Ana Kurauchi^{1}, Helio Junji Shimozako^{1,2*}, Eduardo Massad^{1,3}
^{1}School of
Medicine, University of São Paulo, and LIM01HCFMUSP,
São Paulo, Brazil
^{2}NeuroMat Research, Innovation and
Dissemination Center for Neuro mathematics, University of São Paulo, São Paulo,
Brazil
^{3}School of Applied
Mathematics, Fundação Getúlio Vargas, Rio de
Janeiro, Brazil
^{*}Corresponding authors: Helio Junji Shimozako, NeuroMat Research, Innovation and Dissemination Center for Neuro mathematics, University of São Paulo, São Paulo, Brazil. Email: hjunji21@usp.br
Received Date: 18 August,
2018; Accepted Date: 06 September, 2018; Published Date: 12
September, 2018
Optimizing a
strategy for Dengue control based on vaccination is very challenging, since the
dynamics of Dengue can be influenced by the coexistence of more than one
serotype (DENV1, DENV2, DENV3 and DENV4), allowing the appearance of
heterologous infections. In addition, although the effect of crossimmunity
lasts a short period, the real protective power of the immune response is still
not entirely clear. The aim of this study is to develop an optimization model
to control Dengue transmission considering a multistrain model. In order to
evaluate such optimization, the vaccination was considered the control
strategy. The Dengue dynamics was modeled according to that one developed by [1]. This model considered a human population, which
can be infected by two strains of Dengue virus (strain 1 and 2). Both strains
are simultaneously present in the system and they determine 10 categories in
human population: susceptible to strains 1 and 2 (
2.
Keywords: Dengue; Mathematical Model; Optimal Control;
Vaccine
1. Introduction
In the last years, the estimations of the number of dengue cases (symptomatic or not) have been three times higher than those of the World Health Organization (WHO). The increase on such estimations has occurred not only due to human and/or geographical changes [2,3]. This discrepancy in such estimations has occurred due to biases related to (i) diagnosis processes (failure or imprecision) and (ii) unreported dengue cases. Consequently, the evaluation of the efficacy of potential vaccines is impaired, since its real impact is masked by such biases [46]. Considering the difficulty of evaluating the real impact of vaccination and the optimal strategy of its application, an interesting alternative would be the use of mathematical models. The use of mathematical models has been described in studies related to infectious diseases dynamics [7]. Regarding the evaluation of the dengue vaccine, the mathematical model can be a powerful tool to reach the optimization model for a strategy that considers the maximum between the reduction of the incidence of cases and the availability of resources. And, optimizing a strategy for dengue control based on vaccination is very challenging, since the dynamics of dengue can be influenced by the coexistence of more than one serotype (DENV1, DENV2, DENV3 and DENV4), allowing the appearance of heterologous infections. In addition, although the effect of crossimmunity lasts a short period, the real protective power of the immune response is still not entirely clear [8]. Similarly, the risk of heterologous infections and the uncertainty of the protective power of the immune response have also been reported in the studies about the impact of the dengue vaccine. As an example, it was observed that vaccination in seropositive individuals reduces the risk of hospitalization or symptomatology. However, all dengue vaccine models (produced by Sanofi^{®}) applied to susceptible individuals have shown that they may increase the risk of dengue outbreaks [9,10].
According to Health Ministry/Minister’s Office (MH/MO) Ordinance no. 204 (2016 February 17^{th}), the reporting cases of dengue fever is compulsory in Brazil. Such data are available at Brazilian Health System Data (DATASUS) [11] and the most recent data refers to 2017 year. The Ministry of Health is the first authority concerning to public health and medical processes. According to such data, the number of cases has been increasing over time. Therefore, those data are a good reference to understand the epidemiological scenery about dengue in Brazil. This fact is directly related to consequences in terms of epidemiology and economy. In simple words, infected people are sources of infection and they reduce the workforce, and also increase the public costs due to medicines and treatments. Although this is a simple way to think about this problem (other variables may interfere too), it illustrates the impact of dengue in the society. In addition to the increase of cases, studies have been carried out in order to develop a vaccine which (1) is effective and safe for dengue, (2) provides immunity against the 4 serotypes and (3) is durable. Although the best animal model is the rhesus monkey, there still are some details that make the correlation between clinical research findings and the respective conduct in humans harder [12,13]. Considering this context, the development of a mathematical model in order to study the impact of vaccination emerges as an alternative to the ethical question (since the use of living beings for experimentation would be avoided) and to estimate quantitative indexes for the impact of the vaccine in the epidemiological, social and economic approaches [6].
