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Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1699 - 1708HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2018.810150

How Relevant is the Asymptomatic Populationin Dengue Transmission?

Oscar A. Manrique A., Steven Raigosa O., Juan C. Osorio A.Julian A. Olarte G., Dalia M. Munoz P., Juan C. Jamboos T.

and Anibal Munoz L.

Grupo de Modelacion Matematica en Epidemiologıa (GMME)Facultad de Educacion, Universidad del Quindıo

Armenia, Quindıo, Colombia

Copyright c© 2018 Oscar A. Manrique A. et al. This article is distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduc-tion in any medium, provided the original work is properly cited.

AbstractA mathematical model is formulated based on ordinary non-linear

differential equations that interprets the dynamics of dengue transmis-sion, where the human population is divided into three compartments:susceptible persons, symptomatic persons and people, with the popula-tion of Aedes aegypti: carrier mosquitoes and non-carrier mosquitoes;simulations of the dynamic system are carried out with the MATLABsoftware.

Keywords: Dengue transmission, mathematical model

1 IntroductionDengue virus (DENV1 to 4) causes more human morbidity and mortalityworldwide than any other arthropod-borne virus. A proof of this are morethan 3.5 billion people at risk yearly [4,7], the 3.97 billion people distributedin 128 countries currently at risk for infection [1, 2] or the almost 400 millionof new cases that occur throughout each year, of which about three quartersare expected to be clinically inapparent [9].

Inapparent dengue is an important component of the overall burden of dengueinfection [6] because they can serve as a previously unrecognized source of

1700 Oscar A. Manrique A. et al.

mosquito infection [4]. Previous studies of inapparent or subclinical infectionhave reported varying ratios of symptomatic to inapparent dengue infection [6].

Subclinical, inapparent, and asymptomatic infections are often used as syn-onyms, and the use of paucisymptomatic is used to designate a DENV infec-tion with few symptoms. We will use subclinical and inapparent to denoteinfections with insufficient symptoms to be detected by the research or na-tional 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 beused when there are no symptoms at all reported by the infected individualduring an active infection, whether inferred by seroconversion or serology [7].

In order to understand better this concept and the importance of the inappar-ent roll, we must state that the infectious dose required to infect mosquito vec-tors when they take a blood meal from a viremic person. This critical param-eter underlying the probability of dengue transmission. Because experimentalvector competence studies typically examine the proportion of mosquitoes thatbecome infected at intermediate or high DENV infectious doses in the bloodmeal but the minimum blood meal titer required to infect mosquitoes is poorlydocumented [8].

Infected individuals may harbor sufficiently high viral loads to infect mosquitoesprior to the onset of symptoms and thereby introduce the virus into the pop-ulation. Potentially more important is the epidemiological significance of in-apparent, subclinical infections [7].

People with inapparent dengue virus infections are generally considered dead-end hosts for transmission because they do not reach sufficiently high viremialevels to infect mosquitoes. Despite their lower average level of viremia, DuongV. et al. (2015), show that asymptomatic people can be infectious to mosquitoes[3].

Understanding the factors influencing the lower infectiousness threshold is epi-demiologically significant because it determines the transmission potential ofhumans with a low viremia, possibly including inapparent infections, and dur-ing 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 detectablesymptoms or before the onset of symptoms are significantly more infectious tomosquitoes than people with symptomatic infections [3].

How relevant is the asymptomatic population in Dengue transmission? 1701

Since a very large number of individuals are infected and since viremic levelsare known to vary by many orders of magnitude in symptomatic patients, itis reasonable to augur that a proportion of asymptomatic cases might reachlevels of viremia sufficient to infect competent mosquitoes. Moreover, in bothDengue and Chikungunya fever, nosocomial infections have been identifiedrepresenting an alternative opportunity for virus introduction in non endemicareas [5].

2 Formulation of the modelThe formulation of the model is framed in the following assumptions, i. Thehuman population is constant, ii. Symptomatic and asymptomatic humanpopulations with different recovery rates are considered and the symptomaticpopulation does not reach the asymptomatic stage or vice versa, iii. The stan-dard incidence in the human population is taken into account, iv. Two formsin the Aedes aegypti population growth, are considered, v. Asymptomatic pop-ulation can transmit the dengue virus to non carrier mosquitoes, vi. The timescale considered will be weekly.

In the model, the total population of people N(t) ), is considered. It is di-vided into the following subpopulations: x1(t) Average number of susceptiblepopulation to a particular serotype of the dengue virus, x2(t) Average numberof infected population with a particular dengue serotype who have symptoms(symptomatics), x3(t) Average number of infected population with a particulardengue serotype that have no symptoms (asymptomatics), x4(t) Average num-ber of recovered immune population to a particular dengue serotype at timet, respectively. In the Aedes aegypti vector population, the total mosquitopopulation, is considered M(t), which is divided into: y1(t) Average numberof non carrier mosquitoes, y2(t) Average number of DENV carrier mosquitoesat a time t, respectively.

The dynamics we proposed begins considering the variation of the susceptiblepopulation is given by µN(t) where µ means the constant birth rate which isequal to the natural death rate of population. So, it is considered that thetotal population is constant, while the susceptible population is diminishedby three factors: 1. Those people who acquire dengue virus and have diseasesymptoms by this virus (fλ1(t)), 2. Those people who are infected by the virusbut do not have manifestations (asymptomatics) of the disease ((1 − f)λ1(t)).Finally, 3. The last factor that causes this population decrease are people whodie naturally. That is, as a consequence of the disease µx1(t), the parame-ters involved are f which is a fraction of symptomatic population infected by


dengue virus, 1 − f which is the fraction of infected population with denguebut asymptomatic.

The term that models the incidence in the human population integrates: Themosquito bite rate (weekly), α, β which is the probability of dengue virus trans-mission from mosquito to human, the susceptible human population, the totalhuman population and the carriers mosquito population of one of the dengueserotypes. Thus, the differential equation corresponding to the variation of thesusceptible population over time is:

dx1(t)dt

= µN(t) − fλ1(t) − (1 − f)λ1(t) − µx1(t) (1)

The dynamics for the populations represented by variables x2(t) and x3(t)are similar, they are increased by people who become infected with the virusand are diminished by who recover from the disease and die naturally, reasonwhereby it is established:

dx2(t)dt

= fλ1(t) − (ω + µ)x2(t) (2)

dx3(t)dt

= (1 − f)λ1(t) − (θ + µ)x3(t) (3)

Likewise, the variation in time of the immune population to a serotype ofdengue virus is given by the differential equation,

dx4(t)dt

= ωx2(t) + θx3(t) − µx4(t) (4)

where θ is the recovery rate of the asymptomatic population and ω is the re-covery rate of the symptomatic population. Now the dynamics presented inthe population of the vector is established as follows:

dy1(t)dt

= ρ− λ2(t) − εy1(t) (5)

dy2(t)dt

= λ2(t) − εy2(t) (6)

where there is an increase in the population of non carrier mosquitoes given byρ(t), which represents the increase in the mosquito population dependent ontime (periodic or constant) and are diminished by those non carrier mosquitoesthat bite a infected person by the virus and transmit it to the mosquito, whichis affected by the parameter σ, which indicates the probability of transmissionof human dengue virus to the mosquito. In the same way, these two popula-tions are affected by the mosquitoes that die. This factor is represented by ε


indicating the natural death rate of Aedes aegypti mosquitoes.

Some auxiliary variables are established to analyze the different incidences:xs(t) is the incidence of population of symptomatic population, xa(t) is theincidence of asymptomatic population and yp(t) is the incidence of mosquitopopulation. Below is the differential equations system for these incidents:

dx1(t)dt

= µN(t) − fλ1(t) − (1 − f)λ1(t) − µx1(t) (7)

dx2(t)dt

= fλ1(t) − (ω + µ)x2(t) (8)

dx3(t)dt

= (1 − f)λ1(t) − (θ + µ)x3(t) (9)

dx4(t)dt

= ωx2(t) + θx3(t) − µx4(t) (10)

dy1(t)dt

= ρ− λ2(t) − εy1(t) (11)

dy2(t)dt

= λ2(t) − εy2(t) (12)

Las ecuaciones de las incidencias estan dadas por:

dxs(t)dt

= fλ1(t) (13)

dxa(t)dt

= (1 − f)λ1(t) (14)

dxT (t)dt

= λ1(t) incidencia total en los humanos (15)

dyp(t)dt

= λ2(t) (16)

Where, human population incidence, is given by λ1(t) = αβx1(t) y1(t)M(t) while the

mosquito population incidence, is indicated by λ2(t) = ασy1(t) (x2(t)+x3(t))N(t) . For

the increase of non carrier mosquitoes, it is considered:

ρ(t) ={

ρ constant caseε

[1 + Γ cos(2π

52 t)]

variable case on time.

and Γ represents the seasonal peak of mosquito growth.


The dynamics considered is governed by the compartment diagram, which isshown in Figure 1.

xs(t) x2(t)

��

ωx2(t)

""

��

µN // x1(t)

��

fλ1(t)

bb

fλ1(t)

||

fλ1(t)<<

(1−f)λ1(t)

""

µx2(t) x4(t)

��xa(t) µx1(t) x3(t)

��

θx3(t)

<<

��

µx4(t)

µx3(t)

ρ(t) // y1(t) λ2(t) //

��

λ2(t)

##

y2(t)

��

SS

εy1(t) εy2

yp(t)

Figure 1: Flowchart that schematically illustrates the dynamics of the trans-mission of dengue disease in the human population and the Aedes aegypti.

