Page 1

A Habitat-Based Model for the Spread of Hantavirus Between

Reservoir and Spillover Species

Linda J. S. Allena,*, Curtis L. Wesleyb, Robert D. Owenc,d, Douglas G. Goodine, David

Koche, Colleen B. Jonssonf, Yong-Kyu Chuf, J. M. Shawn Hutchinsone, and Robert L.

Paigea

a Texas Tech University, Department of Mathematics and Statistics, Lubbock, Texas 79409, U.S.A

b Louisiana State University at Shreveport, Department of Mathematics, Shreveport, Louisiana

71115, U.S.A

c Texas Tech University, Department of Biological Sciences, Lubbock, Texas 79409, U.S.A

d Martín Barrios 2230 c/Pizarro, Barrio Republicano, Asunción, Paraguay

e Kansas State University, Department of Geography, Manhattan, Kansas 66506, U.S.A

f Southern Research Institute, Department of Biochemistry and Molecular Biology, 2000 9th Avenue

South, Birmingham, Alabama 35206, U.S.A

Abstract

New habitat-based models for spread of hantavirus are developed which account for interspecies

interaction. Existing habitat-based models do not consider interspecies pathogen transmission, a

primary route for emergence of new infectious diseases and reservoirs in wildlife and man. The

modeling of interspecies transmission has the potential to provide more accurate predictions of

disease persistence and emergence dynamics. The new models are motivated by our recent work on

hantavirus in rodent communities in Paraguay. Our Paraguayan data illustrate the spatial and temporal

overlap among rodent species, one of which is the reservoir species for Jabora virus and others which

are spillover species. Disease transmission occurs when their habitats overlap. Two mathematical

models, a system of ordinary differential equations (ODE) and a continuous-time Markov chain

(CTMC) model, are developed for spread of hantavirus between a reservoir and a spillover species.

Analysis of a special case of the ODE model provides an explicit expression for the basic reproduction

number, ℛ0, such that if ℛ0 < 1, then the pathogen does not persist in either population but if ℛ0 >

1, pathogen outbreaks or persistence may occur. Numerical simulations of the CTMC model display

sporadic disease incidence, a new behavior of our habitat-based model, not present in other models,

but which is a prominent feature of the seroprevalence data from Paraguay. Environmental changes

that result in greater habitat overlap result in more encounters among various species that may lead

to pathogen outbreaks and pathogen establishment in a new host.

Keywords

Hantavirus; Interspecies pathogen transmission; Basic reproduction number

*Corresponding author: linda.j.allen@ttu.edu.

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting

proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could

affect the content, and all legal disclaimers that apply to the journal pertain.

NIH Public Access

Author Manuscript

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

Published in final edited form as:

J Theor Biol. 2009 October 21; 260(4): 510–522. doi:10.1016/j.jtbi.2009.07.009.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 2

1 Introduction

Successful transmission of a directly transmitted pathogen requires opportunities for contact

between species. These opportunities often occur when the preferred habitat of a species

overlaps or is invaded by a second species. Interspecies interactions, especially among species

competitively utilizing the same resources, often result in aggressive encounters. If a pathogen

is present in a reservoir host, the encounter may result in pathogen transmission to a naive host

or adaptation of the pathogen to create a new reservoir. The reservoir population, the carrier

of the pathogen and the long-term host, often does not exhibit disease symptoms or experience

any additional mortality.

In this investigation, we develop, analyze, and numerically simulate solutions to two new

habitat-based models for the spread of a directly transmitted pathogen between two species.

Our goal is to model the process of interspecies pathogen transmission based on species habitat

preferences. The motivation for the models comes from our recent study of hantavirus in

Paraguay. Hantavirus (Family Bunyaviridae) is a genus of viruses, each generally associated

with a specific rodent species (i.e., mice and rats). Approximately 30 different hantaviruses

exist throughout the world, some of which cause human infection (Mills et al., 1997). Human

infection is incidental, generally due to indirect transmission from contact with infectious

rodent excreta, but may result in hantavirus pulmonary syndrome with a mortality rate as high

as 37% (CDC, 2002). One of the reservoir species for hantavirus in Paraguay is Akodon

montensis (Montane Akodont found in Eastern Paraguay, Northeastern Argentina and

Southeastern Brazil) carrier of Jabora virus (JABV, GenBank # EF492471). Our empirical data

show that although these species exhibit different habitat preferences, the combination of

partial habitat flexibility and temporally variable climatic and soil and vegetation conditions,

results in periodic microgeographic sympatry of Akodon with one or both of the spillover

species.

Mathematical models for the spread of hantavirus in rodents have concentrated primarily on

the dynamics of the reservoir population (Abramson and Kenkre, 2003; Abramson et al.,

2003; Allen et al, 2006a; Allen et al., 2003; Allen et al., 2006b; Sauvage et. al, 2007; Sauvage

et al., 2003; Wesley, 2008; Wesley et al., 2009; Wolf et al., 2006). A multi-species epizootic

model for susceptible and infected hosts was formulated and analyzed by McCormack and

Allen (2007) but this model was not spatially explicit and did not account for differences in

epizootiology of reservoir and spillover species. Our new models take into account habitat

partitioning and important differences in the epizootiology of the reservoir and spillover

populations. The role of the spillover species in pathogen and disease emergence is not well

understood. It has been speculated that the spillover species may contribute to maintenance of

the pathogen in the wild, provided there is spillback infection (McCormack and Allen, 2007)

or the spillover species may be instrumental in the evolution of new hantaviruses (Chu et al.,

2006). Spillover infections occur in hantavirus (Delfraro et al., 2008; Palma et al., 2009;

Klingstrom et al., 2002; Torrez-Martinez et al., 2005; Wiedmann et al., 2005) but are not unique

to hantaviruses (Daszak et al, 2000); they have been documented in other zoonotic diseases

including rabies (Nadin-Davis and Loza-Ruio, 2006; Nel et al., 1997), Nipah virus (Chua,

2003), canine distemper, parvovirus (Fiorello et. al, 2006), and the SARS coronavirus (Holmes,

2003).

The habitat-based models consist of three regions: a preferred habitat for each of the reservoir

and the spillover populations, and a third region of overlap (or boundary region) where

interspecies encounters and pathogen transmission may occur. We formulate two models, the

first model is a deterministic model, a system of ordinary differential equations (ODE), whereas

the second model is a stochastic model, a continuous-time Markov chain (CTMC) model. The

ODE system is analytically tractable in the case that encounters in the boundary region are

Allen et al.Page 2

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 3

brief. In this case an explicit expression for the basic reproduction number, ℛ0, can be

calculated, the threshold for disease outbreaks. The basic reproduction number is one of the

most important parameters in the study of disease ecology. Specifically, it is the number of

secondary infections caused by introduction of one infectious individual into an entirely

susceptible population (see Anderson and May, 1991; Hethcote, 2000). The ODE system does

not capture the few cases that occur due to interspecies interactions in the region of overlap.

Therefore, we formulate a CTMC model for this purpose. Numerical solutions of the stochastic

model illustrate sporadic infection in the spillover species when habitats overlap, a prominent

feature of the seroprevalence data from Paraguay. Analysis and simulation of our new models

show that as the number of encounters in the overlap region and the time spent in the overlap

region increase (which may be triggered by habitat change), there is greater likelihood of

pathogen outbreaks and disease persistence in the reservoir and spillover populations (through

increase in ℛ0). Interspecies encounters and pathogen transmission in the region of overlap

may be the first step in the evolution of a new hantavirus strain.

2 Empirical Data and Motivation for the Model

Recent data collected in Paraguay (2005–2007) have shown cases of hantaviral infection in A.

montensis, the reservoir species for JABV. Spillover infection, presumably of JABV, has been

found in several other mouse species, including Necromys lasiurus (Hairy-tailed Akodont in

Central Brazil, Southeastern Peru, Eastern Paraguay and Northeastern Argentina) and

Oxymycterus delator (Paraguayan Hocicudo in Eastern Paraguay and South Central Brazil).

2.1 Habitat Characteristics

Our field work was conducted in the Mbaracayú Biosphere Reserve in eastern Paraguay, which

lies in the western-most portion of the Interior Atlantic Forest. Vegetation composition in this

area is typical of the mixture of intact, disturbed, and deforested areas found in eastern Paraguay

(Fernández Soto and Mata Olmo, 2001).

