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

Authors: Jafaruddin; Sapto W. Indratno; Nuning Nuraini; Asep K. Supriatna; Edy Soewono
Journal: Computational and Mathematical Methods in Medicine
 (2015)

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## Abstract

Estimating the basic reproductive ratioR0 of dengue fever has continued to be an ever-increasing challenge among epidemiologists. In this paper we propose two different constructions to estimate R0 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 R0: 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 R0 accommodates the take-off rate differences between mosquitoes and humans. Our second construction of the R0 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.


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## Body

## 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 femaleAedes 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 ofAedes 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 R0 is constructed from the stability condition of the disease-free equilibrium. The general concept of R0 can be seen in [7, 8] which was adapted to various infectious disease models [9, 10]. The difficulty in using R0 for measuring the level of dengue endemicity in the field is that R0 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 R0 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 R0 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 constructR0 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 R0 estimation and used them for estimating the value of R0 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 Section2 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 R0 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,R0 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 R0 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 m=0 for system in [6]. Modification of the dengue transmission model in [6], is schematically represented by the diagram in Figure 1, whereAh: recruitment rate of host population;Nh: number of host populations;Sh: susceptible host population size;Ih: infected host population size;Rh: recovered/immunes host population size;ph: successful transmission probability within host population;μh: birth/death rate of host population;γ: recovery rate of host population;Av: recruitment rate of vector population;Nv: number of vector populations;Sv: susceptible vector population size;Iv: infected vector population size;pv: transmission probability within the vector population;μv: death rate of vector population;b: biting rate of vector population.The corresponding model of dengue transmission in the short period dengue infection is given in [17] as follows:(1)dShdt=Ah-bphShIvNh-μhSh,dIhdt=bphShIvNh-γIh-μhIh,dRhdt=γIh-μhRh,dSvdt=Av-bpvSvIhNh-μvSv,dIvdt=bpvSvIhNh-μvIv.Figure 1
The diagram of dengue transmission model.Here, we give a short explanation for system (1) as follows. The effective contact rate to human bph is the average number of contacts per day that would result in infection, if the vector is infectious. Furthermore, the effective contact rate bpv 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 Nh and Nv are assumed to be constant then from system (1) ones obtain Sh+Ih+Rh=Nh=Ah/μh and Sv+Iv=Nv=Av/μv. Consequently, we obtain a disease-free equilibrium point of system (1), which is given by (2)E0=Sh0=Ahμh,Ih0=0,Rh0=0,Sv0=Avμv,Iv0=0,and the endemic equilibrium point in the form of (3)E1=Sh∗,Ih∗,Rh∗,Sv∗,Iv∗,where(4)Sh∗=μvγ+μhNh2+μvNhNv2phμhNh+bphNv,Ih∗=μvμhNhb2phpv/μvγ+μhNv/Nh-1bpvμh+bphρ,Rh∗=μvNhγb2phpv/μvγ+μhNv/Nh-1bpvμh+bphρ,Sv∗=phμhNh+bpvphNvbph1+bpvμh,Iv∗=μvμhNhγ+μhb2phpv/μvγ+μhNv/Nh-1bphμvγ+μh+μhbpv.Next, we derive the basic reproductive ratioR0. R0 is often conveniently found using a next generation method [7, 8, 18, 19]. We obtain the next generation matrix of system (1), at E0, as follows (5)NGM=0bphμvbpvγ+μhNvNh0.Based on the definition in [20], NGM represents that the number of bph/μv is the generation factor of dengue transmission, from mosquito to human. It means that one mosquito infects bph humans per unit of time during its infection period 1/μv, and the number of (bpv/γ+μh)(Nv/Nh) is the generation factor of dengue transmission, from human to mosquito. This represents that one human infects bpv(Nv/Nh) mosquitoes during one’s infection period 1/(μh+γ). NGM in (5) have two eigen values, that are ±(b2phpv/μvγ+μh)(Nv/Nh). Therefore the basic reproductive ratio of system (1) is the largest eigen value of NGM, that is (6)R0=b2phpvμvγ+μhNvNh.It is clear that the endemic equilibrium E1 in (3) exists if R0>1. In the next section we propose two constructions of an R0 estimation, at the initial growth of dengue infection, based on incidence data.


