Elsevier

Mathematical Biosciences

Volume 286, April 2017, Pages 1-15
Mathematical Biosciences

The transovarial transmission in the dynamics of dengue infection: Epidemiological implications and thresholds

https://doi.org/10.1016/j.mbs.2017.01.006Get rights and content

Highlights

  • Dengue transmission model with transovarial transmission.

  • Two thresholds obtained from three different approaches.

  • Spectral radius of the next generation matrix, Routh–Hurwitz criteria and M-matrix.

  • First threshold: The gross reproduction number.

  • Second threshold: The product of the fractions of susceptible populations.

Abstract

The anthropophilic and peridomestic female mosquito Aedes aegypti bites humans to suck blood to maturate fertilized eggs, during which dengue virus can be spread between mosquito and human populations. Besides this route of transmission, there is a possibility of dengue virus being passed directly to offspring through transovarial (or vertical) transmission. The effects of both horizontal and transovarial transmission routes on the dengue virus transmission are assessed by mathematical modeling. From the model, the reproduction number is obtained and the contribution of transovarial transmission is evaluated for different levels of horizontal transmission. Notably, the transovarial transmission plays an important role in dengue spread when the reproduction number is near one. Another threshold parameter arises, which is the product of the fractions of the susceptible populations of humans and mosquitoes. Interestingly, these two threshold parameters can be obtained from three different approaches: the spectral radius of the next generation matrix, the Routh–Hurwitz criteria and M-matrix theory.

Introduction

Dengue virus is a flavivirus transmitted by arthropod of the genus Aedes. As a result of being pathogenic for humans and capable of transmission in heavily populated areas, dengue virus (an arbovirus) can cause widespread and serious epidemics, which constitute one of the major public health problems in many tropical and subtropical regions of the world where Aedes aegypti and other appropriate mosquito vectors are present [1].

The incidence of dengue is clearly dependent on abiotic factors such as temperature and precipitation, which affect directly the population dynamics of mosquitoes with serious implications for dengue transmission. By using estimated entomological parameters dependent on temperature, and including the dependency of these parameters on rainfall, the seasonally varying population size of mosquito A. aegypti was evaluated by a mathematical model [2]. This model considered only the horizontal transmission, but the transovarial (or vertical) transmission can play some role in dengue epidemics, which must be assessed.

There is evidence that transovarial (the transfer of pathogens to succeeding generations through invasion of the ovary and infection of the eggs) transmission can occur in some species of Aedes mosquitoes [3], [4], [5], [6], [7], [8], but the role of transovarial transmission in the maintenance of dengue epidemics is not clearly understood [5], [9]. Moreover, the transovarial transmission of dengue virus in A. aegypti has been observed at a relatively low rate [3], [8].

In this paper, the transovarial transmission is included in the modeling. The effects of both horizontal and transovarial routes of dengue transmission are analyzed by obtaining the gross reproduction number, denoted by Rg. This is a threshold parameter encompassing model parameters related to the horizontal and transovarial transmission. The reproduction number Rg is obtained by using three different methods aiming the comparison among them: evaluating the spectral radius of the next generation matrix [10], and determining the conditions that assure to Jacobian matrix eigenvalues with negative real part, which can be assessed by two methods: Routh–Hurwitz criteria and M-matrix theory [11], [12], [13].

In simple directly transmitted infection modeling, there is a well established relationship between the fraction of susceptible humans (s) and the basic reproduction number (R0) in the endemic steady state [14], [15]: s*=1/R0. Similarly, in dengue transmission modeling considering only horizontal transmission, the inverse of R0 is the product of the fractions of susceptible humans and mosquitoes, denoted by χ0. But, if transovarial transmission is included in this dengue transmission, then χ0 cannot be let as the inverse of R0, and an additional threshold quantity must arise. The appearance however of two thresholds also arise for directly transmitted infections modeling. For instance, a well understood two thresholds occur in diseases with secondary infection, such as tuberculosis: a threshold and a sub-threshold [16]. But two thresholds, one for the gross reproduction number and other for the fraction of susceptible individuals, can occur: Driessche and Watmough [10], in their analysis of a tuberculosis transmission including treatment, did not realize the existence of these two thresholds.

The paper is structured as follows. In Section 2, model for dengue transmission encompassing transovarial transmission is presented, and in Section 3, the model is analyzed, determining the equilibrium points, and performing the stability analysis of the disease free equilibrium point. Section 4 presents the discussion about the effects of the transovarial transmission on dengue transmission and on the gross reproduction number Rg and the product of the fractions of susceptibles χ0, and the interpretation of these two thresholds for tuberculosis with failure in the treatment. Conclusions are given in Section 5.

Section snippets

A model for dengue transmission

Dengue virus circulates due to the interaction between human and mosquito populations in urban areas. A unique serotype of dengue virus is being considered in the modeling. A model incorporating two or more serotypes of dengue virus (currently, there are four serotypes) becomes complex. For instance, disregarding co-infection with two or more serotypes, the number of classes of infectious mosquitoes and humans are increased, besides the complexity resulting by the incorporation of the periods

Analysis of the model

The system of Eq. (1) is dealt with determining the equilibrium points, and assessing the stability of these points.

Discussion

Actually, during a year, mosquito population varies broadly due to seasonality, but human population varies smoothly. Non-autonomous modeling deals with varying mosquito and human populations, from which the time dependent effective reproduction number can be obtained (see, for instance, [2]). In this paper, a model considering constant sizes of human and mosquito populations was developed in order to obtain and analyze the steady states. Based on this autonomous model, epidemiological

Conclusions

Dengue transmission modeling incorporating transovarial transmission was analyzed. Qualitative analysis showed that the horizontal (or basic) reproduction number R0 plays the major role in the dynamics of dengue propagation. However, when this number is small, especially near 1, the transovarial transmission (assessed by the transovarial reproduction number Rv=α) enhances strongly the dynamics of dengue infection.

One of the effects is the outbreak of dengue epidemics even for R0 < 1, if the

Acknowledgments

Thanks to anonymous reviewer and Editor for providing comments and suggestions, which contributed to improving this paper. This work was supported by grant from FAPESP (Projeto temático, 09/15098-0).

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