Within this landscape of mixed habitat types, rodents were sampled on two mark-recapture

grids, R3A and R3B, representing contrasting potential habitat for A. montensis (see Figure 1).

Site R3A is largely deforested, with its natural cover replaced by native and exotic graminoids

and forbs. This site is highly disturbed by human activities and is intensively managed for

pasturing and grazing. Reforestation is suppressed and graminoid cover maintained by frequent

prescribed burning. Large ungulate grazers, primarily domestic cattle (Bos taurus) are present

on this site year-round. Vegetation in the site is dominated by species of the genus

Andropogon, warm temperature/tropical grasses used as grazing forage. Other common

vegetation genera include Merostachys (bamboo) along the fringes of pastures and Xyris (a

forb) in lower, wetter areas. Although dominated by herbaceous species, R3A retains a few

islands of woody vegetation and trees, especially along the edges of the deforested areas. These

forest remnants are better microhabitats for Akodon, and most captures of this species were

along the northwest and southeast corners of the grid, the edges closest to the woodlands.

Site R3B contrasts with R3A in that its dominant cover consists of native forest and its

associated vegetation community. Although native cover remains, the site shows evidence of

recent human disturbance, especially selective logging and nearby road construction. These

disturbances have resulted in fragmentation of the native vegetation cover, producing

numerous internal edges and gaps in the forest canopy. These edges and gaps are associated

with a dense understory at the forest floor, favorable habitat for A. montensis (Pardiñas and

D’Elía, 2003). Dominant vegetation genera in areas of intact or mostly intact forest include

Cedrela and Balfourodendron. In disturbed areas, a lower canopy dominated by Sorocea

bonplandii (a lower, woody shrub-like tree) is common, with herbaceous species and

Bromelia common in the understory, and Merostachys frequently in forest gaps.

Allen et al. Page 3

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 4

2.2 Empirical Data

Each sampling grid consisted of an 11×11 square array of trap stations, set 10 m apart. One

standard Sherman Live Trap was set on the ground at each station. Where vegetation structure

permitted (R3B only), another trap was placed 2–3 m above ground, in branches or vines, to

sample other more arboreal species. However, although A. montensis is preferentially a forest-

dweller, it only rarely climbs in the vegetation (651 of 661 [98.5 %] of our captures of this

species on R3B were on the ground). Both grids were sampled in nine sessions from February

2005 through May 2007, using standard mark-recapture techniques for small mammals

(Wilson et al., 1996). Each session included eight consecutive nights (seven in February 2005).

Animals were individually marked with a subdermally-implanted Passive Integrated

Transponder (PIT) tag, which can be read by passing the electronic reader near the animal’s

body. Date, grid, trap station, and the PIT tag number were recorded, animals were identified

to species, and sex and weight were recorded. Blood, saliva, urine, and feces were collected

the first time an animal was captured during each sampling session, for assay of hantavirus

antibody and viral RNA presence. Animals were then released at the site of capture. All field

protocols were approved by the Texas Tech University Animal Care and Use Committee.

A total of 582 captures was recorded on R3B, and 332 on R3A. Species captured on R3B

included A. montensis, Calomys callosus, Oligoryzomys fornesi, O. nigripes, and Oryzomys

megacephalus; on R3A, A. montensis, C. callosus, C. tener, N. lasiurus, O. nigripes, and O.

delator were encountered. In this report we consider only the populations of A. montensis, N.

lasiurus, and O. delator, as exemplifying the scenario being modeled. A. montensis has been

described as the primary reservoir of JABV. RNA-positive individuals of Akodon were

encountered on R3B and R3A, and seropositive individuals of N. lasiurus and O. delator were

trapped and identified on R3A. Each of these species is widely distributed in the central

Southern Cone of South America (Carleton and Musser, 2005), and may be locally abundant

in their preferred habitat. Habitat preferences differ somewhat among the three, which is critical

to this field situation and to the model which we present herein. N. lasiurus and O. delator

prefer grasslands, with Necromys preferring dry soil and Oxymycterus preferring wet or even

saturated soils. A. montensis preferentially inhabits disturbed woodlands, also venturing into

old-fields and grasslands which include forbs and brushy growth (Redford and Eisenberg,

1992; Goodin et al., manuscript). Our data from nine sampling sessions through 27 months

support these descriptions of habitat preferences, and further indicate that the microgeographic

separation among these three species is partial and temporally variable and often incomplete,

with overlap (microsympatry) occurring sporadically between A. montensis and one or both of

the spillover species. Figure 2 illustrates this situation on site R3A, and also indicates the

presence of seropositive individuals for each of these species in close proximity to the others.

At site R3B, A. montensis was captured but neither N. lasiurus nor O. delator. Based on 2005–

2007 data, 21 out of 84 A. montensis males (25%) tested for hantavirus showed positive titers

for antibodies or RNA, including 12 (14.3%) that were only antibody-positive, eight (9.5%)

that were antibody-positive and RNA-positive, and one (1.2%) that was only RNA-positive.

Only four out of 68 A. montensis females (5.9%) tested for hantavirus showed positive titers

for antibodies with no detectable viral RNA. In other studies of hantavirus ecology, male

seroprevalence was higher than female seroprevalence (Bernshtein et al., 1999; Childs, et al.,

1994; Glass et al., 1998; Klein et al., 2001; McIntyre et al., 2005; Mills et al., 1997; Yahnke

et al., 2001).

3 Model Derivation

Based on the empirical data, we model only male rodents in the spread of hantavirus and use

two infectious stages for the reservoir host, a highly infectious stage and a persistent stage, I

and P. The highly infectious stage represents animals that are RNA-positive and may or may

Allen et al.Page 4

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 5

not have antibodies, and the persistent stage represents animals that are only antibody-positive.

This latter persistent stage is less infectious than the highly infectious stage. Two infectious

stages were assumed in models for Puumala hantavirus in bank voles (Sauvage et al., 2007;

Sauvage et al., 2003; Wolf et al., 2006).

First, two basic models for the reservoir and the spillover species, each within their own habitat,

are formulated. Then these two models are merged into a habitat-based model, where

interspecies pathogen transmission may occur in a region of overlap of the two habitats (such

as R3A).

3.1 Basic Model

The disease stages for the reservoir species include susceptible, Sr, exposed or latently infected,

Er, highly infectious, Ir, and persistently infectious, Pr. The subscript r refers to the reservoir

species. The total population density is Nr = Sr + Er + Ir + Pr. The per capita birth rate and

survival to the adult reproductive stage is br. There is no vertical transmission. Disease-related

deaths are not known to occur in the reservoir host (Mills et al., 1997). The natural death rate

depends on population density, a strictly increasing function of the population density, 0 ≤ dr

(0) < br and limNr→∞dr(Nr) > br. The transmission coefficients for the two infectious stages

are βI and βP, respectively. The models described below assume density-dependent

transmission (pseudo mass action incidence) as in other hantavirus models (Abramson and

Kenkre, 2003; Abramson et al., 2003; Allen et al., 2006a; Allen et al., 2006b). The models and

results can be easily generalized to frequency-dependent transmission which in some cases

may provide a better fit to data (Begon et al., 1999). In stable environments, frequency-

dependent transmission may be appropriate, but in the overlap region, where rodents occupy

the region for only a short period of time, encounters are most likely density-dependent. The

average length of the exposed and highly infectious periods are I/δr and 1/γr, respectively. All

parameters are assumed to be positive unless noted otherwise. The model for the reservoir

population takes the following form:

(3.1)

The total population density satisfies the differential equation

(3.2)

From the assumptions on dr, it follows that there exists a unique positive constant Kr, the

carrying capacity, such that br = dr(Kr) and limt→∞ Nr(t) = Kr. With frequency-dependent

transmission the terms Sr(βIIr + βpPr) are replaced with Sr(βIIr + βpPr)/Nr.

The spillover species responds differently to hantavirus infection. Presumably, the infection is

only short-term, an acute stage, A, and therefore, we assume no disease-related deaths occur.

But this assumption can be modified. The model is an SEAR model, where animals pass

through the stages of being susceptible, latent, infectious, and finally recovered. A subscript

s is used to identify the spillover species and distinguish it from the reservoir species. The

differential equations for hantaviral infection in the spillover species are similar to the reservoir

Allen et al.Page 5

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 6

species but transmission of hantavirus occurs only from the infectious stage As and γS is the

recovery rate. The model for the spillover species takes the following form:

(3.3)

We assume ds(Ns) satisfies similar assumptions as dr(Nr), so that the total male population

density for the spillover species Ns satisfies a differential equation similar to equation (3.2).

Likewise, there exists a unique positive constant Ks, the carrying capacity of the spillover

population, such that limt→∞ Ns(t) = Ks.

3.2 A Habitat-Based Epizootic Model

The reservoir and spillover species generally have preferred habitats as shown by the data. The

spillover species is rarely found in the habitat where the reservoir species is dominant and vice

versa (see Figure 2). However, contact between these two species occurs in a boundary or

overlap region adjacent to their habitats, where densities of the two species may be relatively

low. Encounters between infectious and susceptible animals in this boundary region may result

in interspecies transmission of hantavirus. Figure 3 is a schematic of the three regions, the

preferred habitats for the reservoir and the spillover species and the boundary region.

These habitats are connected via movement to and from the preferred habitat and the boundary

region. Time spent in this boundary region is short for both species. The majority of the

population is susceptible, especially in the case of the spillover species.

Suppose the per capita rate of movement pi into the boundary region is low i.e., pi is small, and

that the per capita rate of movement po out of this boundary region is high, i.e., Po is large.

The same movement rates are assumed for each species. Thus, for each of the differential

equations in (3.1) and (3.3) movement into and out of the boundary region is included.

Subscripts a and b on the differential equations denote the reservoir and spillover species,

respectively, in the boundary region.

3.2.1 ODE Model—Based on the preceding assumptions, the differential equations for the

reservoir species in its preferred habitat take the following form:

(3.4)

Similar differential equations apply to the spillover species, where terms for movement into

and out of the preferred habitat are added to the differential equations in (3.3).

Because rodents are in the boundary region for a short period of time, on the order of days, no

births nor deaths occur in this region. We assume that the carrying capacity in the preferred

habitat remains constant. That is, the stable preferred habitat density for the reservoir species

is Kr which can be thought of as a population source for the boundary region. The carrying

capacity in the boundary region may increase or decrease relative to Kr if pi increases or po

Allen et al.Page 6

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 7

decreases, respectively. The differential equations for the reservoir species in the boundary

region are

(3.5)

and for the spillover species they are

(3.6)

Rodents may change their disease status while in the boundary, e.g., Ea → Ia or Eb → Ab. An

inherent assumption in the models is that the time spent in each of the disease states is

exponentially distributed. For example, the probability that an initially exposed reservoir host

transitions to the infectious stage while in the boundary region is

(3.7)

Other models based on more general probability distributions such as the gamma distribution

provide alternative formulations (Feng et al., 2007; Lloyd, 2001a, 2001b). More data are

required to determine the form of the distributions. In this investigation, we consider the

simplest form, an exponential distribution. If po ≫ maxi∈{a,b}{δi,γi} and if the number of

rodents exposed to the infection in the boundary region is relatively small, then transitions

between disease stages in the boundary region may have little impact on the disease dynamics.

In the analysis section 4, an explicit expression for the basic reproduction number is derived

when these transition rates are set to zero: δa = 0 = δb and γa = 0 = γb.

The total male population densities in the preferred habitat and in the boundary region are Nr

and Na for the reservoir species and Ns and Nb for the spillover species. Thus, the differential

equations for the total male population densities are

(3.8)

Initial conditions are nonnegative and strictly positive in the preferred habitat; Nr(0) > 0 and

Ns(0) > 0.

Allen et al. Page 7

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 8

3.2.2 CTMC Model

The ODE model can be easily extended to a CTMC model which includes variability in the

birth, death, transmission, and movement processes. In the CTMC model, the 16 random

variables are integer-valued taking on values in the set {0, 1, 2,…}. There are 38 different

events, that include births, deaths, transmission, and movement. Let (t) be a vector of 16

discrete random variables associated with the CTMC process:

Based on the 38 events, the infinitesimal transition probabilities can be defined, Prob{Δ (t)|

(t)}, where Δ (t) = (t + Δt) − (t) for Δt sufficiently small (Allen, 2003; Karlin and Taylor,

1975). For example, the probability of a birth in the reservoir population is

where the time step Δt is chosen so that the possibility of more than one transition or change

in Δt units of time is negligible. The 38 events and their corresponding transition probabilities

are described in the Appendix.