## 3. Construction forR0 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 ratioR0, 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 R0, in previous works [11, 12, 14, 21, 22], have been conducted based on the assumption that K(t)∝eλt varies, where K(t) is the cumulative number of dengue cases, and λ is called the force of infection [12, 14]. Chowell et al. [11] called λ the initial growth rate of dengue epidemic. Here, we will call λ the take-off rate of dengue infection. In the following section, we build two constructions to estimate R0 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 ofR0 estimation from equation (3) in [12]. The construction of R0 estimation at E0 is based on the assumptions that the numbers of infected human population Ih and infected mosquito population Iv grow exponentially at the same rate at the short time period relative to each other: (7)Iht≈Ih0eλt,Ivt≈Iv0eλt,where Ih0 and Iv0 are constant and λ 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 R0>1. 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 Sh~Nh and Vv~NV at the early state of an epidemic, we get (8)λγ+μh+1Ih0=bphγ+μhIv0,(9)λμv+1Iv0=bpvμvNvNhIh0.Multiplying (8) and (9), and using (6), one gets the construction of the R0 estimation as follows: (10)R0F2=λμh+γ+1λμv+1=b2phpvμvμh+γNvNh=R02,which is equivalent to (11)R0F=λ2μvμh+γ+μv+μh+γλμvμh+γ+1.The value of R0F in (11) is used to estimate R0 in (6), where λ is estimated from the dengue incidence data. Note that since λ>0, then R0F>1. Note also that a small increase of the value λ in the interval 0<λ<1 causes a significant increase of the value R0F.Next we introduce our construction ofR0 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 (12)Iht≈Ih0eλt,Ivt≈Iv0ekλt,where k>0 is the rate of infected mosquitoes per human index. Using the same process as in the derivation of R0F, one obtains (13)R0MF2=λμh+γ+1kλμv+1=b2phpvμvμh+γNvNh=R02or, equivalently, (14)R0MF=kλ2μvμh+γ+μv+kμh+γλμvμh+γ+1.Note that R0F is a special case from R0MF; that is, if k=1 then R0F=R0MF. In reality R0MF is more realistic than R0F. From (11) and (14) we obtain(15)R0MF2R0F2=1+k-1λλ+μv.The comparison between R0MF and R0F is shown as the level set of (15) in Figure 2.Figure 2
Level set forR0MF2/R0F2 in (15) for data set {μv=1/30,λ∈[0,0.20],k∈[0,20]}.Figure2 shows that the variation in take-off rates might significantly change the values of R0F and R0MF. 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 R0, obtained from R0F and R0MF, are affected by biasness; that is, the derivations of R0F and R0MF 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 ofR0 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,(16)Iht=Iv0aeηt+be-ηt+Ih0ceηt+de-ηt,Ivt=Ih0peηt+qe-ηt+Iv0reηt+se-ηt,Ih0,Iv0=Ih0,Iv0,Ih′0,Iv′0=ηIv0,ηIh0.We obtain (17)
a
=
1
2
,
b
=
1
2
,
c
=
1
2
,
d
=
1
2
,
p
=
1
2
,
q
=
1
2
,
r
=
1
2
,
s
=
1
2such that the solution of (16) is obtained; namely, (18)Iht=Ih0coshηt+Iv0sinhηt,Ivt=Iv0coshηt+Ih0sinhηt.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 Ih and Iv in (1) by simple algebraic manipulation; we then obtain a homogeneous system of equations in cosh⁡(ηt) and sinh⁡(ηt), namely,(19)Acoshηtsinhηt=00,where coefficient matrix is(
20
)A=Ih0η-bphIh0+γ+μhIv0Iv0η-bphIv0+γ+μhIh0Iv0η-bpvρIv0+μvIh0Ih0η-bpvρIh0+μvIv0.A nontrivial solution from (19) exists if |A|=0, which gives (21)η2Nh-bpvNv+Nhbphη-μvγ+μhNh+b2pvNvphas the coefficient of Ih02 and Iv02 must be equal to zero. Equivalently, from (21), we obtain (22)R0A2=-η2μvγ+μh+bph+pvNv/Nhμvγ+μhη+1=b2phpvμvμh+γNvNh=R02.Therefore we have (23)R0A=-η2μvγ+μh+bph+pvρμvγ+μhη+1,where ρ=Nv/Nh is the known mosquitoes per person index. We obtain (23), as a new construction for estimation R0, as a function of η. The construction of R0 in (23) is related to a short time interval where linear growth may still take place. Here, we will estimate the value of η during the early state of infection, in which a linear growth of infection is assumed to take place.Note thatR0A increases with respect to η, 0≤η≤b(ph+pvρ)/2. Based on the definition that η is probability, per unit of time, a susceptible becomes infected at the beginning of a dengue epidemic (see [7]), with assumption that b(ph+pvρ)/2<1. This assumption means that the maximum take-off rate of dengue virus infection is defending on the index mosquito per human. This condition implies(24)2η-bphbpv≤ρ≤2-bphbpv.R
