An Outbreak Vectorhost Epidemic Model with Spatial Structure: The 2015 Zika Outbreak in Rio De Janeiro
Abstract
Abstract
An Outbreak Vectorhost Epidemic Model with Spatial Structure: The 2015 Zika Outbreak in Rio De Janeiro W.E. Fitzgibbon, J.J. Morgan and G.F. Webb
Department of Mathematics, University of Houston,77204 Houston, TX, USA Department of Mathematics, University of Houston,77204 Houston, TX, USA Department of Mathematics, Vanderbilt University, 37240 Nashville, TN, USA Email: W.E. Fitzgibbon  wfitzgib@Central.UH.EDU; J.J. Morgan  jjmorgan@Central.UH.EDU; G.F. Webb  glenn.f.webb@vanderbilt.edu; W.E. Fitzgibbon  wfitzgib@Central.UH.EDU; J.J. Morgan  jjmorgan@Central.UH.EDU; G.F. Webb  glenn.f.webb@vanderbilt.edu; Corresponding author
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 epidemic outbreak that initiates from a small number of cases in a small subregion of the geographical setting.
Methods:
Partial differential equations are formulated to describe the interaction of the model compartments.
Results:
The partial differential equations of the model are analyzed and proven to be wellposed. The epidemic outcomes of the model are correlated to the spatially dependent parameters and initial conditions of the model.
Conclusions:
A version of the model is applied to the 20152016 Zika outbreak in the Rio de Janeiro Municipality in Brazil.
An Outbreak Vectorhost Epidemic Model with Spatial Structure: The 2015 Zika Outbreak in Rio De Janeiro W.E. Fitzgibbon, J.J. Morgan and G.F. Webb
Department of Mathematics, University of Houston,77204 Houston, TX, USA Department of Mathematics, University of Houston,77204 Houston, TX, USA Department of Mathematics, Vanderbilt University, 37240 Nashville, TN, USA Email: W.E. Fitzgibbon  wfitzgib@Central.UH.EDU; J.J. Morgan  jjmorgan@Central.UH.EDU; G.F. Webb  glenn.f.webb@vanderbilt.edu; W.E. Fitzgibbon  wfitzgib@Central.UH.EDU; J.J. Morgan  jjmorgan@Central.UH.EDU; G.F. Webb  glenn.f.webb@vanderbilt.edu; Corresponding author
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 epidemic outbreak that initiates from a small number of cases in a small subregion of the geographical setting.
Methods:
Partial differential equations are formulated to describe the interaction of the model compartments.
Results:
The partial differential equations of the model are analyzed and proven to be wellposed. The epidemic outcomes of the model are correlated to the spatially dependent parameters and initial conditions of the model.
Conclusions:
A version of the model is applied to the 20152016 Zika outbreak in the Rio de Janeiro Municipality in Brazil.
1 Background
The model describes an outbreak epidemic with hostvector population dynamics in a geographical region. The epidemic outbreak begins at time 0 in a small subregion in which the epidemic disease is not yet present. The goal of the model is to aid understanding of how the introduction of a very small number of cases in a specific location in the geographic region will result in a dissipation or a sustained and growing epidemic. The focus of the study is on the importance of spatial effects in these possible outcomes. If the equations of the model do not depend on spatial considerations, then a corresponding system of ordinary differential equations can be analyzed for their asymptotic behavior (Appendix).
The geographical region is denoted by . The background population of uninfected hosts in has geographic density , which is unchanging in time in the demographic and epidemic context of the outbreak. Thus, the model is viewed as applicable to an early phase of the epidemic, during which the epidemic does not alter the basic geographic and demographic population structure of hosts, and the susceptible host population is not altered significantly by immunity to reinfection.
1.1 Compartments of the Model
The model consists of the following compartments:

The density of infected hosts at time at , with initial condition .

The density of uninfected vectors at time at , with initial condition .

The density of infected vectors at time at , with initial condition .
The initial state consists of a relatively small number of infected hosts located at time in a small subregion of . This input corresponds to an arrival of infected hosts from outside . This input is assumed to have a threshold level, which may include multiple cases produced from arriving cases. The background uninfected mosquito population has an initial state , which decreases as the infected vector population increases. The initial population of infected vectors in is assumed proportional to .
