How Relevant is the Asymptomatic Population in Dengue Transmission?

Transcription

1 Applied Mathematical Sciences, Vol. 12, 2018, no. 32, HIKARI Ltd, How Relevant is the Asymptomatic Population in Dengue Transmission? Oscar A. Manrique A., Steven Raigosa O., Juan C. Osorio A. Julián A. Olarte G., Dalia M. Muñoz P., Juan C. Jamboos T. and Anibal Muñoz L. Grupo de Modelación Matemática en Epidemiología (GMME) Facultad de Educación, Universidad del Quindío Armenia, Quindío, Colombia Copyright c 2018 Oscar A. Manrique A. et al. This article is 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 A mathematical model is formulated based on ordinary non-linear differential equations that interprets the dynamics of dengue transmission, where the human population is divided into three compartments: susceptible persons, symptomatic persons and people, with the population of Aedes aegypti: carrier mosquitoes and non-carrier mosquitoes; simulations of the dynamic system are carried out with the MATLAB software. Keywords: Dengue transmission, mathematical model 1 Introduction Dengue virus (DENV1 to 4) causes more human morbidity and mortality worldwide than any other arthropod-borne virus. A proof of this are more than 3.5 billion people at risk yearly [4,7], the 3.97 billion people distributed in 128 countries currently at risk for infection [1, 2] or the almost 400 million of new cases that occur throughout each year, of which about three quarters are expected to be clinically inapparent [9]. Inapparent dengue is an important component of the overall burden of dengue infection [6] because they can serve as a previously unrecognized source of

2 1700 Oscar A. Manrique A. et al. mosquito infection [4]. Previous studies of inapparent or subclinical infection have reported varying ratios of symptomatic to inapparent dengue infection [6]. Subclinical, inapparent, and asymptomatic infections are often used as synonyms, and the use of paucisymptomatic is used to designate a DENV infection with few symptoms. We will use subclinical and inapparent to denote infections with insufficient symptoms to be detected by the research or national surveillance program and/or to incite the infected individual to consult, but for which there is evidence, either by seroconversion or detection of virus, that the individual was infected with DENV. Asymptomatic infections will be used when there are no symptoms at all reported by the infected individual during an active infection, whether inferred by seroconversion or serology [7]. In order to understand better this concept and the importance of the inapparent roll, we must state that the infectious dose required to infect mosquito vectors when they take a blood meal from a viremic person. This critical parameter underlying the probability of dengue transmission. Because experimental vector competence studies typically examine the proportion of mosquitoes that become infected at intermediate or high DENV infectious doses in the blood meal but the minimum blood meal titer required to infect mosquitoes is poorly documented [8]. Infected individuals may harbor sufficiently high viral loads to infect mosquitoes prior to the onset of symptoms and thereby introduce the virus into the population. Potentially more important is the epidemiological significance of inapparent, subclinical infections [7]. People with inapparent dengue virus infections are generally considered deadend hosts for transmission because they do not reach sufficiently high viremia levels to infect mosquitoes. Despite their lower average level of viremia, Duong V. et al. (2015), show that asymptomatic people can be infectious to mosquitoes [3]. Understanding the factors influencing the lower infectiousness threshold is epidemiologically significant because it determines the transmission potential of humans with a low viremia, possibly including inapparent infections, and during the onset and resolution of the viremic period of acutely infected individuals [8]. Moreover, at a given level of viremia, DENV-infected people with no detectable symptoms or before the onset of symptoms are significantly more infectious to mosquitoes than people with symptomatic infections [3].

3 How relevant is the asymptomatic population in Dengue transmission? 1701 Since a very large number of individuals are infected and since viremic levels are known to vary by many orders of magnitude in symptomatic patients, it is reasonable to augur that a proportion of asymptomatic cases might reach levels of viremia sufficient to infect competent mosquitoes. Moreover, in both Dengue and Chikungunya fever, nosocomial infections have been identified representing an alternative opportunity for virus introduction in non endemic areas [5]. 2 Formulation of the model The formulation of the model is framed in the following assumptions, i. The human population is constant, ii. Symptomatic and asymptomatic human populations with different recovery rates are considered and the symptomatic population does not reach the asymptomatic stage or vice versa, iii. The standard incidence in the human population is taken into account, iv. Two forms in the Aedes aegypti population growth, are considered, v. Asymptomatic population can transmit the dengue virus to non carrier mosquitoes, vi. The time scale considered will be weekly. In the model, the total population of people N(t) ), is considered. It is divided into the following subpopulations: x 1 (t) Average number of susceptible population to a particular serotype of the dengue virus, x 2 (t) Average number of infected population with a particular dengue serotype who have symptoms (symptomatics), x 3 (t) Average number of infected population with a particular dengue serotype that have no symptoms (asymptomatics), x 4 (t) Average number of recovered immune population to a particular dengue serotype at time t, respectively. In the Aedes aegypti vector population, the total mosquito population, is considered M(t), which is divided into: y 1 (t) Average number of non carrier mosquitoes, y 2 (t) Average number of DENV carrier mosquitoes at a time t, respectively. The dynamics we proposed begins considering the variation of the susceptible population is given by µn(t) where µ means the constant birth rate which is equal to the natural death rate of population. So, it is considered that the total population is constant, while the susceptible population is diminished by three factors: 1. Those people who acquire dengue virus and have disease symptoms by this virus (fλ 1 (t)), 2. Those people who are infected by the virus but do not have manifestations (asymptomatics) of the disease ((1 f)λ 1 (t)). Finally, 3. The last factor that causes this population decrease are people who die naturally. That is, as a consequence of the disease µx 1 (t), the parameters involved are f which is a fraction of symptomatic population infected by