0
A
/
R
0
F ratio in the early phase of dengue epidemics could be significantly larger or smaller than one depending on the value of ρ, in (24), and the value of λ. In Section 4, we determine η as a function of λ at the beginning of a dengue epidemic. We will further analyze the level set of the ratio R0A/R0F. Considering that the value of R0 estimation depends on the value of take-off rate λ, 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. LetI(t) be the number of new cases of infection; we then have I(t)=dIh/dt (see in [14]). Therefore, from (7), it is (25)It=λIh0eλt,withI0=I0=λIh0and, from (12), it is (26)It=ηIv0cosh⁡ηt+Ih0sinh⁡ηt,withI0=I0=ηIv0.By using the Taylor expansion around λ=0 for (25) and around η=0 for (26), we obtain (27)It≈λI0t+I0,It≈η2λI0t+I0,respectively. Consequently, we have a relation that λ=η. Thus η at R0A in (23) can be replaced by λ. By comparing (11) and (23), we have (28)R0A2R0F2=1+λλ+μvbph+pvρ-λλ+μh+γ-1.Also η in (24) can be replaced by λ such that we obtain (29)2λ-bphbpv≤ρ≤2-bphbpv.The ratio R0A2/R0F2≥1, if and only if ρ≥(2λ+μh+γ-bph)/bpv, and R0A2/R0F2<1, if and only if ρ<(2λ+μh+γ-bph)/bpv. Because of (2λ-bph)/bpv<(2λ+μh+γ-bph)/bpv, we have (2λ+μh+γ-bph)/bpv, which belongs to the range of ρ for small λ.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 dayI(t)=dIh/dt and cumulative number of new cases K(t), where we have the initial value K(0)=K0=I0=I(0)=λIh0 from (7) and K(0)=K0=I0=I(0)=ηIv0 from (18). Therefore, by using integral equation K(t)-K(0)=∫0t‍I(τ)dτ with I(τ) from the first equation in (7) and I(τ) from the first equation in (18), we obtain(30)It=λKt+1-λK0,It=η2Kt+1-η2K0,respectively. It is clear that η2=λ so that η in R0A at (23) can be replaced by λ. If (23) is divided by (11), we obtain (31)R0A2R0F2=1+λλ+μvbph+pvρ-λλ+μh+γ-λ.Also η in (24) can be replaced by λ such that we obtain (32)2λ-bphbpv≤ρ≤2-bphbpv.We note that the ratio R0A2/R0F2≥1 if and only if ρ≥(λ3/2+λ1/2(μv+μh+γ+1)-bph)/bpv=ρ0 and R0A2/R0F2<1 if and only if ρ<ρ0. Note also that ρ0<(2-bph)/bpv for small λ.Figures3 and 4 show the level sets for some values of the ratio R0A2/R0F2, for both scenarios of construction of λ. It shows that for small λ, R0A2/R0F2 increases faster, with respect to ρ, in the second scenario (Figure 4) than in the first scenario (Figure 3). Summary of the construction of R0 estimation for the first and the second scenario of λ is given in Table 1.Table 1
Summary of the construction ofR0 and λ estimation for the first scenario.
The first scenario forλ
R
0 estimation
Based on [12] assumption
R
0
F
2
=
λ
2
μ
v
μ
h
+
γ
+
μ
v
+
μ
h
+
γ
λ
μ
v
μ
h
+
γ
+
1
The first construction
R
0
M
F
2
=
k
λ
2
μ
v
μ
h
+
γ
+
μ
v
+
k
μ
h
+
γ
λ
μ
v
μ
h
+
γ
+
1
The second construction
R
0
A
2
=
-
λ
2
μ
v
γ
+
μ
h
+
b
p
h
+
p
v
ρ
μ
v
γ
+
μ
h
λ
+
1
The t.o.r.λ equation
I
(
t
)
=
λ
I
0
t
+
I
0
The second scenario forλ
R
0 estimation
Based on [12] assumption
R
0
F
2
=
λ
2
μ
v
μ
h
+
γ
+
μ
v
+
μ
h
+
γ
λ
μ
v
μ
h
+
γ
+
1
The first construction
R
0
M
F
2
=
k
λ
2
μ
v
μ
h
+
γ
+
μ
v
+
k
μ
h
+
γ
λ
μ
v
μ
h
+
γ
+
1
The second construction
R
0
A
2
=
-
λ
μ
v
γ
+
μ
h
+
b
p
h
+
p
v
ρ
μ
v
γ
+
μ
h
λ
+
1
The t.o.r.λ equation
I
(
t
)
=
λ
K
(
t
)
+
(
1
-
λ
)
K
0Figure 3
Level setR0A2/R0F2 in (28) for data set {b=1,ph=0.01,pv=0.25,γ=1/8,μh=1/(3×70),μv=1/30,λ∈[0,0.20],ρ∈[0,20]}.Figure 4
Level setR0A2/R0F2 in (31) for data set {b=1,ph=0.01,pv=0.25,γ=1/8,μh=1/(3×70),μv=1/30,λ∈[0,0.20],ρ∈[0,10]}.The magnitude ofR0A in Figure 5 indicates that the second scenario is rising faster than R0A 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 R0, 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 R0 than the first one at the beginning of the short period of dengue virus infection.Figure 5
The comparison ofR0 estimation.In the next section, we apply least square method to determine estimation ofλ, I0, and K0 from dengue incidence data based on (27) and (30). We calculated λ, I0, and K0 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 t=0 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 ofR0 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.λ, the value basic reproductive ratio, and their implication. Calculation of λ 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.Table 2
Bioepidemiological data of human and mosquito.
