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An outbreak vectorhost epidemic model with spatial structure: the 2015–2016 Zika outbreak in Rio De Janeiro
Theoretical Biology and Medical Modellingvolume 14, Article number: 7 (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.
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 geographical region is denoted by Ω⊂R ^{2}. The background population of uninfected and susceptible hosts in Ω has geographic density H _{ u }(x,y), which is assumed 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 local geographic and demographic population structure of hosts. The model consists of the following compartments:

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
The equations of the model in the case that transmission from vectors to hosts is yearround are
In addition, the following boundary and initial conditions are satisfied:
The spatially dependent parameters of the model are as follows: λ(x,y) is the loss rate of the infected host population (due to recovery or other removal). β(x,y) is the breeding rate of the vector population. μ(x,y) is the loss rate of the vector population due to environmental crowding. σ _{1}(x,y) is the transmission rate of uninfected hosts and σ _{2}(x,y) is the transmission rate of uninfected vectors. The transmission terms for both hosts and vectors are assumed to be of densitydependent form, rather than frequencydependent form [25]. A comparison of the two forms for spatially dependent models is given in [26]. Since we assume, during the early phase of the epidemic, that the populations of infected hosts (infected vectors) are relatively small fractions of the populations of uninfected hosts (uninfected vectors), the two forms are essentially the same. δ _{1}(x,y) and δ _{2}(x,y) are the diffusion rates of the infected hosts and infected vector populations, respectively.
In the Appendix we prove the wellposedness of the model.
The local basic reproduction number
Define the local basic reproduction number of the model (1), (2), (3) as follows:
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
If the vector population is seasonal, then equations of the model must be modified to account for seasonality. We assume that the vector population breeding term β(t,x,y) is dependent on time. We assume that in addition to the vector loss parameter μ(x,y) corresponding to carrying capacity, there is a timeindependent vector loss term μ _{1}(x,y), corresponding to the average vector lifespan 1/μ _{1}(x,y). The modified equations are
The 2015–2016 Zika outbreak in Rio de Janeiro municipality
We apply the model (1), (4), (5) to the 2015–2016 Zika epidemic in Rio de Janeiro, Brazil. 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 (Fig. 1). The period November through July can be viewed as the seasonal Zika transmission period of the epidemic in the 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.
Explanations for our assumptions on specific parameter values are as follows: The time units for our simulations are weeks. The spatial units are kilometers and Ω=(−25,25)×(−12,12). The boundary conditions for Ω are a reasonable simplification of the costal boundaries and the less populated northern boundary of Rio de Janeiro Municipality. The average length of the infectious period of infected people is approximately 1 to 2 weeks and we set λ(x,y)=1.0 [33, 34]. The average lifespan of female Aedes aegypti mosquitoes is approximately two weeks in an urban environment [35, 36], and we set μ _{1}(x,y)=0.5. The total uninfected host population is approximately approximately 6,000,000, with geographical density function H _{ u }(x,y)=50.0+10^{2} (1.0+ sin(0.02π x) cos(0.03π y)) (Fig. 2 a), which corresponds approximately to the population density distribution in Fig. 1.
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].
For simplicity, we assume that the mosquito lifespan is independent of spatial location, and also independent of time in the season, although for some Aedes species, in some environments, the lifespan is correlated to temperature [35]. We take the time dependent mosquito breeding function as \(\beta (t,x,y) = 300.0 \, emg(t,\bar {\mu },\bar {\sigma },\bar {\lambda)}\), where emg is the shifted exponentially modified gaussian
Here Erfc is the complementary error function. The parameters are \(\bar {\mu } = 2.0\), \(\bar {\sigma } = 5.0\), \(\bar {\lambda } = 0.2\). The graph of the seasonal mosquito breeding function β(t) is given is Fig. 3. The assumptions on the parameters of the mosquito population yield a very rapidly rising population at the beginning of the season, which quickly stabilizes to maximal capacity of approximately 14 million, and then declines gradually to very low levels from midway through the season to the end of the season. The total mosquito population, both uninfected and infected, remains mostly uniformly spatially distributed throughout the Municipality throughout the season. The infected mosquito population spatial distribution is very similar to the spatial distribution of infected people. During much of mosquito season, the ratio of total mosquitoes to total people is approximately 2 to 1, which agrees with the ratio in [15].
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.