1.2 Equations of the Model
The equations of the model in the case that transmission from vectors to hosts is yearround are
1.3 Wellposedness of the Model
Theorem. Let be a bounded domain in with smooth boundary such that lies locally on one side of . Let , , , , , and let . There exists a unique global classical solution , to (1.2),(1.2),(1.2), satisfying boundary conditions
and initial conditions
Proof. We first observe that a unique classical solution exists in on a maximal interval of existence ([1], [2], [3]). Standard arguments ([3]) guarantee that remain nonnegative for . Moreover, the classical solution can be globally defined if we can establish uniform a priori bounds. Set and add equations (1.2) and (1.2) to obtain
Theorem 1 in [4] guarantees the existence of a unique global classical solution to Equation (1.3) satisfying
Further, in [4] it is proved that there exists , , such that . We note that the disease free equilibrium of (1.2),(1.2),(1.2) is . From [4] there exists such that , which implies . Then, since in (1.2), there exists such that . Consequently, the solution exists globally on .
2 The Spatially Structured Basic Reproduction Number
Define the basic reproduction number of the model (1.2),(1.2),(1.2) as
is interpreted as the average number of new cases generated by a single case at a given location in . Our motivation for this definition is the basic reproduction number of the spatially independent model (Appendix). Simulations of the spatially dependent model show that for certain parameterizations of equations (1.2),(1.2),(1.2), the solutions have the following behavior: (1) If everywhere in , then the populations of both infected hosts and infected vectors extinguish, and the populations converge to the disease free equilibrium. (2) If in some subregion , then the populations of both infected hosts and infected vectors converge to an endemic equilibrium independently of the initial conditions, and even if the average value of in all of is .
3 The 2015 Zika Outbreak in Rio de Janeiro Municipality
We apply a version of the model (1.2),(1.2),(1.2) to the 2015 Zika outbreak in Rio de Janeiro Municipality in Brazil. Because disease transmission in the Municipality is seasonal, equations (1.2) and (1.2) must be modified to account for seasonality. We assume that there is a mosquito population breeding term , depending on time. We also assume that there is an ongoing mosquito loss term , corresponding to the average mosquito lifespan , independent of the carrying capacity loss term . The modified equations are
The host population are the people in the Municipality, which in 2016 is approximately 6,000,000, in a geographical region of approximately 1,200 square kilometers (Source: Instituto Brasileiro de Geografia e Estatistica). The vector population is the female Aedes aegypti mosquito. The Municipality comprises 33 subdistricts, with population densities ranging from 1,000 to 50,000 inhabitants per square kilometer (Figure 1).
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 ([5], [6]). The Brazilian Health Ministry ([7],[8]) reported that Rio de Janeiro State (population approximately 16,000,000) registered a count of 25,930 cumulative cases from January 1, 2016 to April 1, 2016 (with incidence of 156.7 cases per 100,000 inhabitants), and 32,312 cumulative cases by April 23, 2016 (with incidence of 195.2 cases per 100,000 inhabitants). The Ministry ([9],[10]) reported no new cases in the State from April 24, 2016 to May 7, 2016. In [11] 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 period can be viewed as the 20152016 seasonal mosquito transmission period of the epidemic in the Municipality.
3.1 Parameterization of the Rio de Janeiro Model
We simulate this case data for the Rio de Janeiro Municipality with the following parameterization: The time units are weeks. The spatial units are kilometers and . The average length of the infectious period of infected people is approximately 1 to 2 weeks and we set ([12]). The average lifespan of female Aedes aegypti mosquitoes is approximately two weeks in an urban environment ([13],[14],[15]), and we set . The total uninfected host population is , with geographical density function (Figure 2A), which corresponds approximately to the population density distribution in Figure 1.
Set and the density dependent mosquito loss function (Figure 2B), which corresponds to higher levels of mosquito control in the eastern region of the Municipality, where the population density is highest. Here is the probability density function in of the normal distribution function with mean and standard deviation . Set the transmission parameters , (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). Set , which is a simplified estimate of human dispersal in urban settings. Set , which is consistent with an estimated adult mosquito dispersal of per day ([15]).