Symbol
Definition
Value
b
Biting rate (day−1)
1
p
h
Effective contact rate to human
0.01
p
v
Effective contact rate to mosquito
0.25
μ
h
Human mortality rate (day−1)
1
365
×
70
μ
v
Mosquito mortality rate (day−1)
1
30
γ
Recovering rate of human (day−1)
1
8
ρ
Mosquito per human index
0–7.96
k
Proportion of mosquito per human index
0–2Figure 6
A time series of the number of daily new cases of dengue infection in the SB Hospital is from Nov. 05, 2008, to Dec. 2012. The cut off period of new cases of dengue is inside the solid box. Every t.o.p. contains i.t.o.p. of dengue infection.Figure6 shows the five possibilities for the t.o.p. The t.o.p. is defined by the minimum to maximum values of r-square (r2). The number of days for every i.t.o.p. of incidence data depends on the value of r2 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 r2 before the eighth day. Next we determine the rate of the i.t.o.p. of dengue infection (t.o.r.) λ for every t.o.p. The value of λ 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 ofλ and I0 by using least square method for (27), and, in the second scenario, we calculate the values of λ and K0 by using least square method for (30), for every i.t.o.p. Those values of λ, I0, and K0 are summarized in Table 3.Table 3
The value ofλ and I0 from the first scenario and the value of λ and K0 for the second scenario, respectively.
Take-off period
Initial take-off period
The values ofλ and I0 for the first scenario
The values ofλ and K0 for the second scenario
I: 103 days
I: 5 days
λ
=
0.093,    I0=5.4
λ
=
0.091,  K0=5.22
II: 187 days
II: 6 days
λ
=
0.009,   I0=11.53
λ
=
0.019,  K0=11.17
III: 13 days
III: 6 days
λ
=
0.190,   I0=3.62
λ
=
0.121,  K0=3.78
IV: 17 days
IV: 5 days
λ
=
0.038,   I0=16
λ
=
0.048,  K0=15.54
V: 85 days
V: 7 days
λ
=
0.171,   I0=2.93
λ
=
0.105,  K0=3.1Using take-off rate values of infectionλ in Table 3, we calculate the value of ρ, R0F, R0MF, and R0A, which are given in Tables 4–7 as well as the intervals of ρ^, R^0MF, R^0, and R^0A and their important notes, which are given in Tables 8–13.Table 4
The values ofR0F for a given value of λ at the i.t.o.p. and the corresponding values of R0MF for different values of k for the first scenario.
Then days at the i.t.o.p.
λ
R
0
F
R
0
M
F
R
0
M
F
R
0
M
F
R
0
M
F
R
0
M
F
k
=
0
k
=
0.75
k
=
1
k
=
1.25
k
=
2
II: 7 days
0.009
[
1.17
]
1.04
1.14
[
1.17
]
1.20
1.29
IV: 5 days
0.038
[
1.66
]
1.14
1.55
[
1.66
]
1.77
2.06
I: 5 days
0.093
[
2.56
]
1.32
2.32
[
2.56
]
2.79
3.38
V: 7 days
0.171
[
3.80
]
1.54
3.38
[
3.80
]
4.18
5.16
III: 6 days
0.190
[
3.89
]
1.59
3.64
[
3.89
]
4.52
5.58Table 5
The values ofR0F for a given value of λ at the i.t.o.p. and the corresponding values of R0MF for different values of k for the second scenario.
Then days at the i.t.o.p.
λ
R
0
F
R
0
M
F
R
0
M
F
R
0
M
F
R
0
M
F
R
0
M
F
k
=
0
k
=
0.75
k
=
1
k
=
1.25
k
=
2
II: 7 days
0.019
[
1.35
]
1.07
1.29
[
1.35
]
1.41
1.58
IV: 5 days
0.048
[
1.83
]
1.17
1.69
[
1.83
]
1.96
2.30
I: 5 days
0.091
[
2.54
]
1.32
2.30
[
2.54
]
2.77
3.35
V: 7 days
0.105
[
2.76
]
1.36
2.49
[
2.76
]
3.01
3.66
III: 6 days
0.121
[
3.02
]
1.40
2.71
[
3.02
]
3.30
4.03Table 6
The values ofR0F for every value of λ at each initial t.o.p. and the values of R0A for every change in the value of λ and ρ for the first scenario for calculation of λ.
Then days at the i.t.o.p.
λ
R
0
F
R
0
A
R
0
A
R
0
A
R
0
A
R
0
A
ρ
=
0.67
ρ
=
0.89
ρ
=
1.33
ρ
=
1.96
ρ
=
2.11
II: 7 days
0.009
[
1.17
]
[
1.17
]
1.22
1.32
1.45
1.48
IV: 5 days
0.038
[
1.66
]
1.50
[
1.66
]
1.94
2.27
2.34
I: 5 days
0.093
[
2.56
]
1.70
2.03
[
2.56
]
3.17
3.30
V: 7 days
0.171
[
3.80
]
1.12
1.89
2.84
[
3.80
]
4.00
III: 6 days
0.190
[
4.10
]
0.67
1.73
2.83
3.89
[
4.10
]Table 7
The values ofR0F for every value of λ at each initial t.o.p. and the values of R0A for every change in the value of λ and ρ for the second scenario for calculating of λ.
Then days at the i.t.o.p.
