An outbreak vectorhost epidemic model with spatial structure: the 2015–2016 Zika outbreak in Rio De Janeiro
 W. E. Fitzgibbon^{1},
 J. J. Morgan^{1} and
 G. F. Webb^{2}Email author
https://doi.org/10.1186/s129760170051z
© The Author(s) 2017
Received: 25 June 2016
Accepted: 7 March 2017
Published: 27 March 2017
Abstract
Background
A deterministic model is developed for the spatial spread of an epidemic disease in a geographical setting. The disease is borne by vectors to
susceptible hosts through crisscross dynamics. The model is focused on an outbreak that arises from a small number of infected hosts imported into a subregion of the geographical setting. The goal is to understand how spatial heterogeneity of the vector and host populations influences the dynamics of the outbreak, in both the geographical spread and the final size of the epidemic.
Methods
Partial differential equations are formulated to describe the spatial interaction of the hosts and vectors. The partial differential equations have reactiondiffusion terms to describe the crisscross interactions of hosts and vectors. The partial differential equations of the model are analyzed and proven to be wellposed. A local basic reproduction number for the epidemic is analyzed.
Results
The epidemic outcomes of the model are correlated to the spatially dependent parameters and initial conditions of the model. The partial differential equations of the model are adapted to seasonality of the vector population, and applied to the 2015–2016 Zika seasonal outbreak in Rio de Janeiro Municipality in Brazil.
Conclusions
The results for the model simulations of the 2015–2016 Zika seasonal outbreak in Rio de Janeiro Municipality indicate that the spatial distribution and final size of the epidemic at the end of the season are strongly dependent on the location and magnitude of local outbreaks at the beginning of the season. The application of the model to the Rio de Janeiro Municipality Zika 2015–2016 outbreak is limited by incompleteness of the epidemic data and by uncertainties in the parametric assumptions of the model.
Keywords
Background
The Zika virus is a mosquito borne flavivirus that was first isolated in Uganda in 1947 [1]. Subsequently, it has become prevalent in parts of Africa, Asia, and Central and South America. The geographic distribution of the virus has been steadily increasing since 2015 and its further geographic spread to additional countries that are home to competent mosquito vectors is highly probable. As of September 15, 2016, the World Health Organization reports that local circulation of the virus has been reported by 72 countries and territories. Although there have been reports of transmissions through sexual contact [2], Zika virus appears to be primarily spread through the human population through bites from Aedes mosquitos. The virus incubates in a human host over an asymptomatic period lasting from three to twelve days and once fully developed, the virus disease persists for about a week. It is characterized by low grade fever, rash, joint pain, and conjunctivitis (red eyes). Typically it is mild and seldom requires hospitalization. However the virus has two severe complications which make it a menace to public health. The virus has been linked to an increased risk of GuillianBarre syndrome which is a severe autoimmune disorder [3]. Perhaps even more serious is its linkage to microcephaly birth defects in newborn babies [4].
Zika epidemics are both yearround and seasonal, dependent upon the yearround prevalence or seasonality of the resident mosquito populations. A recent study [5] describes in detail the potential spread of Zika epidemics into African and AsianPacific regions by the importation of infected people. The generation of Zika epidemics by the importation of infected people into yearround or seasonal environments is a major public health concern. Recent mathematical models have been developed to understand these concerns [6–13]. We develop a model that describes both yearround and seasonal hostvector epidemic population dynamics in a geographical region. The disease is borne by vectors to susceptible hosts through crisscross dynamics in a region of spatially distributed vectors and hosts. The epidemic outbreak begins with the arrival of a small number of viremic hosts in one or more locales in which the disease is not yet present. Our goal is to aid understanding of how the introduction of a small number of infected hosts, in a specific location in a geographic region, will result in a dissipated or a sustained epidemic. The focus of the study is examine the influence of spatial effects on these possible outcomes.
