Computational and Mathematical Methods in Medicine

Computational and Mathematical Methods in Medicine / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 206131 | 14 pages | https://doi.org/10.1155/2015/206131

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

Academic Editor: Chung-Min Liao
Received22 Dec 2014
Revised06 Jul 2015
Accepted07 Jul 2015
Published25 Aug 2015

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 [35]. 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 [1114] 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:

Here, we give a short explanation for system (1) as follows. The effective contact rate to human is the average number of contacts per day that would result in infection, if the vector is infectious. Furthermore, the effective contact rate defines the average number of contacts per day that effectively transmit the infection to vectors.

First of all, we determine equilibrium points for the system (1). Because and are assumed to be constant then from system (1) ones obtain and . Consequently, we obtain a disease-free equilibrium point of system (1), which is given by and the endemic equilibrium point in the form of where

Next, we derive the basic reproductive ratio . is often conveniently found using a next generation method [7, 8, 18, 19]. We obtain the next generation matrix of system (1), at , as follows

Based on the definition in [20], NGM represents that the number of is the generation factor of dengue transmission, from mosquito to human. It means that one mosquito infects humans per unit of time during its infection period , and the number of is the generation factor of dengue transmission, from human to mosquito. This represents that one human infects mosquitoes during one’s infection period . NGM in (5) have two eigen values, that are . Therefore the basic reproductive ratio of system (1) is the largest eigen value of NGM, that is It is clear that the endemic equilibrium in (3) exists if . In the next section we propose two constructions of an estimation, at the initial growth of dengue infection, based on incidence data.

3. Construction for Estimation

Predicting a basic reproductive ratio is not easily done in the field. In the real-life situation, information about the number of mosquitoes, precisely the ratio between mosquito and human population, is limited. Based on the incidence data, we will estimate some related parameters, for the construction of the basic reproductive ratio , from dengue incidence data.

This construction is motivated by previous work in [12], which assumes that the infective growth rate is linear at the early state of infection. The exponential growth of the infective compartment, at the early state of infection, is commonly found in the estimation of the basic reproductive ratio. All estimation of , in previous works [11, 12, 14, 21, 22], have been conducted based on the assumption that varies, where is the cumulative number of dengue cases, and is called the force of infection [12, 14]. Chowell et al. [11] called the initial growth rate of dengue epidemic. Here, we will call the take-off rate of dengue infection. In the following section, we build two constructions to estimate in (6), which is derived from a dynamical system of the host-vector dengue transmission model in system (1).

First of all we assume the construction of estimation from equation (3) in [12]. The construction of estimation at is based on the assumptions that the numbers of infected human population and infected mosquito population grow exponentially at the same rate at the short time period relative to each other: where and are constant and is the take-off rate of the initial growth dengue epidemic. Moreover, the number of nonsusceptible hosts and vectors can be assumed to be negligible and by assuming (7) is “like solution” of linearized system (1) for . These assumptions are also used in [11, 14] for calculating the force of infection from the spatial data of a dengue epidemic. Next, by substituting (7) with (1), and assuming that and at the early state of an epidemic, we get Multiplying (8) and (9), and using (6), one gets the construction of the estimation as follows: which is equivalent to The value of in (11) is used to estimate in (6), where is estimated from the dengue incidence data. Note that since , then . Note also that a small increase of the value in the interval causes a significant increase of the value .

Next we introduce our construction of estimation. The first construction is derived under the assumption that, at the beginning of dengue infection, the take-off rate of the host and vector varies differently; that is where is the rate of infected mosquitoes per human index. Using the same process as in the derivation of , one obtains or, equivalently, Note that is a special case from ; that is, if then . In reality is more realistic than . From (11) and (14) we obtainThe comparison between and is shown as the level set of (15) in Figure 2.

Figure 2 shows that the variation in take-off rates might significantly change the values of and . Both estimations are derived under the assumption that the pure exponential growth phase, of an epidemic, increases or has a fluctuating increase. Accordingly the estimates of , obtained from and , are affected by biasness; that is, the derivations of and use the extreme assumption in the early dengue epidemic that the dynamics of an infected human are not yet intervened by the presence of an infected mosquito’s influence, and vice versa.

The second construction of estimation is done in order to revise the first construction. We assume that the exponential growth of the infective host, at the early state of infection, is slightly affected by the growth of the infective vector, and vice versa. The main problem here is to find the best fit interval (here we use the terminology take-off period) where exponential growth occurs. Statistically, this situation can be related to the linear phase fit in a short time period.

We define the initial growth in a period of time as the initial value problem; that is, We obtain such that the solution of (16) is obtained; namely,

The main idea from (18) is to extract the relevant parameter(s) during the early state of infection, in which linear growth rate of infection is assumed to take place. Next we substitute (18) with the expression for the derivative of and in (1) by simple algebraic manipulation; we then obtain a homogeneous system of equations in and , namely,where coefficient matrix isA nontrivial solution from (19) exists if , which gives as the coefficient of and must be equal to zero. Equivalently, from (21), we obtain Therefore we have where is the known mosquitoes per person index. We obtain (23), as a new construction for estimation , as a function of . The construction of in (23) is related to a short time interval where linear growth may still take place. Here, we will estimate the value of during the early state of infection, in which a linear growth of infection is assumed to take place.

Note that increases with respect to , . Based on the definition that is probability, per unit of time, a susceptible becomes infected at the beginning of a dengue epidemic (see [7]), with assumption that . This assumption means that the maximum take-off rate of dengue virus infection is defending on the index mosquito per human. This condition implies

ratio in the early phase of dengue epidemics could be significantly larger or smaller than one depending on the value of , in (24), and the value of . In Section 4, we determine as a function of at the beginning of a dengue epidemic. We will further analyze the level set of the ratio . Considering that the value of estimation depends on the value of take-off rate , we discuss a method to formulate the take-off rate in the following section.