Due to climatic variations within the year-dry, wet, and transitional (D, W, T) periods-there

may be greater overlap of the habitats during certain periods of the year. One way of modeling

this variability in the overlap region is to modify the rate of movement into or out of the

boundary region for each species, depending on their habitat preferences during each of these

periods. For example, A. montensis is seen in the overlap region more frequently in period T

than in periods D or W, when densities of O. delator are high. In the model, we do not

specifically include this seasonal variability but we do consider the effects of changes in pi and

po on the basic reproduction number Ro, the threshold for disease outbreaks. More data are

required to predict whether certain periods are more likely to result in spillover infection.

4 Model Analysis

There are three types of equilibria for the ODE habitat-based model: an extinction equilibrium,

where the population density is zero, a unique disease-free equilibrium (DFE), where all the

infectious and recovered states are zero but the susceptible states are positive, and enzootic

equilibria (EE), where some infectious states have positive values. It can be easily shown that

the extinction equilibrium is unstable; the population persists. For example, it follows from

equations (3.8) that in the preferred habitats, the population densities approach a constant value,

their respective carrying capacities,

The preferred habitats serve as a population source for the boundary region. The densities in

the boundary region depend on the densities of these source populations and the movement

rates into and out of this region. In particular, in the boundary region, the reservoir and spillover

population densities are

Allen et al.Page 8

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 9

respectively. As the ratio pi/po increases, so do the population densities in the boundary region,

whereas the densities in the preferred habitats will approach their respective carrying-

capacities. Hence, the total reservoir population density (preferred + boundary) is Kr + Ka and

the total spillover population density is Ks + Kb. The equilibrium values for the unique DFE

are Sr = Kr, Sa = Ka, Ss = Ks, and Sb = Kb; all of the other equilibrium values are zero. We

assume initial densities are less than or equal to their respective carrying capacities, 0 ≤ Ni(0)

≤ Ki, i = r, s, a, b with Nr(0) > 0 and Ns(0) > 0. Whether the DFE is stable depends on the basic

reproduction number for the habitat-based model.

We calculate reproduction numbers for each of the preferred habitats and an approximation to

the overall basic reproduction number for the habitat-based model (3.4)–(3.6), ℛ0. If ℛ0 > 1,

then it is likely that the disease persists in the reservoir and spillover populations. The

reproduction numbers can be calculated using the next generation matrix approach (van den

Driessche and Watmough, 2002). For the general system (3.4)–(3.6), it is possible to show that

the basic reproduction number is a positive root of a fourth degree polynomial but it is difficult

to obtain a simple analytical expression for ℛ0. The simplifying assumption

(4.9)

leads to an explicit expression for ℛ0 (shown below). This explicit expression is a close

approximation to the overall basic reproduction number, if the time spent in the boundary is

short relative to the time spent in each of the disease states. This expression is very useful in

interpreting the contributions to disease outbreaks by the reservoir and the spillover species

and in making comparisons to other reproduction numbers.

Assume the condition (4.9) holds. First, the reproduction number for the reservoir species

(assuming the spillover species is not present) is

Second, the reproduction number for the spillover species (assuming the reservoir species is

not present) is

If there are no interspecies interactions so that pi = 0 = po, then Ka = 0 = Kb. Each of the

reproduction numbers simplifies to well-known reproduction numbers for SEIP or SEAR

models (3.1) or (3.3), respectively. In this case, it is straightforward to calculate the enzootic

equilibrium for the reservoir host,

Allen et al. Page 9

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 10

(4.10)

whenever the reproduction number for the SEIP model (3.1) is

At equilibrium, the proportion of animals that are RNA- or antibody-positive, (Īr + P̄r)/Kr, is

(4.11)

To derive an expression for the basic reproduction number for the habitat-based model when

(4.9) holds, we first define an expression which depends on interspecies or crossover

transmission. That is, let the intraspecies transmission parameters be zero, βI = βP = βA = 0 and

βa1 = βa2 = βb3 = 0 and the interspecies transmission parameters, βb1, βb2, and βa3, be nonzero.