We formulate a crisscross reactiondiffusion partial differential equations model to describe the spatial evolution of an epidemic. Crisscross reactiondiffusion models for the circulation of disease between vectors and hosts have been used to describe the spatial spread of malaria [14], the spatial spread of Dengue outbreaks [15, 16], and the spatial spread of other diseases by many authors [17–24]. We apply our model to the 2015–2016 Zika seasonal outbreak in the urban area of Rio de Janeiro Municipality in Brazil. We numerically simulate the model to analyze varied scenarios of Zika seasonal epidemics in Rio de Janeiro, dependent upon the input of local spatial outbreaks at the beginning of the season and the timelimitation of seasonality.
Methods

The density of infected hosts H _{ i }(t,x,y) at time t at (x,y)∈Ω, with initial condition H _{ i0}(x,y).

The density of uninfected vectors V _{ u }(t,x,y) at time t at (x,y)∈Ω, with initial condition V _{ u0}(x,y).

The density of infected vectors V _{ i }(t,x,y) at time t at (x,y)∈Ω, with initial condition V _{ i0}(x,y).
Equations of the model
In the Appendix we prove the wellposedness of the model.
The local basic reproduction number
R _{0}(x,y) is interpreted as the average number of new cases generated by a single case at a given location (x,y) in Ω. An analysis of local reproduction numbers for spatially dependent models is given in [27] and in [28]. Our motivation for this definition is the basic reproduction number R _{0} of the spatially independent model (Appendix). Simulations of the spatially dependent model show the following behavior: (1) If R _{0}(x,y)<1 everywhere in Ω, then the populations of both infected hosts and infected vectors extinguish, and the populations converge to the disease free equilibrium. (2) If R _{0}(x,y)>1 in some subregion Ω _{0}⊂Ω, then the populations of both infected hosts and infected vectors may converge from an initial local outbreak to an endemic equilibrium in Ω, even if the average value of R _{0}(x,y) in all of Ω is <1.
Equations of the model when the vector population is seasonal
The 2015–2016 Zika outbreak in Rio de Janeiro municipality
A small number of cases were recorded in the Municipality into the summer of 2015, with the highest number of cases in the eastern region of the Municipality [29, 30]. The Brazilian Health Ministry [31] reported that Rio de Janeiro State (population approximately 16,000,000) registered a count of 60,176 cumulative cases from January 1, 2016 to August 13, 2016 (incidence of approximately 364 cases per 100,000 inhabitants). In [32] the weekly case data for Rio de Janeiro Municipality is given from November 1, 2015 through April 10, 2016, during which time the reporting of cases became mandatory. The cumulative number of reported cases in the Municipality during this period was 25,400 [32] (incidence of approximately 423 cases per 100,000 inhabitants).
Parameterization of the Rio de Janeiro model
We simulate the model (1), (4), (5) for Rio de Janeiro Municipality with some parameters assumed. The available epidemic data used for comparison to our simulations for the Rio de Janeiro Municipality 2015–2016 Zika outbreak is very limited. Further, the number of unreported cases, necessarily unknown, is a limitation of the applicability of the model for this application. A more precise fitting of parameters μ, σ, and β requires much higher data accuracy specific to the Zika epidemic in the Municipality. Our purpose is to provide a qualitative description of a typical vectorborne epidemic spatial outbreak, and our simulation of this particular outbreak, with its limitations on parameterization, serves this purpose.
We set the density dependent mosquito loss function μ(x,y)=0.0015(1.0+100 g a u s s(20.0,30.0,x)×g a u s s(0.0,30.0,y)) (Fig. 2 b), which corresponds to higher levels of mosquito control in the eastern region of the Municipality, where the population density is highest. Here g a u s s(m,s d,x) is the probability density function in x of the normal distribution function with mean m and standard deviation sd. Set the transmission parameters σ _{1}(x,y)=0.00000049, σ _{2}(x,y)=0.78 (we assume that individual mosquitoes bite multiple people, people receive multiple bites, and the probability of infection of mosquitoes is much higher than the probability of infection of people).