λ
R
0
F
R
0
A
R
0
A
R
0
A
R
0
A
R
0
A
ρ
=
0.61
ρ
=
1.01
ρ
=
1.47
ρ
=
1.60
ρ
=
1.74
II: 7 days
0.019
[
1.35
]
[
1.35
]
2.27
2.99
3.16
3.34
IV: 5 days
0.048
[
1.83
]
—
[
1.83
]
3.06
3.32
3.59
I: 5 days
0.091
[
2.54
]
—
—
[
2.54
]
2.96
3.37
V: 7 days
0.105
[
2.76
]
—
—
2.27
[
2.76
]
3.23
III: 6 days
0.121
[
3.02
]
—
—
1.87
2.48
[
3.02
]Table 8
Summary of theR^0 and R^0MF for the first scenario of the value of λ and ρ in (29).
Then days at the i.t.o.p.
λ
ρ
^
R
^
0
M
F
R
^
0
Important notes
from (29)
fork∈(0,2]
II: 7 days
0.009
(
0.034,7.96
)
(
1.04,1.29
]
(
0.02,4.77
)
R
0
F
∈
R
^
0
M
F
⊂
R
^
0
IV: 5 days
0.038
(
0.26,7.96
)
(
1.14,2.06
]
(
0.16,4.77
)
R
0
F
∈
R
^
0
M
F
⊂
R
^
0
I: 5 days
0.093
(
0.701,7.96
)
(
1.32,3.38
]
(
0.42,4.77
)
R
0
F
∈
R
^
0
M
F
⊂
R
^
0
V: 7 days
0.171
(
1.33,7.96
)
(
1.54,5.16
]
(
0.80,4.77
)
R
0
F
∈
R
^
0
∩
R
^
0
M
F
III: 6 days
0.190
(
1.48,7.96
)
(
1.59,5.58
]
(
0.89,4.77
)
R
0
F
∈
R
^
0
∩
R
^
0
M
F
Note. The value of R0F from Table 6.Table 9
Summary of theR^0 and R^0MF for the second scenario of the value of λ and ρ in (32).
Then days at the i.t.o.p.
λ
ρ
^
R
^
0
R
^
0
M
F
Important notes
from (32)
fork∈(0,2]
II: 7 days
0.019
(
1.07,7.96
)
(
0.64,4.77
)
(
1.07,1.58
)
R
0
F
∈
R
^
0
M
F
⊂
R
^
0
IV: 5 days
0.048
(
1.71,7.96
)
(
1.02,4.77
)
(
1.17,2.30
)
R
0
F
∈
R
^
0
M
F
⊂
R
^
0
I: 5 days
0.091
(
2.37,7.96
)
(
1.43,4.77
)
(
1.32,3.35
)
R
0
F
∈
R
^
0
∩
R
^
0
M
F
V: 7 days
0.105
(
2.55,7.96
)
(
1.53,4.77
)
(
1.36,3.66
)
R
0
F
∈
R
^
0
∩
R
^
0
M
F
III: 6 days
0.121
(
2.74,7.96
)
(
1.65,4.77
)
(
1.40,4.03
)
R
0
F
∈
R
^
0
∩
R
^
0
M
F
Note. The value of R0F from Table 7.Table 10
Summary of theR0 and R0A for the first scenario of the value of λ and ρ in (29).
Then days at the i.t.o.p.
λ
ρ
^
R
^
0
R
^
0
A
Important notes
from (29)
II: 7 days
0.009
(
0.034,7.96
)
(
0.02,4.77
)
(
1.01,2.33
)
R
0
F
∈
R
^
0
A
⊂
R
^
0
IV: 5 days
0.038
(
0.26,7.96
)
(
0.16,4.77
)
(
1.16,4.32
)
R
0
F
∈
R
^
0
A
⊂
R
^
0
I: 5 days
0.093
(
0.701,7.96
)
(
0.42,4.77
)
(
1.75,6.59
)
R
0
F
∈
R
^
0
∩
R
^
0
A
V: 7 days
0.171
(
1.33,7.96
)
(
0.80,4.77
)
(
2.83,8.71
)
R
0
F
∈
R
^
0
∩
R
^
0
A
III: 6 days
0.190
(
1.48,7.96
)
(
0.89,4.77
)
(
3.10,9.13
)
R
0
F
∈
R
^
0
∩
R
^
0
A
Note. The value of R0F from Table 6.Table 11
Summary of theR^0 and R^0A for the second scenario of the value of λ and ρ in (32).
Then days at the i.t.o.p.
λ
ρ
^
R
^
0
R
^
0
A
Important notes
from (32)
II: 7 days
0.019
(
1.07,7.96
)
(
0.64,4.77
)
(
2.37,7.93
)
R
0
F
∈
R
^
0
∖
R
^
0
A
IV: 5 days
0.048
(
1.704,7.96
)
(
1.02,4.77
)
(
3.52,9.70
)
R
0
F
∈
R
^
0
∖
R
^
0
A
I: 5 days
0.091
(
2.37,7.96
)
(
1.43,4.77
)
(
4.79,11.14
)
R
0
F
∈
R
^
0
∖
R
^
0
A
V: 7 days
0.105
(
2.55,7.96
)
(
1.53,4.77
)
(
5.12,11.46
)
R
0
F
∈
R
^
0
∖
R
^
0
A
III: 6 days
0.121
(
2.74,7.96
)
(
1.65,4.77
)
(
5.48,11.78
)
R
0
F
∈
R
^
0
∖
R
^
0
A
Note. The value of R0F from Table 7.Table 12
The value ofR0F is related to R^0MF, R^0, and R^0A for the first scenario.