Example 1. In Example 1 the outbreak begins at time 0 on November 1, 2015 in a small eastern location of the Municipality, where R _{0}(x,y) is very high. The total number of infected people at time 0 is 10 (H _{ i0}=10), with spatial distribution centered at x _{0}=15 and y _{0}=0, R _{0}(15,0)≈2.27. At time 0 the total number of uninfected mosquitoes is 120,000, distributed uniformly throughout the Municipality. The total number of infected mosquitoes at time 0 is 100, with spatial distribution V _{ i }(0,x,y)=10.0H _{ i }(0,x,y). The simulation of the model (1), (4), (5) over the time period November 1, 2015 to May 21, 2016 is graphed in Figs. 4, 5 and 6. The simulation agrees qualitatively with the weekly reported case data for Rio de Janeiro Municipality in [32] (Fig. 4). The spatial distribution of infected people expands from a very small number of initial cases in a small eastern subregion of the Municipality, and disperses throughout the eastern region of the Municipality (Fig. 5). In Fig. 6 we graph the total number of infected people and the total cumulative number of infected people throughout the season. The simulation agrees qualitatively with the weekly reported case data for Rio de Janeiro Municipality given in [32], with approximately 25,500 reported cases between week 44, 2015 and week 15, 2016.
Example 2. We repeat the simulation with the only change from Example 1 the location of the initial outbreak. We take the initial outbreak location as the center of the Municipality with x _{0}=0 and y _{0}=0, R _{0}(0,0)≈1.26. The total number of infected people at time 0 is 20 (H _{ i0}=20). The infected population again expands from the initial location and disperses throughout the eastern region of the Municipality, but at approximately onetenth of the number of infected cases as in Example 1 (Figs. 7 and 8). The reason is that R _{0}(x,y) is much lower in this initial location than the initial location in Example 1, and the rise of the epidemic is much slower than in Example 1. In Fig. 8 (bottom) we repeat the example with the center of the outbreak location at x _{0}=−10, y _{0}=0, R _{0}(−10,0)≈0.52. The infected cases decrease rapidly to 0, because R _{0}(x,y) is even lower in the region of the outbreak.
Example 3. We again repeat the simulation with the only change from Example 1 the location of the initial outbreak. We take the initial outbreak with two locations in the center of the Municipality with
The total number of infected people at time 0 is 30. The infected population again expands from the initial location and disperses throughout the eastern region of the Municipality, but at approximately onethird of the number of infected cases as in Example 1 (Figs. 9 and 10). The reason is that the R _{0}(x,y) is again lower in the initial outbreak locations than the initial outbreak location in Example 1.
Example 4. We again repeat the simulation with the only change from Example 1 the location of the initial outbreak. We take the initial outbreak with three locations in the eastern region of the Municipality with
The total number of infected people at time 0 is 45. The infected population again expands from the initial locations and disperses throughout the eastern region of the Municipality, with the number of total cumulative infected cases the same as in Example 1 (Figs. 11 and 12). The reason is that R _{0}(x,y) is high in the third location, as it is in the initial location in Example 1.
Example 5. We provide a simulation of the model Eqs. (1), (2), (3) to illustrate that the solutions may approach an endemic steady state even if the average value of R _{0}(x,y)<1 in the spatial domain. We use the same parameters as in Rio de Janeiro Municipality, except that σ _{1}(x,y)=0.0000001, σ _{2}(x,y)=0.1, μ(x,y)=0.00005(1.0+100 g a u s s(−20.0,10.0,x)×g a u s s(0.0,10.0,y)), δ _{1}=0.1, δ _{2}=0.3, and β is set at the constant value 0.5 (the mosquito population is assumed to be present yearround rather than seasonal). The average value of R _{0}(x,y) in the whole region is ≈0.984. The results are illustrated in Figs. 13 and 14. For initial data in the eastern region (where R _{0}(x,y)>1), the number of infected cases increases and converges to an endemic steady state. For initial data in the western region (where R _{0}(x,y)<1), the number of infected cases first decreases, and then increases to the same endemic state. The simulations indicate the importance of spatial heterogeneity in epidemic models, especially for outbreak scenarios. The importation of a small number of infected cases to isolated localities, may at first dissipate in subregions with R _{0}(x,y)<1, but later rise and spread to subregions with R _{0}(x,y)>1, and establish endemicity in the greater geographical region.