The diffusion terms for the infected people, uninfected mosquitoes, and infected mosquitoes in the model are understood as idealizations of the indirect spatial spread of the Zika virus infection agent. The spatial spread of the virus is dependent on the direct spread of infected people and uninfected/infected mosquitoes. The spatial movement of people in an urban setting is extremely complex, and a major challenge for epidemic modeling. We set the infected people diffusion parameter δ _{1}=0.2, which provides a simplified way of describing the movement of infected people, in the context of the epidemic, with respect to the spatial spread of the virus. We set the mosquito diffusion parameter δ _{2}=0.2, which is consistent with an estimated adult mosquito dispersal of 30−50 m per day [36].
We set the initial outbreaks in variable locations in the Municipality. For the initial spatial distribution of infected people we set H _{ i }(0,x,y)=H _{ i0} g a u s s(x _{0},1.0,x)×g a u s s(y _{0},1.0,y), centered at (x _{0},y _{0}). The initial number of infected people at the location (determined by H _{ i0}) is viewed as small and above a threshold level capable of outbreak. It includes imported cases (first order) and possibly some cases generated by first order cases (higher order).
Results
Simulations of the model for Rio de Janeiro
We provide four simulations of the model with initial outbreaks in different locations in the Municipality.
Discussion and conclusions
The model (1), (2), (3) describes crisscross vectorhost transmission dynamics of an epidemic outbreak in a geographical region Ω, where the vector population is present yearround. The outbreak occurs with a small number of infected hosts in a small subregion of the much larger geographical region Ω. The diffusion terms describe the ongoing average spatial spread of the disease microbial agent within infected vectors and infected hosts in the geographical region. The focus of the model is to describe the geographical spread from an initial localized immigration into the region, in terms of the epidemiological properties of the outbreak vectorhost transmission dynamics.
We prove that the partial differential equations model (1), (2), (3) is mathematically wellposed, and compare its properties to an analogous ordinary differential equations model in the spatially independent case (Appendix). The outcomes of the model depend on the spatially distributed local reproduction number R _{0}(x,y). In the case of yearround vector settings, simulations indicate that the connection of R _{0}(x,y) to the outcome of an outbreak is as follows: if R _{0}(x,y)<1 everywhere in Ω, then the epidemic will extinguish; if R _{0}(x,y)>1 in some subregion of Ω, then the epidemic has the possibility to spread from an initial outbreak to an endemic equilibrium in Ω, even if the average value of R _{0}(x,y)<1 throughout all of Ω.
The model Eqs. (1), (2), (3) are modified to incorporate seasonality of the vector population in Eqs. (1), (4), (5), and applied to the 2015–2016 Zika outbreak in Rio de Janeiro Municipality. Simulations of the model (Examples 1 and 4) provide qualitative agreement with the reported case data in the Municipality [32]. We argue that the assumption of an unchanging number for the susceptible population is reasonable for the Zika outbreak in Rio de Janeiro Municipality. The justification for this assumption is based on current demographic data for the Municipality [37]. Between 2010 and 2016 the population increased from approximately 6.000,000 at approximately 0.49% per year. The total number of reported cases during the 2015–2016 outbreak is less than 1% of the susceptible population, which is not significantly depleted during the outbreak.
A limitation of our model is the difficulty of estimating the number of unreported cases, and in some examples of Zika epidemics the ratio of reported cases to unreported cases has been quite high. In one study, the Federated States of Micronesia in 2007, the number of reported cases was 108 and the number of unreported cases (estimated through seroconversion testing) was estimated at 74% of the total population of 7,391 [38]. In another study, the French Polynesia outbreak in 2013–2014, the number of reported cases was estimated at 7–17% of the total number of infections, with 94% of the total population infected [33]. The setting for Rio de Janeiro Municipality is very different, however, and the demographic changes in Rio de Janeiro Municipality in one year could offset a relatively higher ratio of unreportedtoreported cases, given that the reported cases represented approximately 0.4% of the population [31, 32]. Additionally, the probability of Zika reinfection is not yet fully known. Whether Zika could become established as an endemic disease in a larger urban population thus remains unclear [33]. Our model simulations are based on the number of reported cases, but we note that if the ratio of unreported to reported cases is significantly higher, then the parameters must be adjusted.
