On the Dynamics of Dengue Virus type 2 with Residence Times and Vertical Transmission

A two-patch mathematical model of Dengue virus type 2 (DENV-2) that accounts for vectors' vertical transmission and between patches human dispersal is introduced. Dispersal is modeled via a Lagrangian approach. A host-patch residence-times basic reproduction number is derived and conditions under which the disease dies out or persists are established. Analytical and numerical results highlight the role of hosts' dispersal in mitigating or exacerbating disease dynamics. The framework is used to explore dengue dynamics using, as a starting point, the 2002 outbreak in the state of Colima, Mexico.


Introduction
Dengue, a re-emerging vector-borne disease, is caused by members of the genus Flavivirus in the family Flaviviridae with four active antigenically distinct serotypes, DENV-1, DENV-2, DENV-3, and DENV-4 [22]. The pathogenicity of dengue can range from asymptomatic, mild dengue fever (DF), to dengue hemorrhagic fever (DHF), and dengue shock syndrome (DSS) [22,34]. Although infection with a dengue serotype does not usually protect against other serotypes, it is belief that secondary infections with a heterologous serotype increase the probability of DHF and DSS [12,33]. According to the World Health Organization, 40% of the global population is at risk for dengue infection with an estimate of 50 to 100 million infections yearly including 500,000 cases of DHF. It has been estimated that about 22,000 deaths, mostly children under 15 years of age, can be attributed to DHF [72]. In the United States, approximately 5% or more of the Key West population in Florida was exposed to dengue during the 2009-2010 outbreak [15] while the Hawaii Department of Health reported 190 cases during the 2015 outbreak on Oahu, the first outbreak since 2011. Since dengue is not endemic in Hawaii, health authorities have suggested that the recent outbreak may have been started by infected visitors [57]. Dengue is highly prevalent and endemic in Southeast Asia, which has experienced a 70% increase in cases since 2004 [40]; Mexico, also an endemic country, reported during the 2002 Mathematical models describing the dynamics of interaction between host and vector go back to Ross [64], Lotka [44] and MacDonald [45]; first used to study vector-host dynamics in the context of Malaria [11,30,65]. Variations of such framework have been applied to dengue ( for a review see [67]). Further applications of modeling variations in the context of Malaria include, [27,48,53,55] and in the context of dengue [13,20,30,51,56].
The potential role of vertical transmission in dengue endemic regions or in fluctuating environments has been explored in [1,26,56]. The role in the displacement of DENV-2 American via DENV-2 Asian vertical transmission has also been addressed [51]. The role of host movement has also been explored in the context of dengue [2] in a formulation that does not account for the the effective population size. In this paper, the role of vertical transmission and movement via residence times are explored via a two-patch model involving non-mobile vectors and mobile hosts. This paper is organized as follows: The derivation of the model is presented in Section 2; Analytical results are collected in Section 3; The results of numerical simulations are collected in Section 4; Section 5 explores the possible role of movement on joint dynamics of dengue in Colima and Manzanillo in the presence of host mobility; Concluding remarks are collected in Section 6.
2 Derivation of the model A single patch model is derived and embedded into a two-patch model via a residence-times matrix in order to study the impact of host mobility on dengue disease dynamics. Conditions for dengue eradication and persistence in the population are computed.

Single patch model
We consider a population of host composed of susceptible (S h ), exposed (E h ), infectious (I h ) and recovered (R h ) individuals interacting with a vector population composed of susceptible (S v ), exposed (E v ) and infected (I v ) vectors. The dynamics of dengue follows an SEIR structure for the host population and an SEI type for the vector population. The birth rate for the host population is µ h , assumed to be equal to the death rate, that is, hosts' demographic differentials are conveniently ignored, that is, the host population is assumed to be constant. Susceptible hosts are infected, by infectious mosquitoes, at the rate aβ vh Iv N h where a is the biting rate and β vh is the infectiousness of human to mosquitoes. The exposed population develops symptoms becoming infectious at the rate ν h . Infectious individuals recover at the per-capita rate γ. Susceptible mosquitoes become infected, via interactions with infectious hosts, at the rate aβ hv I h N h . Recent studies place significant importance to the connection between DENV-2 and DHF cases [19,25,49,61,66,74] and on DENV-2 vertical transmission [47]. Hence, it is assumed that a fraction q of the mosquitoes are "born" infected entering directly the infectious class. The natural per-capita vector mortality is µ v .
The model describing the dynamics of DENV-2 is given by the following system of differential equations: In the absence of selection, that is, differences in birth and death rate and in the absence of vertical transmission, Model (1) turns out to be isomorphic to model considered by Chowell et al in [18]. Model (1) is well defined supporting a sharp threshold property, namely, the disease dies out if the basic reproduction number R 0 is less than unity, persisting whenever R 0 > 1 where