Then days  at the i.t.o.p.
λ
R
0
F
R
^
0
M
F
Important notes
fork∈(0,2]
II: 7 days
0.009
1.17
(
1.04,1.29
]
R
0
F
∈
R
^
0
M
F
∩
R
^
0
∩
R
^
0
A
IV: 5 days
0.038
1.66
(
1.14,2.06
]
R
0
F
∈
R
^
0
M
F
∩
R
^
0
∩
R
^
0
A
I: 5 days
0.093
2.56
(
1.32,3.38
]
R
0
F
∈
R
^
0
M
F
∩
R
^
0
∩
R
^
0
A
V: 7 days
0.171
3.80
(
1.54,5.16
]
R
0
F
∈
R
^
0
M
F
∩
R
^
0
∩
R
^
0
A
III: 6 days
0.190
3.89
(
1.59,5.58
]
R
0
F
∈
R
^
0
M
F
∩
R
^
0
∩
R
^
0
A
Note. R^0A and R^0 are from Table 10.Table 13
The value ofR0F is related to R^0MF, R^0, and R^0A for the second scenario.
Then days  at the i.t.o.p.
λ
R
0
F
R
^
0
M
F
Important notes
fork∈(0,2]
II: 7 days
0.019
1.35
(
1.07,1.58
]
R
0
F
∈
R
^
0
M
F
∩
R
^
0
∖
R
^
0
A
IV: 5 days
0.048
1.83
(
1.17,2.30
]
R
0
F
∈
(
R
^
0
M
F
∩
R
^
0
)
∖
R
^
0
A
I: 5 days
0.091
2.54
(
1.32,3.35
]
R
0
F
∈
(
R
^
0
M
F
∩
R
^
0
)
∖
R
^
0
A
V: 7 days
0.105
2.76
(
1.36,3.66
]
R
0
F
∈
(
R
^
0
M
F
∩
R
^
0
)
∖
R
^
0
A
III: 6 days
0.121
3.02
(
1.40,4.03
]
R
0
F
∈
(
R
^
0
M
F
∩
R
^
0
)
∖
R
^
0
A
R
^
0
A and R^0 are from Table 11.The interpretation of the result in Tables9–13, for instance, on R0F at t.o.p. III, is that each infective person infected 3.89 other persons via the mosquito vector at the start of the epidemic when the population was susceptible. It is similar meaning for other constructions of R0 estimation. The maximum basic reproductive ratio, R0F found was 4.10, R0MF found was 5.58, and R0A found was 9.13, which should be compared with the maximum value which was obtained  103 in [12]. Furthermore, R0F is a special case for both R0MF and R0A, in terms of magnitude. Furthermore, R0MF magnitude is a special case of R0A magnitude. R0A magnitude therefore generalized both R0F and R0MF magnitudes. Table 4 shows that the value of R0F increases with an increasing λ. Also, R0MF increases with increasing λ and k. The special cases are R0F=R0MF for k=1, R0MF<R0F for k<1, and R0F<R0MF for k>1. Thus, if we have the value of k from the real-life situation, then the value of R0MF is more realistic for estimating R0 rather than R0F.From Tables4 and 5, with respect to R0F, we obtain that the average secondary case of dengue infection is at most four persons during the time period of dengue infection. Meanwhile, the value of R0MF depends on the value of k. We show that for k=1, then R0F=R0MF (values in bracket). The increases in the value of λ and k cause an increase in the value of R0MF. While the value in the square brackets in Tables 5 and 7 shows the relationship between the value of λ and ρ, so that R0F=R0A. We also found that an increase in the value of λ, for a particular value of ρ, does not always cause an increase in the value of R0A. Particular information for the four t.o.p. in Tables 6 and 7 shows that, to obtain the four secondary infection cases from one primary infection, the number of mosquitoes should be at least twice as large as the human population, whereas in Tables 4 and 5 we do not obtain information about the ratio of the number of mosquitoes and humans. Therefore, R0A provides more complete information than R0F and R0MF.Next, From Tables6 and 7, there is the value of ρ such that R0F=R0A for every dengue infection i.t.o.p. (see the value in brackets of Tables 6 and 7). For k=1, there are a certain pair of ρ and λ for every i.t.o.p., such that R0A=R0MF=R0F (Table 4). For the first scenario, if ρ is relatively small then the value of R0A is inversely proportional to the value of λ. For instance, if ρ=0.67 (see t.o.p. II, Table 6) then R0A increases or decreases with an increasing λ. The magnitude relation between λ and R0 is not the case in other values of ρ. Also, we obtained that R0F is a special case of R0A for a certain pair of ρ and λ. Thus, if we have the value of ρ from the real-life situation then the value of R0A is more realistic than R0F when estimating R0 from the real-life dengue epidemic.Next, in the first scenario, the value ofR0F is not only contained in R^0MF and R^0A but also contained within R^0. Also, the value of R0F belongs to the intersection of R^0MF and R^0A. This is also the case in the first scenario, where R0F have good precision but not as good a predictor when compared with R^0MF and R^0A. Whilst R^0MF and R^0A are good predictors, they do not have a good precision (see Tables 7–13). In the second scenario, the value of R^0F is contained within the range of R^0MF and R^0 values but not contained in R^0A range. Furthermore, the value of R0F is lower than range’s lower end of values of R^0A (see Tables 10–13). Meanwhile, it is not enough just to pay attention to R^0 value of the model in (1). Therefore, from the aspect of the efficiency and effectiveness of dengue control, the government should pay attention to the basic reproductive ratio estimation of R0A that is reported here, in order to improve the quality control of the spread of dengue fever.The important notes in Tables10–13 show that R0F value (see Table 4 or Tables 5–7) is included in the intervals R^0MF, R^0A, and R^0 for the first scenario. It means that the value of R0F can still be used