A limitation of our model is that it does not take into account the possibility of sexual transmission of Zika. It is noted in [2], however, that sexual transmission is a small percentage of total transmission, and may not initiate or sustain an outbreak. Another limitation of our model is that we assume the uninfected mosquito population is uniformly geographically distributed at the beginning of the season, since there is no detailed temporal geographic mosquito data available for Rio de Janeiro Municipality. We note that current investigations are developing such data for geographical regions, which could be implemented eventually for spatial models of vector borne epidemics as described by our model. One such investigation is Project Premonition [39], developed by Microsoft to autonomously locate, robotically collect, and computationally analyze mosquito populations for pathogenicity in geographical environmental regions.
The model simulation suggests that the Zika epidemic in Rio de Janeiro Municipality may rise each season from initial outbreak locations, with very small numbers of infected people, and spread through a larger region of the Municipality. Although the epidemic subsides at the end of the season, the final size of the epidemic at the end of the season depends on the initial outbreak locations of infected cases in the region, when geographic heterogeneity and timelimited seasonality are taken into account. The local reproduction number R _{0}(x,y) indicates that the most effective interventions decrease the infection rates σ _{1}(x,y), σ _{2}(x,y), increase the isolation of infected people λ(x,y), increase the mosquito removal rate μ(x,y), and control the importation of infected people, all concentrated in regions of high density population H _{ u }(x,y) and in the beginning of the season.
For the Zika epidemic in Rio de Janeiro Municipality the model suggests that the outbreak in the 2015–2016 season will occur again in the 2016–2017 season, and in future seasons. The importation of infected cases into the Municipality at the beginning of the season is inevitable, because of the general influx of people into this major metropolitan center of Brazil. Some of these cases will not generate a further spread of cases, but some will, with consideration of spatially variable factors. The reduction of future, and more extensive, seasonal outbreaks of Zika in the Municipality requires higher level monitoring of the people arriving in the region and higher level mosquito control measures throughout the region, again with consideration of spatially variable factors.
Appendix
Wellposedness of the model
The model equations without spatial dependence
with initial conditions H _{ i }(0)=H _{ i0}, V _{ u }(0)=V _{ u0}, V _{ i }(0)=V _{ i0}. Set the basic reproduction number R _{0}=H _{ u } σ _{1} σ _{2}/λ μ. We note that R _{0} is independent of the vector reproduction rate β. The epidemic size of the epidemic, however, is proportional to β, as seen in their formulas below. The behavior of solutions of Eqs. (7), (8), (9) can be classified as follows:
Declarations
Acknowledgments
Not applicable.
Funding
No funding bodies were utilized in the design, analysis, and writing of the manuscript.
Availability of data and materials
The data in the manuscript is published by the Brazilian Ministry of Health, as given in the References. The authors agree to provide upon request computer codes for the numerical simulations in the manuscript.
Authors’ contributions
All authors conceived and developed the study. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
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Authors’ Affiliations
References
 World Health Organization. Zika virus. 2016;Sept 16. http://www.who.int/mediacentre/factsheets/zika/en/.
 Gao D, Lou Y, He D, et al. Prevention and control of Zika as a mosquitoborne and sexually transmitted disease: A mathematical modeling analysis. Sci. Rep. 2016;17(6).Google Scholar
 CaoLormeau VM, Blake A, Mons S, et al. GuillainBarré Syndrome outbreak associated with Zika virus infection in French Polynesia: a casecontrol study. Lancet. 2016; 387:1531–1539.View ArticlePubMedGoogle Scholar
 Nishiura H, Mizumoto K, Rock KS, et al. A theoretical estimate of the risk of microcephaly during pregnancy with Zika virus infection. Epidemics. 2016; 15:66–70.View ArticlePubMedGoogle Scholar
 Bogoch II, Brady OJ, Kraemer MU, et al. Potential for Zika virus introduction and transmission in resourcelimited countries in Africa and the AsiaPacific region: a modelling study. Lancet Infect. Dis. 2017. (Epub ahead of print).Google Scholar
 Zinszer K, Morrison K, Brownstein JS, et al. Reconstruction of Zika virus introduction in Brazil. Emerg. Infect. Dis. 2017. (Epub ahead of print).Google Scholar
 Carlson CJ, Dougherty ER, Getz W. An ecological assessment of the pandemic threat of Zika virus. PLoS Negl. Trop. Dis. 2016;eCollection.Google Scholar
 Robert CJ, Christofferson RC, Silva NJ, et al. Modeling mosquitoborne disease spread in U.S. urbanized areas: The case of Dengue in Miami. PLoS One. 2016;11(8).Google Scholar
 Huff A, Allen T, Whiting K, et al. FLIRTing with Zika: A web application to predict the movement of infected travelers validated against the current Zika virus epidemic. PLoS Curr. 2016;10(8).Google Scholar
 Chowell G, HincapiePalacio D, Ospina J, et al. Using phenomenological models to characterize transmissibility and forecast patterns and final burden of Zika epidemics. PLoS Curr. 2016;31(8).Google Scholar
 Goubert C, Minard G, Vieira C, et al. Population genetics of the Asian tiger mosquito Aedes albopictus, an invasive vector of human diseases. Heredity. 2016; 117(3):125–134.View ArticlePubMedPubMed CentralGoogle Scholar
 Majumder MS, Santillana M, Mekaru SR, et al. Utilizing nontraditional data sources for near realtime estimation of transmission dynamics during the 20152016 Colombian Zika virus disease outbreak. JMIR Public Health Surveill. 2016;1(2).Google Scholar
 Massad E, Tan SH, Khan K, et al. Estimated Zika virus importations to Europe by travellers from Brazil. Glob Health Action. 2016;17(9).Google Scholar
 Bailey NTJ. The Mathematical Theory of Epidemics. London: Charles Griffin and Co. Ltd; 1957.Google Scholar
 Manore C, Hickmann S, Xu S, et al. Comparing Dengue and Chikungunya emergence and endemic transmission in A. aegypti and A. albopictus. J. Theoret. Biol. 2014; 356:174–191.View ArticleGoogle Scholar
 Ho SM, Speldewinde P, Cook A. Predicting arboviral disease emergence using Bayesian networks: a case study of dengue virus in Western Australia. Epidemiol. Infect. 2016; 145(1):1–13.Google Scholar
 Capasso V. Global Solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 1978; 35(20):274–284.View ArticleGoogle Scholar
 Webb GF. A reactiondiffusion model for a deterministic diffusive epidemical model. J. Math. Anal. Appl. 1981; 84:150–161.View ArticleGoogle Scholar
 Fitzgibbon WE, Martin CB, Morgan J. A diffusive epidemic model with crisscross dynamics. J. Math. Anal. Appl. 1994; 184:399–414.View ArticleGoogle Scholar
 Fitzgibbon WE, Parrott ME, Webb GF. Diffusion Epidemic models with incubation and crisscross dynamics. Math. Bios. 1995; 128(12):131–155.View ArticleGoogle Scholar