Heterogeneity through virtual dispersal
The single patch model is the building block for the two-patch model used in this study. Within each patch, in the absence of host mobility, dengue dynamics are modeled via System 1. A metapopulation approach, an Eulerian perspective, is most often applied to the study of vector-borne diseases involving host mobility ( [2,4,28]). Here, a Lagrangian approach is used instead to model the movement of individuals between patches (see [7,8]). It is assumed that vectors don't move between patches since vecors Ae. aegypti and Ae. albopictus do not travel more than few tens of meters over their lifetime [2,58]; moving 400-600 meters at most [9,54], respectively. In short, we neglect vector's dispersal, which fits well the simulations involving two cities in the state of Colima, Mexico.
And so, the effective infectious population in Patch 1 is p 11 I h,1 + p 21 I h,2 and, consequently, the proportion of infectious individuals in Patch 1, is The dynamics of susceptible mosquitoes in Patch 1 are modeled as follows: The complete dynamics of DENV-2, with the host moving between patches, is given by the following system, Since the total populations of hosts and vectors are constant in each patch, System (4) has the same qualitative dynamics as, The parameters of Model 5 are described in Table 1.
is a compact positively invariant for the System (5).

Proof.
The positive orthant is clearly positively invariant. Since the host population is constant, then the inequality , is positively invariant; the set is a compact set.

Equilibria and stability analysis
This section characterizes the equilibrium dynamics of Model (5).

The disease free equilibrium and the basic reproduction number
The disease free equilibrium is which is used to compute the basic reproduction number via the next generation method [24,71]. The basic reproduction number R 0 is defined by the expression (See Appendix A, for details), R 2 0 = ρ(M vh M hv ), that is, the spectral radius of the matrix of M vh M hv , where is called the host-vector network configuration [38]. The result of local asymptotic stability if R 2 0 < 1 and instability if R 2 0 > 1 has been established in [71]. The following theorem gives the global result of the DFE.
We use the comparison theorem [68] to prove the GAS of the DFE.
We define an auxiliary system via the right hand side of Equations (6)-(7) and the infected compartments of Equation (5) as follows: where the matrices F and V in (8) were just generated using the next generation method. System (8) is linear and its dynamics is well known. Since V is a Metzler matrix and F a nonnegative matrix Since, all the variables in System (5) are nonnegative, the use of a comparison theorem [68] leads to, Therefore, by using the asymptotic theory of autonomous systems [14], System (5) has the qualitative dynamics of the following limit system:Ṡ for which the equilibrium (N h,1 , N h,2 ) is globally asymptotically stable. If R 0 > 1, the instability of the DFE follows from [24,71].
Let φ t be semi-flow induced by the solutions of (5) and Therefore a global attractor for φ t exists . The DFE is the unique equilibrium on the manifold ∂X 0 and is GAS on ∂X 0 . Moreover ∪ x∈M ∂ ω(x) = {E 0 } and no subset of M forms a cycle in ∂X 0 . Finally since the DFE is unstable on X 0 if R 0 > 1, we deduce that System (5)  Proof. We will use a result by Hethcote and Thieme [37] to prove the uniqueness of the endemic equilibrium. An endemic equilibrium (S h,1 ,S h,2 ,Ē h,1 ,Ē h,2 ,Ē v,1 ,Ē v,2 ,Ī h,1 ,Ī h,2 ,Ī v,1 ,Ī v,2 ) satisfies: The first equation of (9) implies that Hence, we deduce that, from System (9), that Let . The function F (x) is continuous, bounded, differentiable and F (0 R 6 ) = 0 R 6 . The function F is monotone if the corresponding Jacobian matrix is Metzler, i.e all off-diagonal entries are nonnegative. We have: Since,m ij vh ≥ 0 andm ij hv ≥ 0 for all i, j = 1, 2, hence all off diagonal entries of the Jacobian matrix are nonnegative and so, the function F (x) is monotone, moreover, This matrix is irreducible wheneverM vh (0)M hv (0) andM hv (0)M vh (0) are irreducible. The latter is guaranteed since M vh M hv and M hv M vh (from the next generation matrix) are both irreducible. Hence, an application of Theorem 2.1 in [37] implies that Model (10)