The reproduction number for interspecies pathogen transmission is defined as

Note that

in the overlap region, Kb and Ka. The preceding definition can be used to define the basic

reproduction number for the habitat-based model (3.4)–(3.6):

depends on the ratio pi/po directly and indirectly through the carrying capacities

(4.12)

(Derivation of this formula is given in the Appendix.) It follows that

interspecies pathogen transmission increases the basic reproduction number. A similar

relationship was shown in a multi-species SI model of McCormack and Allen (2007).

;

The local stability of the DFE follows directly from the results of van den Driessche and

Watmough (2002). Global stability of the DFE when ℛ0 < 1 and condition (4.9) holds can be

verified by construction of a Liapunov function. The full system (3.4)–(3.6) consists of 16

differential equations which makes it difficult to find an explicit closed form solution for an

enzootic equilibrium (EE). However, existence and uniqueness of a positive EE can be verified

when ℛ0 > 1 and condition (4.9) holds. It is shown, in the Appendix, that the EE for the full

system (3.4)–(3.6) is a fixed point of Er = f(Er, Es) and Es = g(Er, Es). Then the existence of a

unique positive EE follows by applying a theorem (Hethcote and Thieme, 1985) on existence

Allen et al.Page 10

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 11

and uniqueness of a positive fixed point. The following theorem summarizes the preceding

results.

Theorem 4.1

A basic reproduction number ℛ0 exists for system (3.4)–(3.6) such that if ℛ0 < 1, the DFE is

locally asymptotically stable. If condition (4–9) holds for system (3.4)–(3.6), then

i.

ℛ0 has the form given in (4.12),

if ℛ0 < 1, then the DFE is globally asymptotically stable, and

iii. if ℛ0 > 1; then the DFE is unstable and there exists a unique positive enzootic

equilibrium.

ii.

5 Numerical Examples

Selection of parameters values is based on estimates from the literature for hantavirus (δi and

γi, i = r, s) and on trapping and demographic data for the reservoir species A. montensis and

for the spillover species N. lasiurus or O. delator (bi, Ki, i = r, s and equilibrium ratios). The

reservoir species A. montensis and one spillover species (N. lasiurus) are widely distributed in

many of our study sites sampled in Paraguay, whereas the other spillover species (O.

delator) is less widespread and generally less abundant. We choose carrying capacities of Kr

= 100 and Ks = 50 for the reservoir and spillover species in their respective habitats. Although

the values for Kr and Ks are not known, the selected values are close to the estimates for minimal

number known alive based on data from the trapping regions. The basic time unit in the model

is one year. We assume total number of births per female per year that survive to the adult

reproductive stage is six for the reservoir and the spillover species (several litters per year).

Assuming an equal sex ratio, for the male population the number of males that survive to

reproductive age is br = 3 = bs. Population growth is assumed to satisfy a logistic growth

assumption so that for the reservoir and spillover species,

so that dr(Kr) = br and ds(Ks) = bS. We assume an average duration of two weeks for the latent

period E and for the spillover infectious stage A, and an average duration of three months for

the highly infectious stage for the reservoir species (Bernshtein et al., 1999; Lee et al., 1981;

Padula et al., 2004). Hence, δr = 26 = δs, γs = 26, and γr = 4, e.g., 1/26 year ≈ two weeks. In

addition, δa = δr, δb = δs, γa=γr, and γb = γS. Animals enter the boundary region several times

per year and stay only a short time. The average length of time in the boundary region is less

than the average length of time for the latent period or acute infectious period, 1/po ≤1/δi and

1/po ≤ 1/γi, i = a, b. In the numerical examples, we let pi = 8 and po = 52 which means, on

average, each animal may make eight visits per year to the boundary region, spending about

one week in the boundary region. The probability there is a transition from an exposed to an

infectious state while in the boundary region is 1/3 (see equation (3.7)). Even though this

probability is not small, the basic reproduction number given by (4.12) is a good approximation

to the overall basic reproduction number for our parameter values (shown below). The

parameter values are reasonable but are chosen for illustrative purposes (a range of values, pi

∈ [2, 25] and po ∈ [26,364], are considered later in this section). These parameter choices give

population densities in the boundary region of Ka ≈ 15 and Kb ≈8.