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
Theorem. Let Ω be a bounded domain in R ^{2} with smooth boundary ∂ Ω such that Ω lies locally on one side of ∂ Ω. Let β, μ, λ, σ _{1}, \(\sigma _{2}, \delta _{1}, \delta _{2} \, \in C_{+}^{0}(\overline {\Omega })\), and let \(H_{u}, I_{0}, V_{u0},V_{i0} \in C_{+}^{1}(\overline {\Omega })\). There exists a unique global classical solution \(\{H_{i}(t),V_{u}(t),V_{i}(t)\} \in C_{+}^{1}(\overline {\Omega }), \, t \geq 0\), to (1), (2), (3), satisfying boundary conditions
and initial conditions
Proof. We first observe that a unique classical solution {H _{ i }(t),V _{ u }(t),V _{ i }(t)} exists in \(C^{1}(\overline {\Omega })\) on a maximal interval of existence [0,T _{ max }) [40–42]. Standard arguments [42] guarantee that {H _{ i }(t),V _{ u }(t),V _{ i }(t)} remain nonnegative for t∈[0,T _{ max }). Moreover, the classical solution can be globally defined if we can establish uniform a priori bounds. Set M(t,x,y)=V _{ u }(t,x,y)+V _{ i }(t,x,y) and add Eqs. (2) and (3) to obtain
Theorem 1 in [24] guarantees the existence of a unique global classical solution \(M(t) \in C_{+}^{1}(\overline {\Omega })\) to Eq. (6) satisfying
Further, in [24] it is proved that there exists \(\overline {M} \in C_{+}^{0}(\overline {\Omega })\), \(\overline {M} \neq 0\), such that \({\lim }_{t \rightarrow \infty }M(t) = \overline {M} \in C_{+}^{0}(\overline {\Omega })\). We note that the disease free equilibrium of (1), (2), (3) is \((0,\overline {M},0)\). From [24] there exists N _{1}>0 such that \({max}_{t \geq 0} \ M(t) \_{C_{+}^{0}(\overline {\Omega })} < N_{1}\), which implies \(\ V_{i}(t) \_{C_{+}^{0}(\overline {\Omega })}, \, \ V_{u}(t) \_{C_{+}^{0}(\overline {\Omega })} < N_{1}\). Then, since λ>0 in (1), there exists N _{2}>0 such that \(\ H_{i}(t) \_{C_{+}^{0}(\overline {\Omega })} < N_{2}\). Consequently, the solution exists globally on [0,∞).
The model equations without spatial dependence
The Eqs. (1), (2), (3) without spatial dependence are
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:
Proposition If R _{0}<1, then the only steady states of (7), (8), (9) in \(R_{+}^{3}\) are s s _{0}=(0,0,0), which is unstable in \(R_{+}^{3}\), and s s _{1}=(0,β/μ,0), which is proportional to β and locally exponentially asymptotically stable in \(R_{+}^{3}\). If R _{0}<1, H _{ i }(0)>0, and V _{ i }(0)=0, then (H _{ i }(t),V _{ u }(t),V _{ i }(t)) converges to \((0,\overline {M},0)\). If R _{0}>1, then s s _{0} and s s _{1} are unstable in \(R_{+}^{3}\) and there is another steady state in \(R_{+}^{3}\),
which is proportional to β and locally exponentially asymptotically stable in \(R_{+}^{3}\).
Proof. Set M(t)=V _{ u }(t)+V _{ i }(t) and \(\overline {M} = \beta / \mu \). Then M ^{′}(t)=β M(t)−μ M(t)^{2} and \({\lim }_{t \rightarrow \infty } M(t) = \overline {M}\). It can be verified that the steady states of (7), (8), (9) in \(R_{+}^{3}\) are s s _{0},s s _{1}, and s s _{2}. The Jacobian of (7), (8), (9) at s s _{0} is
with eigenvalues {−λ,β,0}, which means that (0,0,0) is unstable. If H _{ i }(0)>0 and V _{ i }(0)=0, then (7) implies \(H_{i}^{\prime }(0) < 0\). Assume there is a smallest positive time t ^{∗} such that \(H_{i}^{\prime }(t^{\ast }) = 0\). Then (7) implies H _{ i }(t ^{∗})=(σ _{1} H _{ u }/λ)V _{ i }(t ^{∗}). If R _{0}<1, then (9) implies
Then (7) implies
which implies H _{ i }(t) is strictly decreasing at t ^{∗}, yielding a contradiction. Thus, H _{ i }(t) is strictly decreasing for all t≥0. Let \(H_{i,\infty } = {\lim }_{t \rightarrow \infty } H_{i}(t) \geq 0\). Assume H _{ i,∞ }>0. Then (9) implies l i m _{ t→∞ } V _{ i }(t)=λ H _{ i.∞ }/σ _{1} H _{ u }>0. Equation (8) then implies \({\lim }_{t \rightarrow \infty }V_{u}(t) = \beta \overline {M} / (\sigma _{2} H_{i,\infty } + \mu \overline {M})\). Then \((H_{i,\infty },\beta \overline {M} / (\sigma _{2} H_{i,\infty } + \mu \overline {M}),\lambda H_{i,\infty } / (\sigma _{1} H_{u}))\) is a steady state of (7), (8), (9). If R _{0}<1, then H _{ i,∞ }=0, yielding a contradiction. Thus, H _{ i,∞ }=0.
The eigenvalues of the Jacobian of (7), (8), (9) at s s _{1}
are
Thus, J(0,β/μ,0) is unstable if R _{0}>1 and locally exponentially asymptotically stable if R _{0}<1.
The Jacobian of (7),(8),(9) at s s _{2} is
with eigenvalues
Since −(R _{0} β+λ)^{2}+(R _{0} β−λ)^{2}+4β λ=−4(R _{0}−1)β λ<0 if R _{0}>1, the eigenvalues of the Jacobian at s s _{2} are strictly negative if R _{0}>1, which means that s s _{2} is locally exponentially asymptotically stable if R _{0}>1.
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Keywords
 Crisscross dynamics
 Local reproduction number
 Zika epidemic