Simulations
Simulations are carried out in order to highlight the effects of residence times on disease dynamics. The simulations have a dual goal, first, to illustrate the theoretical results of this manuscript and secondly to illustrate the impact of host mobility across high and low-risk dengue areas.
The basic reproduction reproduction number R 0 (P) is a function of the residence times matrix P.
Simulation baseline values, except for those involving the entries of P are as follows: The values of the parameters ν h and ν v are taken from [1,2]. The infectiousness parameters (β hv and β vh ) and vector's natural mortality rate are taken from [17]. Host and vector population are N h,1 = 400, 000 N h,2 = 300, 000 N v,1 = 35, 000, N v,2 = 30, 000 Patch 1 is the high-risk and Patch 2 is the low-risk and so, it is assumed that a 1 > a 2 . Figure 1 represents the dynamics of Patch 1 (Fig 1(a)) and Patch 2 (Fig 1(b)) infected hosts while Fig 2 collects the vector dynamics in both patches. Since Patch 1 is high-risk, the number of infected host should decrease as p 12 increases; see Fig 1(a). Figure 1(b) shows the Patch 2 infected host population, which it is decreasing, as p 21 and p 12 increase. Disease prevalence among Patch 2 residents remains very small when compared to that in Patch 1. In Fig 2, Patch 1 (Fig 2(a)) and Patch 2 (Fig 2(b)) vector dynamics are seen to follow the hosts' endemicity pattern.
For all the different values of p ij chosen in Fig 1 and Fig 2, the host-vector configuration matrix or equivalently, the products M hv M vh and M vh M hv , are irreducible. Moreover the basic reproduction number R 0 is greater than one, hence the the disease is, in both patches, at an endemic level.
(a) The level of infected host in Patch 1 seems to decrease as p12 increases (and hence p11 decreases).
(b) The level of infected host in Patch 2 seems to decrease with respect to p22. (a) Asymptotically, the level of infected vectors in Patch 1 seems to decrease as p12 increases (and hence p11 decreases).
(b) Asymptotically, the level of infected vectors in Patch 2 seems to decrease with respect to p22.

