a single infected individual, in a virgin population, during the infection period [7]. The general host-vector dengue transmission model was conducted in [6]. Here, we assume that there are no alternative hosts available, as blood sources for system in [6]. Modification of the dengue transmission model in [6], is schematically represented by the diagram in Figure 1, where recruitment rate of host population; number of host populations; susceptible host population size; infected host population size; recovered/immunes host population size; successful transmission probability within host population; birth/death rate of host population; recovery rate of host population; recruitment rate of vector population; number of vector populations; susceptible vector population size; infected vector population size; transmission probability within the vector population; death rate of vector population; biting rate of vector population.The corresponding model of dengue transmission in the short period dengue infection is given in [17] as follows: