Elsevier

Applied Mathematics and Computation

Volume 347, 15 April 2019, Pages 566-577
Applied Mathematics and Computation

Analysis of transmission dynamics for Zika virus on networks

https://doi.org/10.1016/j.amc.2018.11.042Get rights and content

Abstract

Transmission of Zika virus (ZIKV) is a serious problem in public health, which can make the babies suffer from microcephaly if their mothers are infected by ZIKV during pregnancy. In this paper, we develop a model of ZIKV transmission in Colombia on complex networks which considers both sexual transmission among humans and the transmission by an infective vector in the process of propagation. We estimate the basic reproduction number R0 and prove that the disease-free equilibrium is globally asymptotically stable when R0<1. In addition, we study the effects of sexual transmission and the transmission route by an infective vector on the process of propagation. Invasion regions of ZIKV were shown in two-parameters space. The obtained results may provide new insights for the control of ZIKV.

Introduction

A member of the family Flaviviridae, genus Flavivirus, Zika virus (ZIKV) is a single-stranded positive sense RNA virus, and it is an arthropod-borne virus (arbovirus) distributed throughout much of Africa and Asia. Other mosquitoborne flaviviruses include yellow fever, dengue, St. Louis encephalitis, West Nile and Japanese encephalitis viruses [1]. For the above-mentioned mosquitoborne flaviviruses except ZIKV, there were quite a lot of studies in some respects of nature cell biology, biodynamics, disease control and prevention and so forth [2], [3], [4], [5], [6]. However, other medically important members of the mosquitoborne flaviviruses, such as ZIKV, have received far less attention from researchers. ‘Zika’ being the ugandan language means weeds which was firstly isolated from the serum of a pyrexial rhesus monkey caged in the canopy of Zika Forest near Lake Victoria, Uganda in April 1947; a second isolation from the mosquito Aedes Africanus followed at the same forest in January 1948 [1], [6], [7], [8], [9], [10], [11], [12], [13]. Since then, The Zika virus has been isolated in humans being Uganda and Tanzania in 1952. In 1954, Nigeria has first confirmed three cases of human Zika virus infection [14]. But the virus has rarely infected humans before, there were only 14 human cases of Zika virus disease documented from which the virus was discovered until 2006. In 2007, an unprecedented outbreak of ZIKV occurred on Yap Island, Micronesia [15]. More recently, ZIKV has been rapidly spreading across the South Pacific since Zika virus causes epidemics in French Polynesia, New Caledonia, Easter Island and the Cook Islands in 2013. At present, Zika virus is prevalent in countries and regions such as the Americas, Africa, southeast Asia and the Pacific island countries. In Colombia, 37 reporting areas have Zika virus disease cases in which 17 areas had more than 1000 infected cases. There are 1342 reported cases per 100,000 population on San Andres and Providencia, islands in the Caribbean Sea where the incidence of Zika virus disease is the highest [16], [17], which implies that the situation is serious in Columbia.

ZIKV is transmitted mainly through infected mosquitoes which are primarily Aedes aegypti, secondarily Aedes albopictus, being the same as mosquitoes that transmit dengue, chikungunya and yellow fever. Other mechanisms of transmission contain sexual transmission, maternal-fetal transmission (not breastfeeding) and blood transfusion [8]. In view of the virus pathes of propagation, we consider both sexual transmission among humans and the transmission by an infective vector. Clinical manifestations of human ZIKV disease are similar to many arboviral infections that occur without serious complications and may include a self-limiting febrile illness, arthralgia, myalgia, headache and maculopapular rash, conjuctivitis, retro-orbital eye pain, lymphadenopathy and diarrhoea [1], [8]. And these symptoms occur about three to twelve days after the mosquito vector bite, which signifies that ZIKV disease have incubation period. Taking into account of the latent period, we propose a dynamical model of ZIKV transmission to research mathematically the overall situation.