Colima City and Manzanillo Dengue Inspired Simulation Study
Ae. aegypti was declared eradicated in Mexico in 1963. Not surprisingly, all four dengue serotypes (DENV-1, DENV-2, DENV-3 and DENV-4) re-emerged two years after local the 1963 eradication [23]. Further, DHF cases have steadily increased since 1994 [52]. Dengue is endemic in Mexico with approximately 60% of year-round cases reported in southern part of the country; a region is characterized by a warm and humid climate [21]. Colima, located on the central Pacific Coast (see Figure 4), is also a reservoir of Dengue. In 2002, the State of Colima reported 4,040 cases dengue in all of its 10 municipalities; 495 progressing to DHF [19,25]. DENV-2 was isolated from patients during this outbreak [25]. The increase in DHF cases in Mexico has been linked to the introduction of DENV-2 Asian, previously isolated in 2000 and again in 2002 [43].
The dynamics of dengue are explored in the context of this 2002 State of Colima outbreak. The first reported (index) case was identified as that of a 10-year-old female in the municipality of Manzanillo on January 11, 2002. Dengue infection spread throughout the whole state with the most affected municipalities being Colima city, the capital of the state, and Manzanillo, an important tourist destination in the coast [25]. The city of Colima reported approximately 1,167 dengue cases, with 169 cases progressing to DHF while Manzanillo, reported 1,334 dengue cases, with 123 progressing to DHF in 2002 [18]. The city of Colima and Manzanillo are linked via high levels of travel and tourism. Both cities account for approximately 47% of the state population. We apply a two-patch model to explore the role that movement, modeled via the matrix p ij , may have had on dengue disease transmission during this 2002 outbreak. The estimated population of Manzanillo and Colima City were N h,1 = 1, 355 and N h,2 = 1, 184, respectively, and the initial mosquito populations were choosen to best fit the data. They were approximately 308 and 738 in Manzanillo and Colima City, respectively. Note that the host population is not the actual population of the cities but rather the population at risk in each of the corresponding cities. The population at risk is much smaller that the actual population because in the same city there are social groups practically disconnected to others by geographic, cultural and social factors. Entomological parameters were estimated using [73] and taking into account the mean temperature in each region [18]. The remaining parameters used to study the outbreak in Colima, Mexico were obtained from the literature [1,17,29,73]: In order to assess, within our staged scenarios, the impact of migration during the 2002 dengue outbreak, we fit the two-patch model using the incidence data for Manzanillo and the city of Colima reported by the Mexican Social Security Institute (IMSS) during the outbreak (see Figure 5). The data fitting for cumulative dengue cases given by the model using 'scipy.optimize.curve fit' library of python v2.7 programming language, is shown in Figure 7. Model results show that dengue spreads more quickly in the city of Colima when the proportion of visits from Manzanillo's infected residents is high, see the left panel of Figure 6 compared with Figure 7. Alternatively, susceptible Colima City residents would acquire dengue infections over a longer time frame in Manzanillo, introducing the disease over a slower time scale in their home residence, the city of Colima. Of course, the absence of movement leads to no dengue cases in Manzanillo; an outbreak occurring only in Colima p 11 = p 22 = 1.0, see center panel of Figure 6; equal movement, p 11 = p 22 , would cause the outbreak in Colima to grow faster, as can be seen in the right panel of Figure 6. Hence, limiting the movement of the Manzanillo population seems like a good strategy while limiting the movement of the Colima population wouldn't be as effective. In the latest scenario, the economic cost would be high since Manzanillo is a tourist destination.    We can also observe in Figure 8 (on the left), that the effect of reducing the transit from Manzanillo to Colima city led only to a delay in the appearance of the outbreak in Colima. This indicates that the outbreak in Colima followed its own local dynamics and that transit between these two cities only led to delays in the introduction of the dengue virus without affecting the local outbreak dynamics. When the average visiting time spent in a place where the disease prevalence is low (small value of p ij , i = j) then the only way of reducing an outbreak would require strict migration control, that is, complete travel avoidance to the high risk zone. In Figure 8 (on the right), we see that with only a small fraction of visitors from Manzanillo to Colima, the outbreaks in both cities occur almost simultaneously. Model simulations re-affirm the views that the rate of host movement and time spent in endemic geographic regions are important for the spread of dengue between two patches. The question then becomes, why aren't then these residence times estimated? In the first scenario (on the left), the blue lines represent no transit control and the red lines represent a reduction of 90% in movement from Manzanillo city to Colima. In the second scenario (on the right), the blue lines represent no movement control and the red lines represent an increment of movement from Colima to Manzanillo city of 1%.

Conclusion
The persistence of vector-bone diseases, such as dengue, is connected to factors that include the presence ecological conditions that favor high vector densities, vector-host interactions, the spatial movement of humans, and of course, the effectiveness of control measures [46,69]. In this paper, a two-patch host-vector model was used to study the role of movement on the transmission dynamics of dengue, especially DENV-2. We focus on the applications of our framework to scenarios where dengue is endemic and where vertical transmission has been documented. A residence times matrix P is used to model host mobility. This modeling approach provides a framework for exploring spatial vector-borne disease dynamics and control within relatively "close' environments. Analytical results were derived and the conditions for which the disease dies out or persists have been identified; conditions that depend on whether the basic reproduction number R 0 (P) is less or greater than unity and the connectivity of patches.
Using data from the 2002 DENV-2 outbreak in Colima, Mexico, we compare the overall prevalence in the cities of Colima and Manzanillo as a function of pre-selected P matrices. Our model shows that reducing traveling from to Colima city, considered high-risk and the place of the 2002 outbreak onset, causes a slight delay in the spread of the disease. In order to completely prevent an outbreak in Colima city, migration between Colima city and Manzanillo must be stopped. Manzanillo a tourist destination implies that transit from Colima city to Manzanillo is expected to peak during certain seasons. The model suggests that dengue would become endemic in both patches almost simultaneously. The twopatch model highlights the role of human spatial movement on disease transmission and control. The strength of this effect depends on the proportion of time commuters to high or low risk spend in each patch.