4. Formulation of the Take-Off Rate

Here we derive two scenarios for the derivation of the take-off rate (t.o.r.) at the beginning of every take-off period (t.o.p.) of dengue infection. Both scenarios are based on the same assumptions, but with a slightly different implementation for real-life dengue epidemics. Therefore, the main problem here is to find the best fit interval (here we use the terminology “take-off period”) where exponential growth occurs.

The first scenario is done under the assumption that, during early infection, recovery and natural death do not yet occur. Note that the original data taken from the hospital is in the form of daily new cases. Let be the number of new cases of infection; we then have (see in [14]). Therefore, from (7), it is and, from (12), it is By using the Taylor expansion around for (25) and around for (26), we obtain respectively. Consequently, we have a relation that . Thus at in (23) can be replaced by . By comparing (11) and (23), we have Also in (24) can be replaced by such that we obtain The ratio , if and only if , and , if and only if . Because of , we have , which belongs to the range of for small .

The second scenario is conducted under the same assumption as the first scenario. But, here, we regress between the number of new cases in a day and cumulative number of new cases , where we have the initial value from (7) and from (18). Therefore, by using integral equation with from the first equation in (7) and from the first equation in (18), we obtainrespectively. It is clear that so that in at (23) can be replaced by . If (23) is divided by (11), we obtain Also in (24) can be replaced by such that we obtain We note that the ratio if and only if and if and only if . Note also that for small .

Figures 3 and 4 show the level sets for some values of the ratio , for both scenarios of construction of . It shows that for small , increases faster, with respect to , in the second scenario (Figure 4) than in the first scenario (Figure 3). Summary of the construction of estimation for the first and the second scenario of is given in Table 1.


The first scenario for estimation

Based on [12] assumption
The first construction
The second construction
The t.o.r. equation

The second scenario for estimation

Based on [12] assumption
The first construction
The second construction
The t.o.r. equation

The magnitude of in Figure 5 indicates that the second scenario is rising faster than of the first scenario at the beginning of the growing epidemic of dengue fever, although the rate of infection is very small. This means that the presence of mosquitoes in early dengue infection directly affects the human dengue infection. Furthermore, Figure 5 shows that the second scenario of construction of , which is summarized in Table 1, is increasing faster than the first scenario. Therefore, we obtain that the second scenario is a more realistic construction of than the first one at the beginning of the short period of dengue virus infection.

In the next section, we apply least square method to determine estimation of , , and from dengue incidence data based on (27) and (30). We calculated , , and by using least square method which differs from the estimation that has been conducted in [14]. They assume that the number of cases of dengue at is equal to one, but, here, we estimate from the dengue incidence data. In the next section, we examine the dependence of the basic reproductive ratio on the take-off rate, at the beginning of the take-off period of dengue infection, for daily cases of dengue from SB Hospital in Bandung Indonesia.

5. Application of Estimation to the Real-Life Dengue Epidemic

This section presents the application of the method to the data of dengue incidence from SB hospital, the estimation value of the t.o.r. , the value basic reproductive ratio, and their implication. Calculation of is based on the assumption that at the beginning of the infection natural death and recovery have not yet occurred. This assumption can be taken before the eighth day of incidence. Data is divided into five take-off periods (t.o.p.) of infections, which is represented inside the solid box in Figure 6. Each t.o.p. contains an initial take-off period (i.t.o.p.) of dengue infection. Here, we define the i.t.o.p. as being within the range of the fourth and the seventh days of the dengue incidence. The correspondents of the bioepidemiological parameter from human and mosquito are given in Table 2.


Symbol Definition Value

Biting rate (day−1)1
Effective contact rate to human0.01
Effective contact rate to mosquito0.25
Human mortality rate (day−1)
Mosquito mortality rate (day−1)
Recovering rate of human (day−1)
Mosquito per human index 0–7.96
Proportion of mosquito per human index0–2

Figure 6 shows the five possibilities for the t.o.p. The t.o.p. is defined by the minimum to maximum values of -square (). The number of days for every i.t.o.p. of incidence data depends on the value of criteria (approximately at the 4–7th day after contact with the infected mosquito). Meanwhile, the i.t.o.p. is defined by the minimum to maximum values of before the eighth day. Next we determine the rate of the i.t.o.p. of dengue infection (t.o.r.) for every t.o.p. The value of is calculated at intervals of four to seven days for the dengue of dengue infection by using two different scenarios in respect to the incidence data and the value of parameter in Table 2.

In the first scenario we calculate the values of and by using least square method for (27), and, in the second scenario, we calculate the values of and by using least square method for (30), for every i.t.o.p. Those values of , , and are summarized in Table 3.


Take-off periodInitial take-off period The values of and for the first scenarioThe values of and for the second scenario

I: 103 daysI: 5 days , ,
II: 187 daysII: 6 days, ,
III: 13 daysIII: 6 days, ,
IV: 17 daysIV: 5 days, ,
V: 85 daysV: 7 days, ,

Using take-off rate values of infection in Table 3, we calculate the value of , , , and , which are given in Tables 47 as well as the intervals of , , , and and their important notes, which are given in Tables 813.


The days at the i.t.o.p.

II: 7 days
IV: 5 days
I: 5 days
V: 7 days
III: 6 days


The days at the i.t.o.p.

II: 7 days
IV: 5 days
I: 5 days
V: 7 days
III: 6 days


The days at the i.t.o.p.

II: 7 days
IV: 5 days
I: 5 days
V: 7 days
III: 6 days