as a standard for the control of dengue virus infection. However, to obtain the maximum value of R0A=2-12, then the number of mosquitoes would be 2–8 times greater than the number of people, with the take-off rate of 0.121; see Table 11. Therefore, the use of value of ρ, λ, and R0A as a standard value for the control of a dengue endemicity is more realistic than the others.From Figures7–14, not only do we obtain the value of the basic reproductive ratio estimation of the R0A which is greater than one but also we do obtain the value which is less than one; for instance, from the first scenario at λ=0.190 and ρ=0.67 we have R0A=0.67<1 (see Table 7). Also, we can determine the intersection range of a basic reproductive ratio from a different construction of value of λ. Both proposed constructions of R0 are the same when using assumption in [12] for these particular cases. But increasing the number of secondary cases for every t.o.p. will depend on k for the first construction and depend on ρ for the second construction. These results indicate that the greater the number of mosquitoes means the higher the number of new cases of dengue and the shorter the period of the i.t.o.p. of dengue infection. Furthermore, R0F estimates of R0 are too low when compared to R0A. This is very dangerous in application, since the intervention to control dengue that results from R0F estimation will not be able to stop the disease once it exists. Therefore, prevention of dengue incidences is more appropriate if it is done early, and exactly how early is dependent on R0A estimation.Figure 7
The dynamic ofR0MF for the dengue outbreak at the beginning of the t.o.p. as a function of the infected mosquito per human index for λ from the first scenario.Figure 8
The ratio ofR0MF/R0F for the dengue outbreak at the beginning of the t.o.p. as a function of the ratio of the rate of the infected mosquito per human index k for λ from the first scenario.Figure 9
The dynamic ofR0A for the dengue outbreak at the beginning of the t.o.p. as a function of the mosquito per person index for λ from the first scenario.Figure 10
The ratio ofR0A/R0F for the dengue outbreak at the beginning of the t.o.p. as a function of the infected mosquito per human index ρ for λ from the first scenario.Figure 11
The dynamic ofR0MF for the dengue outbreak at the beginning of the t.o.p. as a function of the infected mosquito per human index for λ from the first Scenario.Figure 12
The ratio ofR0MF/R0F for the dengue outbreak at the beginning of the t.o.p. as a function of ratio of the rate of the infected mosquito per human index k for λ from the second scenario.Figure 13
The dynamic ofR0A for the dengue outbreak at the beginning of the t.o.p. as a function of the mosquito per person index for λ from the second scenario.Figure 14
The ratio ofR0A/R0F for the dengue outbreak at the beginning of the t.o.p. as a function of the infected mosquito per human index ρ for λ from the second scenario.We used real dengue epidemic data from a dengue epidemic, between the dates Nov. 25, 2008, and 2012 from SB Hospital. This includes the estimation of the basic reproductive ratio at the beginning of t.o.p. of dengue infection by usingR0F, R0MF, and R0A for R0 estimation. R0 magnitude estimation under the same assumption was conducted in [12], resulting in R0F. We propose R0MF and R0A for comparing with R0F for R0 estimation from dengue incidence data from SB Hospital. As a results, we get R0MF>1.59 from the first scenario (Table 4, Figures 7 and 9) and R0MF>1.40 from the second scenario, respectively (Table 6, Figures 7 and 9). From the first scenario, we obtained the the basic reproductive ratio was lowest in the t.o.p. II [1.01,2.33) and largest in the t.o.p. III [3.10,9.13) (Table 10, Figures 8 and 10). Furthermore, from the second scenario, the basic reproductive ratio was lowest in the t.o.p. II [2 .37,7.93) and largest in the t.o.p. III [5.48,11.78) (Table 11, Figures 8 and 10). We believe that R0MF and R0A are a good R0 estimation at the beginning of the take-off period of infection. Thus, our results can be considered in the control of a dengue fever epidemic, in the dengue endemic regions of Bandung, Indonesia. But the effects of dengue incidence data were obtained from SB Hospital, Bandung, to the value of R0 estimation, and describe the transmission of dengue fever, because only the dengue incidence data of the sameserotype were considered.Some of the results of the basic reproductive ratio estimation for dengue fever in compartmental model were reported in several publications. Favier et al. in [12] estimated a basic reproductive ratio, which ranged widely from 2.0 to 103 for dengue epidemics occurring during the years 1996 to 2003 in 9 Brazilian regions. The upper bound of their range, in estimating the basic reproductive ratio, is significantly higher than the upper bound estimate reported here. The lower bound of their range estimate, of basic reproductive ratio, is relatively smaller than the lower bound estimate reported here for some take-off period but