The aim of this study is to develop an optimization model to control dengue transmission considering a multistrain model. In order to evaluate such optimization, the vaccination was considered the control strategy. Here, the vaccine model produced by Sanofi^{®} (which is recommended by World Health Organization) was taken as reference to the simulations. Therefore, it is expected to evaluate the epidemiologic impact of dengue vaccination as preventive control strategy.
2.
The Model
Firstly, the mathematical model presented in this project is an adaptation of that one proposed by Santos in her doctoral thesis [1], and also added some conceptions from [1416]. Thus, in this project, it is considered:
I.A human population, which is constant over time, is divided into ten
classes: (i) susceptible to strains
II.Individuals from
III.The infection by one serotype confers lifelong immunity to that
serotype;
IV.The human population is homogeneously in contact with two strains of
dengue fever;
The complete system of ordinary diﬀerential equations for the twostrain
epidemiological system is given by system (1) and the dynamics is described as
follows. Susceptible individuals to both strains (
No epidemiological asymmetry between strains is assumed, i.e. infections
with strain
The process of model analysis and results interpretation will be similar to that one developed by [14,16,26,27].
3.
The Reported
Cases
In Brazil, reporting cases of Dengue fever is compulsory (Ordinance
MH/MO no. 204 (2016 February 17th)). Thus, we can assume:
·
An infected human should look for medical treatment
when he/she will become clinically ill;
· Only a fraction of those humans that are clinically ill will be reported to sanitary authorities. The remaining fraction (I) will not look for medical help, even if the clinical symptoms and signs appear; or (II) will not be correctly reported in the hospitals. Now, the equations related to infected categories in system (1) are:
Where
The data provided by Brazilian Ministery of Health represents the
reported cases per year. Thus, since time scale considered in this work is day,
it was estimated an average of human reported cases per day for each year
(dividing the total from each year by 365). Finally, model considers a
normalized population (all population is constant). As a last step, we have to
divide each rate of human reported cases per day by the ofﬁcial population size
of Brazil. The estimated population size of Brazil is available on Brazilian
Institute of Geography and Statistics (BIGS) website [17].
In order to ﬁt and compare the model output to real data, it was also
calculated a normalized average of reported cases per day from each 365 days of
simulation. This simulation was run considering 50 years and the obtained curve
was ﬁtted by simple handling along the timeaxis (for instance, we could assume
the initial day
4.
Numerical Simulation
In order to analyze the dynamics of our model, we simulated the set of equations from (1) considering the parameters on Table 1. It was focused on human reported rate, since the real data can be used as reference (Figure 2) illustrate the reported human cases rate and the trend indicated by the simulation (yearly average).
For the evaluation of the impact of vaccination, it was supposed that
the vaccination campaign would start by 2018. Then, a vaccination rate
Some comments can be introduced about the dynamics presented in Figure
3. Note that, for a vaccination rate of
5.
Analysis of
Vaccination Impact
Considering the evaluation of the efficacy and costeffectiveness of a
dengue vaccination (

(4) 
where
In the case of

(5) 
Here, the result presented in Figure 4 deserve for some clarification.
As described before, (Figure 4) informs how many
vaccines were needing to avoid one reported case over time. For example,
considering 2 years after the introduction of the vaccination (2020 year),
Figure 4 indicates that it was necessary around
6.
Discussion
In this work, the impact of Dengue vaccination was evaluated considering
a mathematical model adapted from [1], which in
turn evaluated the Dengue fever dynamics based on a multistrain model. Here,
the parameters values were modified to day time scale and their values were
changed according to Brazilian scenery. The study presented here provided a new
approach for impact of Dengue vaccine. Since Dengue fever is caused by 4 serotypes,
the usual mathematical models that consider the Dengue transmission by a
generic virus cannot be the best option. Due to this fact, a multistrain model
was adapted to evaluate the Dengue dynamics and the impact of vaccination on
it. To the best of our knowledge, the work presented here is one of the first
in which evaluated by mathematical modeling the impact of Dengue vaccination
considering a multistrain model. Also, considering that such vaccine has a
short period of crossimmunity and the possibility of vaccinated individuals
become serotype 2 infected, the mathematical modeling of this phenomena was
very useful to aid in the comprehension of this dynamics. The results presented
in Figure 3 indicates how the Dengue dynamics would be modified if the vaccine
described by World Health Organization is introduced. It is important to
highlight that the immunization effect of such vaccine is valid only to
individuals from category
Finally, it should be noted that in multistrain models like the current one [28] showed that the strain which is only directly transmitted can invade the endemic state of the strain with mixed transmission. The endemic state of the first strain, however, is neutrally stable to invasion by the second strain.