4 1702 Oscar A. Manrique A. et al. dengue virus, 1 f which is the fraction of infected population with dengue but asymptomatic. The term that models the incidence in the human population integrates: The mosquito bite rate (weekly), α, β which is the probability of dengue virus transmission from mosquito to human, the susceptible human population, the total human population and the carriers mosquito population of one of the dengue serotypes. Thus, the differential equation corresponding to the variation of the susceptible population over time is: dx 1 (t) = µn(t) fλ 1 (t) (1 f)λ 1 (t) µx 1 (t) (1) The dynamics for the populations represented by variables x 2 (t) and x 3 (t) are similar, they are increased by people who become infected with the virus and are diminished by who recover from the disease and die naturally, reason whereby it is established: dx 2 (t) = fλ 1 (t) (ω + µ)x 2 (t) (2) dx 3 (t) = (1 f)λ 1 (t) (θ + µ)x 3 (t) (3) Likewise, the variation in time of the immune population to a serotype of dengue virus is given by the differential equation, dx 4 (t) = ωx 2 (t) + θx 3 (t) µx 4 (t) (4) where θ is the recovery rate of the asymptomatic population and ω is the recovery rate of the symptomatic population. Now the dynamics presented in the population of the vector is established as follows: dy 1 (t) = ρ λ 2 (t) ɛy 1 (t) (5) dy 2 (t) = λ 2 (t) ɛy 2 (t) (6) where there is an increase in the population of non carrier mosquitoes given by ρ(t), which represents the increase in the mosquito population dependent on time (periodic or constant) and are diminished by those non carrier mosquitoes that bite a infected person by the virus and transmit it to the mosquito, which is affected by the parameter σ, which indicates the probability of transmission of human dengue virus to the mosquito. In the same way, these two populations are affected by the mosquitoes that die. This factor is represented by ɛ

5 How relevant is the asymptomatic population in Dengue transmission? 1703 indicating the natural death rate of Aedes aegypti mosquitoes. Some auxiliary variables are established to analyze the different incidences: x s (t) is the incidence of population of symptomatic population, x a (t) is the incidence of asymptomatic population and y p (t) is the incidence of mosquito population. Below is the differential equations system for these incidents: dx 1 (t) dx 2 (t) dx 3 (t) dx 4 (t) dy 1 (t) dy 2 (t) = µn(t) fλ 1 (t) (1 f)λ 1 (t) µx 1 (t) (7) = fλ 1 (t) (ω + µ)x 2 (t) (8) = (1 f)λ 1 (t) (θ + µ)x 3 (t) (9) = ωx 2 (t) + θx 3 (t) µx 4 (t) (10) = ρ λ 2 (t) ɛy 1 (t) (11) = λ 2 (t) ɛy 2 (t) (12) Las ecuaciones de las incidencias están dadas por: dx s (t) dx a (t) dx T (t) dy p (t) = fλ 1 (t) (13) = (1 f)λ 1 (t) (14) = λ 1 (t) incidencia total en los humanos (15) = λ 2 (t) (16) Where, human population incidence, is given by λ 1 (t) = αβx 1 (t) y 1(t) while the M(t) mosquito population incidence, is indicated by λ 2 (t) = ασy 1 (t) (x 2(t)+x 3 (t)). For N(t) the increase of non carrier mosquitoes, it is considered: { ρ constant case ρ(t) = ɛ [ 1 + Γ cos( 2π 52 t)] variable case on time. and Γ represents the seasonal peak of mosquito growth.