the others are relatively higher than the lower bound estimate reported here (Tables 4 and 6). Chowell et al. in [11] estimated the basic reproductive ratio, by using the relationship between the basic reproductive ratio and the growth rate in the early phase of dengue fever transmission in Toceman, Mexico. Their mean estimate of the basic reproductive ratio is 4.22, which is significantly higher than the mean estimate reported here for every take-off period which is obtained by R0F. The weekly number of dengue cases in Peru for the period 1994–2006 was analyzed in [21] which obtained that the basic reproductive ratio had a median 1.76 (0.83–4.46), where these values inside the value of R0 are reported here; see Table 7. During the 2000 epidemic, occurring in 12 cities of the state of Sao Paulo, Brazil, [4] estimated that the basic reproductive ratio was in the range [3.6,12.9]. More recently [14] estimated the reproductive ratio by using the relationship between the basic reproductive ratio and the force of infection of dengue fever transmission in Salvador from periods 1996 to 1997, and 2002. They obtain an estimation of R0=2.85(2.77,3.70) in the periods 1996 to 1997, and R0=2.65(2.50,3.30) for 2002. The newest report of R0 estimation is in [22] for dengue epidemics in 2002 dengue outbreak in Easter Island, Chile. Their mean of the basic reproductive ratio R0 at 27.2 is significantly higher than every value of R0 estimation reported here for five initial take-off periods of dengue infection from two scenarios for estimating of λ. Our estimation results, for the basic reproductive ratio by using R0F range widely, are from 1.17 to 3.89 for the first scenario and from 1.35 to 3.02 for the second scenario which belongs to the range of the first scenario. Our R0MF estimation accommodates for the difference of the take-off rate λ between mosquito and human. If there are no differences in the take-off rate then [12] estimates are exactly the same as ours. Otherwise R0F could overestimate or underestimate depending on the value of k. Tables 4 and 6 show that R0A accommodates differences in ρ. R0F is exactly the same as our second estimates for a certain pair of ρ and λ. Otherwise R0F could overestimate or underestimate depending on the value of ρ, and so forth. Therefore, our R0MF and R0A constructions are more realistic for estimating the value of the basic reproductive ratio, than the R0F.In the first scenario, the control of dengue fever, which is based on the standard valueR0F, is still realistic. Furthermore, better control of dengue fever, when based on the maximum value of R0MF or maximum value of R0A, is shown in Table 6 and Figures 7–10. While for the second scenario, which is shown by Table 6 and Figures 11–14, it is better to choose one value from the interval as a standard value used to control dengue fever. Therefore, the second construction is better choice, for dengue control strategies in Bandung, Indonesia. Nevertheless, the estimation of R0 has limitations, because it is only done from the fourth day until the seventh day, so the effect of the increase in the incidence of dengue R0 is no longer considered. Furthermore, the dengue incidence data used, from SB Hospital, do not distinguish between the latent population and the dengue infected human population, so the estimate of R0 is limited by the data characteristics. In reality, nobody can distinguish between the latent population and the dengue infected population. Furthermore the latent population has a short incubation time period.


## 6. Conclusion

We have conducted two different constructions of basic reproductive ratio estimations based on incidence data. These constructions modifyR0F model by allowing the take-off rates of human and mosquito to vary. In the first construction, we assumed that the take-off rate of an infected mosquito is proportional to the take-off rate of a human. This construction leads to a new form of modified basic reproductive ratio estimation for R0MF was a function of the take-off rate of a human (λ) and the take-off rate of a mosquito (kλ). A sensitivity analysis indicated that a small variation of a mosquito take-off rate could affect significant change in the basic reproductive ratio. In the second construction we assumed that, during the take-off period, the initial growth of infected humans is slightly affected by the growth of infected mosquitoes, and vice versa. This construction leads to a one-parameter coupling of the initial growth of infected human and mosquitoes. The modified basic reproductive ratio estimation for R0A resulted from this construction, and it depends on the mosquitoes per person index ρ. This ratio increases for small take-off rate and decreases for larger take-off rate. A numerical analysis of R0MF/R0F and R0A/R0F ratios shows variation in a relatively large range of values for R0F, in respect to (λ,k) and (λ,ρ) for the first and second scenario for determined λ, respectively.



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*Source: 206131-2015-08-25.xml*