The study presented here provided a general approach about how the
Dengue dynamics can change when the vaccine is introduced. In addition, it also
helps to understand how the densities of each infected category are modified
over time. Since the vaccinated individual
7. Acknowledgments
This work has been supported by the project ZikaPLAN, funded by the European Union's Horizon 2020 research and innovation programme under Grant Agreement No. 734584, by LIM01HFMUSP, CNPq and FAPESP.
Figure 1: Compartment model and flowchart. The noncontinuous line indicates the
influence of a category on the indicated flux. For instance,
Figure 2: Dynamics of reported human cases rate. The available real data are from
2000 to 2017 (bars) and our model was ﬁtted for the same period (line). Observe
that the real data has increased over time and reaches a peak in 2016. Source: Brazilian
Ministry of Health (SINAN) and BIGE.
Figure 3: Impact of vaccination on dynamics of reported human cases rate. The
graph represents a scenery where the vaccination starts in 2018 (the simulation
estimated that the average density of total reported cases per day was around
Figure 4: Result of the
simulations of expression (4) over time, according to each strategy.
Parameter 
Biological meaning 
Value 
Unit 
Source 
μ 
Natural mortality rate 
3.67×105 
day1 
(Brazilian Institute of Geography and Statistics, 2013) [17] 
γ 
Recovery rate 
1.43×101 
day1 
(Halstead, 1990) [18] 
βt 
Infection rate 
2.85×101 
human×day1 
(Fergurson, et al., 1999)[19] 
α 
Temporary crossimmunity rate 
5.48×105 
day1 
(Matheus, et al., 2005) [20] 
ϕ 
Ratio of secondary infections contributing to force of infection 
∈0, 3 
dimensionless 
(Aguiar, et al., 2008)[21] 
ρ 
Import parameter 
0 to 1010 
dimensionless 
(Nagao & Koelle, 2008)[22] 
υ 
Vaccination rate 
variable 
day1 
 
ψ 
Infection rate to y2 due to vaccination 
variable 
day1 
 
τ 
Interval between vaccine doses 
1.82×102 
day 
(World Health Organization, 2017) [23,24] 
η 
Proportion of reported cases 
5.0×102 
dimensionless 
Silva et al. (2Table 1: Parameters and their biological meaning adapted from [1].016) [25] 
Table 1: Parameters and their biological meaning adapted from [1].
Year 
Total of reported cases (Brazilian Ministry of Health, 2018) 
Day average 
Total Brazilian population (BIGS, 2013) 
Normalized day average 
Log (Normalized day average) 
2000 
506 
1.39 
173448346 
7.99E09 
8.1 
2001 
1938 
5.31 
175885229 
3.02E08 
7.52 
2002 
3786 
10.37 
178276128 
5.82E08 
7.24 
2003 
1285 
3.52 
180619108 
1.95E08 
7.71 
2004 
859 
2.35 
182911487 
1.29E08 
7.89 
2005 
1185 
3.25 
185150806 
1.75E08 
7.76 
2006 
1425 
3.9 
187335137 
2.08E08 
7.68 
2007 
1453 
3.98 
189462755 
2.10E08 
7.68 
2008 
1073 
2.94 
191532439 
1.53E08 
7.81 
2009 
1376 
3.77 
193543969 
1.95E08 
7.71 
2010 
2978 
8.16 
195497797 
4.17E08 
7.38 
2011 
3129 
8.57 
197397018 
4.34E08 
7.36 
2012 
2853 
7.82 
199242462 
3.92E08 
7.41 
2013 
2928 
8.02 
201032714 
3.99E08 
7.4 
2014 
2742 
7.51 
202768562 
3.70E08 
7.43 
2015 
4720 
12.93 
204450649 
6.33E08 
7.2 
2016 
1475741 
4043.13 
206081432 
1.96E05 
4.71 
2017 
24742 
677.87 
20766092 
3.26E06 
5.49 
Table 2: Human reported cases in Brazil: average of the normalized rate per day for each year.
7. Massad E (2004) Introdução. Massad E, Menezes RX, Silveira PSP, Ortega NRS, eds. Métodos Quantitativos em Medicina. Barueri: Manole: 570.
23. World Health Organization (2017) Dengue and severe dengue.
30. United Nations (2010) World Urbanization Prospects: The 2009 Revision Population Database. New York: United Nations. Department of Economic and Social Affairs/Population Division.
Citation: Kurauchi A, Shimozako HJ, Massad E (2018) Analysis of the Impact of Vaccination as Control Strategy of Dengue Transmission Dynamics Considering a MultiStrain Model. J Vaccines Immunol: JVII134. DOI: 10.29011/2575789X. 000034