Although the disease is self-limiting, cases of neurologic manifestations and the Guillain-Barre syndrome (GBS) were described in French Polynesia and in Brazil during ZIKV epidemics [18], [19]. What’s more, it’s truth that the ZIKV causes newborn to suffer from microcephaly and congenital malformations. Brazil has experienced an unprecedented epidemic of Zika virus (ZIKV), with 30,000 cases reported to date. ZIKV was first detected in Brazil in May 2015, and cases of microcephaly potentially associated with ZIKV infection were identified in November 2015 [11], [12]. And the incidence of microcephaly in Brazil in 2015 was 20 times higher than in previous years. In consideration of the case, we detected the Zika virus genome in the amniotic fluid of pregnant women [13]. According to the report, Zika virus (ZIKV) infection in pregnant women causes intrauterine growth restriction, spontaneous abortion, and microcephaly. It was confirmed that the virus is transmitted from mother to child. Recent reports from the Ministry of Health of Brazil suggest that cases of microcephaly have increased by a factor of approximately 20 among newborns in the northeast region of the country, which causes peoples attention and concern.

There are many studies in the transmission dynamics of dengue [3], [4], [5], and [5] stated that reducing the contact between vector and hosts (biting rates) is very efficient for disease control by mathematical analysis. Moreover, with respect to Zika and Ebola, Miller developed the models based on their transmission mechanism, predicted the dynamics of simulations in the large population limit, and investigated R0 and final size relations [20]. Therefore, we aim to find some efficient measures to control the epidemic scientifically in this article. In Ref. [21], a SEIR-SEI epidemic model to describe the dynamics of the 2016 Zika virus outbreak in Brazil is developed and calibrated by comparing the system response to real data from the outbreak for the sake of testing the effectiveness of different strategies used to combat the associated diseases. In view of these work, we develop a model of ZIKV in Colombia on complex networks on the basis of mechanism of ZIKV transmission. And then we conduct dynamic analysis of the model by methods in Refs. [22], [23], [24], [25], [26].

This paper is organized as follows. In Section 2, we utilize the quantity of individuals and mosquitoes to construct the SEAIR-SI model with an infective vector on complex networks. We compute the basic reproduction number, and analyze the global asymptotic stability of the disease-free equilibrium in Section 3. In Section 4, we perform numerical simulations and sensitive analysis of the basic reproduction number on various model parameters. We raise strategies of disease control. In Section 5, we give a brief conclusion and discussion.

Section snippets

A dynamical model of ZIKV transmission

We propose an SEAIR-SI model on the basis of the spreading process of ZIKV, in which, individuals may be susceptible (S), exposed (E), asymptomatically infected (A), symptomatically infected (I) or removed (R). In this model, we consider two transmission routes by sexual contacts and mosquitoes.

Based on the transmission of ZIKV, three principles are needed:

  • (i)

    People in the incubation period can not infect others;

  • (ii)

    Susceptible mosquitoes can be infected by an exposed individual because there are ZIKV

Basic reproduction number and global dynamics

It is clear that the system (2) has a unique disease-free equilibrium E0=(0,,04n+5). Following the approach of Van den Driessche and Watmough [27], we note that only compartments ek, ak, ik and m are involved in the calculation of R0. In the disease-free state E0, the rate of appearance of new infections F(3n+4)×(3n+4) and the rate of transfer of individuals out of the compartments V(3n+4)×(3n+4) are given by:F=(0000000000r1000β2p(1)k2β2p(2)knβ2p(n)k0β1p(1)k2β1p(2)knβ1p(n)kr1

Sensitivity analysis

We use Latin hypercube sampling (LHS) algorithm to study parameters sensitivity of R0. In our model, α1, α2, βMH, βHM, μ, σ, β1, β2 determine the value of R0. We show values of PRCC in Table 2 based on variable partial rank correlation coefficient (PRCC) which can describe the correlation between parameters and R0. It can be found from Table 1 that α1, α2, μ, σ are negatively correlated with R0. Moreover, β(βMH, βHM, β1, β2) has positive correlations with R0.

In Fig. 3, we show the distribution

Conclusion and discussion

In Colombia, 37 reporting areas have Zika virus disease cases in which 17 areas had more than 1000 infected cases. There are 1342 reported cases per 100,000 population on San Andres and Providencia, islands in the Caribbean Sea where the incidence of Zika virus disease is the highest [16], which implies that the situation is serious in Columbia. In this paper, we posed a transmission model of ZIKV in Columbia which considered both sexual transmission and the transmission by vectors of ZIKV

Acknowledgments

The project is funded by the National 1000 Young Talent Plan W099102, the Fundamental Research Funds for the Central Universities 3102017jc03007, the National Natural Science Foundation of China under Grant nos. 11501338, 11747142, Program for the Outstanding Innovative Teams (OIT) of Higher Learning Institutions of Shanxi, and China Postdoctoral Science Foundation under Grant no. 2017M621110.

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