6 1704 Oscar A. Manrique A. et al. The dynamics considered is governed by the compartment diagram, which is shown in Figure 1. x s (t) fλ 1 (t) µn x 1 (t) fλ 1 (t) fλ 1 (t) x 2 (t) (1 f)λ 1 (t) x a (t) µx 1 (t) x 3 (t) ωx 2 (t) µx 2 (t) x 4 (t) θx 3 (t) µx 4 (t) µx 3 (t) ρ(t) y 1 (t) λ 2 (t) y 2 (t) ɛy 1 (t) ɛy 2 λ 2 (t) y p (t) Figure 1: Flowchart that schematically illustrates the dynamics of the transmission of dengue disease in the human population and the Aedes aegypti. 3 Simulations To perform the numerical simulations of the process, the parameter values written in table 3, are considered. The simulations are carried out in the MATLAB software using the routine ode45 and the initial conditions x 1 (0) = , x 2 (0) = 10, x 3 (0) = 22, x 4 (0) = 0, y 1 (0) = 50000, y 2 (0) = 15. In figures 2 and 3 are presented the scenarios where the increase is constant and the simulations have a time scale per week. In Figure 2 the dynamics of the six populations considered in the proposed model are illustrated in three different scenarios, the scenarios are given by the value taken by the parameter f, that is, in the three scenarios we can see the fraction of the symptomatic infected population. The black line indicates a value f = 0.25, while the blue and red line correspond to values of f = 0.5 and f = 0.85 respectively.

7 How relevant is the asymptomatic population in Dengue transmission? 1705 Figure 2: Dynamics of populations when f = 0.25 (black line), f = 0.5 (blue line), and f = 0.85 (red line) for 52 weeks. Figure 3 shows the incidence in the symptomatic infected population, as well as the population that does not present symptoms. It shows the total incidence in the human population and the incidence in the mosquito population for different values of f. Figure 3: Dynamics of the incidence when f = 0.25 (black line), f = 0.5 (blue line), and f = 0.85 (red line) for 52 weeks. Figures 4 and 5 show scenarios where the population growth of the mosquito is variable over time. Figure 4 represents the population dynamics considered in the proposed model for three different scenarios, which are generated from considering the value of the parameter f. That is, the three scenarios are the result of varying the fraction of the symptomatic infected population. The black line indicates that f = 0.25, while blue and red correspond to values f = 0.5 and f = 0.85, respectively. The simulations were performed on a weekly time scale.

8 1706 Oscar A. Manrique A. et al. Figure 4: Population dynamics when f = 0.25 (black line), f = 0.5 (blue line), and f = 0.85 (red line) for 52 weeks. The incidences when considering the population growth of non carrier mosquitoes variable in time, are shown in figure 5 for the three values taken by the parameter f. Figure 5: Population dynamics when f = 0.25 (black line), f = 0.5 (blue line), and f = 0.85 (red line) for 52 weeks. The values of the parameters are taken from the literature and are given by:

9 How relevant is the asymptomatic population in Dengue transmission? 1707 Parameter Value θ 0.3 ω 0.33 µ ɛ 0.25 f 0.25, 0.5, 0.85 α 4.9 β 0.75 σ 0.75 ρ 350 (conxtant case) Γ 0.4 Table 1: Values of the parameters per (week) of the mathematical model. 4 Conclusiones When considering two types of infected, symptomatic and asymptomatic populations, two distinct foci are presented for the spread of the dengue infection. The dynamics of both the populations and the incidences are different when there is variability in asymptomatic and symptomatic populations. This can be seen in figures 2, 3, 4 and 5. References [1] S. Bhatt et al., The global distribution and burden of dengue, Nature, 496(7446) (2013), [2] O. J. Brady et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLOS Negl. Trop. Dis., 6(8) (2012), e [3] V. Duong, Asymptomatic humans transmit dengue virus to mosquitoes, PNAS, 112 (2015), no. 47, [4] L. Grange, Epidemiological risk factors associated with high global frequency of inapparent dengue virus infections, Front Immunol., 5 (2014), [5] C. Chastel, Asymptomatic infections in man: a Trojan horse for the introduction and spread of mosquito-borne arboviruses in non-

10 1708 Oscar A. Manrique A. et al. endemic areas?, Bull. Soc. Pathol. Exot., 104 (2011), [6] T.P. Endy, K.B. Anderson, A. Nisalak, I.-K. Yoon, S. Green et al., Determinants of Inapparent and Symptomatic Dengue Infection in a Prospective Study of Primary School Children in Kamphaeng Phet, Thailand, PLOS Negl. Trop. Dis., 5(3) (2011), e [7] L. Grange et al., Epidemiological risk factors associated with high global frequency of inapparent dengue virus infections, Frontiers in Immunology, (2014). [8] A. Pongsiri, A. Ponlawat, B. Thaisomboonsuk, R. G. Jarman, T. W. Scott et al., Differential Susceptibility of Two Field Aedes aegypti Populations to a Low Infectious Dose of Dengue Virus, PLOS One, 9(3), (2014), e Received: November 2, 2018; Published: December 28, 2018