The time dependent mosquito breeding function is , where and is the shifted exponentially modified gaussian
Here is the complementary error function. The parameters are , , . The graph of the seasonal mosquito breeding function is given is Figure 3 ( is independent of and ).
3.2 Initialization of the Rio de Janeiro Model
The outbreak begins at time 0 on November 1, 2015 in a northeastern location of the Municipality with high population density. The total number of infected cases at time is , with spatial distribution . At time the total number of uninfected mosquitoes is , distributed uniformly throughout the Municipality. The total number of infected mosquitoes at time is , with .
3.3 Simulation of the Rio de Janeiro Model
Example 1. The simulation of the model (1.2), (3), (3) over the time period November 1, 2015 to May 21, 2016 is graphed in Figures 4, and 5. The simulation agrees qualitatively with the weekly reported case data for Rio de Janeiro Municipality in [11] (Figure 4). The spatial distributions of infected people expand from a very small number of initial cases in a small subregion of the Municipality, and disperses throughout the eastern region of the Municipality (Figure 5). The mosquito population rises rapidly and reaches carrying capacity at approximately 14 million in earlier 2016. During much of mosquito season, the ratio of mosquitoes to people is approximately to , which agrees with the ratio in ([16]).
Example 2. We repeat the simulation with the only change the location of the initial infected cases. We take (Figure 6). The infected population again expands from the initial location and disperses throughout the eastern region of the Municipality, but at approximately onehalf the number of infected cases as in Example 1. The reason is that the density of susceptible people is lower in this initial location than the initial location in Example 1.
4 Conclusions
The model (1.2),(1.2),(1.2) describes crisscross vectorhost transmission dynamics of an epidemic outbreak in a geographical region . The outbreak occurs with a small number of infected hosts in a small subregion of the much larger geographical region . The diffusion terms describe the ongoing average spatial movement of vectors and 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.
4.1 Summary of the Outcomes of the Outbreak Model
We prove that the partial differential equations model (1.2),(1.2),(1.2) 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 basic reproduction number . If everywhere in , then the epidemic will extinguish. If in some subregion of , then the epidemic will spread and converge to an endemic equilibrium throughout all of , independently of the location of the subregion.
4.2 Summary of the Model Applied to the Zika Outbreak in Rio de Janeiro
The model (1.2),(1.2),(1.2) is modified to allow seasonality of the vector population, and applied to the 20152016 Zika outbreak in Rio de Janeiro Municipality. A simulation of the model provides qualitative agreement with the case data reported by the Brazilian Ministry of Health in [11]. The model simulation suggests that the Zika epidemic in Rio de Janeiro Municipality, will rise each season from initial locations with very small numbers of infected people, and spread throughout the Municipality. The evolution of the epidemic depends on the initial location of infected cases. In general, the evolution of the epidemic depends on the parameterization and initialization of the model, and can be limited only by reduction of the disease parameters throughout the entire region. If the number of actual infected cases is significantly higher than the number of reported cases, then the parameterization must be adjusted accordingly.
Appendix
The equations (1.2),(1.2),(1.2) without spatial dependence are
(7)  
(8)  
(9) 
with initial conditions , , . Set . The behavior of solutions of equations (7),(8), (9) can be classified as follows:
Proposition If , then the only steady states of (7),(8),(9) in are , which is unstable in , and , which is proportional to and locally exponentially asymptotically stable in . If , , and , then converges to . If , then and are unstable in and there is another steady state in ,
which is locally exponentially asymptotically stable in .
Proof. It can be verified that the steady states of (7),(8),(9) in are , and . The Jacobian of (7),(8),(9) at is
with eigenvalues , which means that is unstable.
If and , then (7) implies . Assume there is a smallest positive time such that . Then (7) implies . If , then (9) implies
Then (7) implies which implies is strictly decreasing at , yielding a contradiction. Thus, is strictly decreasing for all . Let . Assume . Then (9) implies . Equation (8) then implies . Then is a steady state of (7),(8),(9). If , then , yielding a contradiction. Thus, .
are
Thus, is unstable if and locally exponentially asymptotically stable if .
with eigenvalues
Since if , the eigenvalues of the Jacobian at are strictly negative if , which means that is locally exponentially asymptotically stable if .
Authors contributions
All authors conceived and developed the study. All authors read and approved the final manuscript.
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