 Fitzgibbon WE, Langlais M, Morgan J. A reaction diffusion system on noncoincident domains modeling the circulation of a disease between two host populations. Dif. Int. Eq. 2004; 17:781–802.Google Scholar
 Fitzgibbon WE, Langlais M, Marpeau F. Modelling the circulation of a disease between two host populations on noncoincident spatial domains. Biol. Invasions. 2005; 7:863–875.View ArticleGoogle Scholar
 Anita S, Fitzgibbon WE, Langlais M. Global existence and internal stabilization for a reaction diffusion system posed on noncoincident domains. Disc. Cont. Dyn. Sys.Series B. 2009; 11(4):805–822.View ArticleGoogle Scholar
 Fitzgibbon WE, Langlais M. Lecture Notes in Mathematics: Biomathematics Subseries In: Magal P, Ruan S, editors. New York: SpringerVerlag: 2008. p. 115–164.Google Scholar
 Thrall PH, Antonovies J, Hall DW. Host and pathogen coexistence in sexually transmitted and vectorborne diseases. Amer. Nat. 1993; 142:543–552.View ArticleGoogle Scholar
 Wu Y, Zou X. Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J. Dif. Eq. 2016; 261(8):4424–4447.View ArticleGoogle Scholar
 Allen LJS, Bolker BM, Lou Y, et al. Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model. Disc. Cont. Dyn. Sys  Series B. 2008; 21:1–20.View ArticleGoogle Scholar
 Peng R. Asymptotic profiles of the positive steady state for an SIS epidemic reactiondiffusion model. Part I. J. Dif. Eq. 2009; 247(415):1096–1119.View ArticleGoogle Scholar
 Brasil P, Calvet GA, Siqueira AM, et al. Zika virus outbreak in Rio de Janeiro, Brazil: Clinical characterization, epidemiological and virological aspects. PLOS Neglected Tropical Diseases. 2016;20(12).Google Scholar
 Honório N, Nogueira R, Codeco C, et al. Spatial evaluation and modeling of Dengue seroprevalence and vector density in Rio de Janeiro, Brazil. PLOS Neglected Tropical Diseases. 2009;3(11).Google Scholar
 da Saúde M, Boletim Epidemiológica, Secretaria de Viglilácia em Saúde. Monitoramento dos cases de dengue, febre de chikungunya e febre pelo virus Zika até a Semana Epidemiológica 32. 2016;47(33).Google Scholar
 Bastos L, Villela D, de Calvalho L, et al. Assessment of basic reproductive number and its comparison with dengue. bioRxiv:055475. Posted online May 25, 2016.Google Scholar
 Kucharsky A, Funk S, Eggo R, et al. Transmission dynamics of Zika virus island populations: A modelling analysis of the 2013–2014 French Polynesia outbreak. PLOS Neglected Tropical Diseases. 2016;10(5).Google Scholar
 Centers for Disease Control. Zika virus. 2016. https://www.cdc.gov/zika/index.html.
 Brady O, Johansson M, Guerra C, et al. Modelling adult Aedes aegypti and Aedes albopictus survival at different temperatures in laboratory settings. Parasites & Vectors. 2013;6(351).Google Scholar
 Otero M, Schweigmann N, Solaria H. A stochastic spatial dynamical model for Aedes aegypti. Bull. Math. Biol. 2008; 70:1297–1325.View ArticlePubMedGoogle Scholar
 World Population. 2016. http://www.population.city/brazil/riodejaneiro/.
 Duffy MR, Chen TH, Hancock WT, et al. Zika virus outbreak on Yap Island, Federate States of Micronesia. N. Eng. J. Med. 2009; 360:2536–2543.View ArticleGoogle Scholar
 Project Premonition. 2016. http://www.microsoft.com/enus/research/project/projectpremonition/.
 Martin RH. Nonlinear Operators and Differential Equations in Banach Spaces. New York: WileyInterscience; 1976.Google Scholar
 Pazy A. Semigroups of Operators and Applications. New York: SpringerVerlag; 1983.Google Scholar
 Smoller J. Shock Waves and Reaction Diffusion Equations. New York: SpringerVerlag; 1994.View ArticleGoogle Scholar