3 SimulationsTo perform the numerical simulations of the process, the parameter valueswritten in table 3, are considered. The simulations are carried out in theMATLAB software using the routine ode45 and the initial conditions x1(0) =348048, x2(0) = 10, x3(0) = 22, x4(0) = 0, y1(0) = 50000, y2(0) = 15.

In figures 2 and 3 are presented the scenarios where the increase is constantand the simulations have a time scale per week. In Figure 2 the dynamics ofthe six populations considered in the proposed model are illustrated in threedifferent scenarios, the scenarios are given by the value taken by the parameterf , that is, in the three scenarios we can see the fraction of the symptomaticinfected population. The black line indicates a value f = 0.25, while the blueand red line correspond to values of f = 0.5 and f = 0.85 respectively.


Figure 2: Dynamics of populations when f = 0.25 (black line), f = 0.5 (blueline), and f = 0.85 (red line) for 52 weeks.

Figure 3 shows the incidence in the symptomatic infected population, as well asthe population that does not present symptoms. It shows the total incidencein the human population and the incidence in the mosquito population fordifferent values of f .

Figure 3: Dynamics of the incidence when f = 0.25 (black line), f = 0.5 (blueline), and f = 0.85 (red line) for 52 weeks.

Figures 4 and 5 show scenarios where the population growth of the mosquitois variable over time. Figure 4 represents the population dynamics consideredin the proposed model for three different scenarios, which are generated fromconsidering the value of the parameter f . That is, the three scenarios are theresult of varying the fraction of the symptomatic infected population. Theblack line indicates that f = 0.25, while blue and red correspond to valuesf = 0.5 and f = 0.85, respectively. The simulations were performed on aweekly time scale.


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 mosquitoesvariable in time, are shown in figure 5 for the three values taken by the pa-rameter 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:


Parameter Valueθ 0.3ω 0.33µ 2.56 × 10−4

ε 0.25f 0.25, 0.5, 0.85α 4.9β 0.75σ 0.75ρ 350 (conxtant case)Γ 0.4

Table 1: Values of the parameters per (week) of the mathematical model.

4 ConclusionesWhen considering two types of infected, symptomatic and asymptomatic pop-ulations, two distinct foci are presented for the spread of the dengue infection.The dynamics of both the populations and the incidences are different whenthere is variability in asymptomatic and symptomatic populations. This canbe 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), 504-507. https://doi.org/10.1038/nature12060

[2] O. J. Brady et al., Refining the global spatial limits of dengue virustransmission by evidence-based consensus, PLOS Negl. Trop. Dis., 6(8)(2012), e1760. https://doi.org/10.1371/journal.pntd.0001760

[3] V. Duong, Asymptomatic humans transmit dengue virus to mosquitoes,PNAS, 112 (2015), no. 47, 14688–14693.https://doi.org/10.1073/pnas.1508114112

[4] L. Grange, Epidemiological risk factors associated with high global fre-quency of inapparent dengue virus infections, Front Immunol., 5 (2014),280. https://doi.org/10.3389/fimmu.2014.00280

[5] C. Chastel, Asymptomatic infections in man: a Trojan horse forthe introduction and spread of mosquito-borne arboviruses in non-


endemic areas?, Bull. Soc. Pathol. Exot., 104 (2011), 213-219.https://doi.org/10.1007/s13149-011-0165-1

[6] T.P. Endy, K.B. Anderson, A. Nisalak, I.-K. Yoon, S. Green et al., Deter-minants of Inapparent and Symptomatic Dengue Infection in a Prospec-tive Study of Primary School Children in Kamphaeng Phet, Thailand,PLOS Negl. Trop. Dis., 5(3) (2011), e975.https://doi.org/10.1371/journal.pntd.0000975

[7] L. Grange et al., Epidemiological risk factors associated with high globalfrequency of inapparent dengue virus infections, Frontiers in Immunol-ogy, (2014). https://doi.org/10.3389/fimmu.2014.00280

[8] A. Pongsiri, A. Ponlawat, B. Thaisomboonsuk, R. G. Jarman, T. W.Scott et al., Differential Susceptibility of Two Field Aedes aegypti Pop-ulations to a Low Infectious Dose of Dengue Virus, PLOS One, 9(3),(2014), e92971. https://doi.org/10.1371/journal.pone.0092971

Received: November 2, 2018; Published: December 28, 2018

How Relevant is the Asymptomatic Population in …...1702 Oscar A. Manrique A. et al. dengue virus,...

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