The transmission parameters βj cannot be estimated directly. Instead, we make some reasonable

assumptions about their relationship to disease transmission in the infectious stages, βI and

βP. The product βIKr is the number of infectious contacts that result in infection by a highly

Allen et al. Page 11

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 12

infectious reservoir animal per year (at equilibrium). Based on the summary data for proportion

of animals RNA- or antibody-positive, 0.25 at site R3B, if we equate formula (4.11) to 0.25

and let br = 3 and δr = 26, this leads to ℛr = 1.39 (when pi = 0 = po). Thus, the value of βI is

chosen so that (which is close to ℛr) is between one and two. We assume that the highly

infectious stage of the reservoir host (RNA-positive and/or antibody-positive) is three times

as infectious as the persistently infectious stage (only antibody-positive), i.e., βI = 3βP

(Bernshtein et al., 1999;Lee et al., 1981;Padula et al. 2004). In addition, we assume the

transmissibility of the pathogen in the acute infectious stage A of the spillover host is the same

as for the persistent stage in the reservoir host, βA = βP. This leads to

region, intraspecific transmissibility remains the same as in the preferred habitats, but

interspecies transmissibility is doubled due to aggressive encounters. In particular,

. In the boundary

(5.13)

(5.14)

The basic parameter values are given in Table 1.

If (4.9) holds and the remaining parameter values are as in Table 1 with pi = 8 and Po = 52, we

obtain , so that ℛ0 ≈ . The approximate reproduction numbers based on

the analysis in section 4 are

The overall basic reproduction number for the parameters in Table 1 is ℛ0= 1.38 which is close

to the approximation 1.42. The disease persists in the habitat-based model. For the parameter

values in Table 1, one sample path of the CTMC model and the solution to the ODE model are

graphed for the two infectious stages of the reservoir species (see Figure 4).

Although the pathogen persists, the infection in the spillover population is very low (straight

line is the ODE equilibrium value); only sporadic infection occurs in the sample path for the

spillover species in the preferred habitat and in the boundary region (Figure 5).

For the ODE model with parameter values given in Table 1, pi = 8, and po = 52, there is a

unique enzootic equilibrium which is locally asymptotically stable:

With no interspecies transmission and no overlap region (pi = 0 = po) the equilibrium values

for the reservoir host, based on the formulas given in (4.10), are (S̄r, Ēr, Īr, P̄r) = (72.1, 2.9,

10.7, 14.3). These latter equilibrium values show that the percentage of highly infectious and

persistently infectious rodents are in close agreement with the summary data for A.

montensis at site R3B, i.e., 25% are infected.

The CTMC simulation with interspecies transmission illustrates the sporadic infection in the

spillover population (as in site R3A) and provides information about the variability in number

Allen et al.Page 12

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 13

of cases. A quasistationary probability distribution is reached in the CTMC model (conditional

on nonextinction). Approximations (estimated from 10,000 sample paths) to the

quasistationary probability distributions for the two infectious stages in the reservoir species

are graphed in Figure 6. The mean values for Ir and Pr are μ̂Ir = 8.8, μ̂Pr= 14.0. In the absence

of interspecies transmission and pi = 0 = po, the mean values are μ̂Ir = 9.2 and μ̂Pr = 12.3.

Encounters that lead to interspecies pathogen transmission can be measured by the magnitude

of

. The greater the habitat overlap, the greater the number of interspecies and intraspecies

encounters which in turn increase the likelihood of pathogen outbreaks and disease persistence.

Changes that affect the overlap region will have the greatest impact on the parameters pi and

po rather than the parameters affecting transmission, births, or deaths. As more animals enter

and stay in the boundary region, that is, if pi increases and po decreases, then

consequently, ℛ0 increases. This increase can be seen in Figure 7. The value of ℛ0, computed

from formula (4.12), is compared to the exact value of ℛ0 based on the parameter values in

Table 1. Both reproduction numbers show similar increases with pi (average number of visits/

year) and 364/po (average number of days in the boundary region). The difference between

these two reproduction numbers is also computed (Figure 7 (c)); the largest relative difference

is 0.12, when pi = 25 and po = 26.

increases and

6 Discussion

Biologically-motivated models for pathogen spread between two species were formulated, an

ODE model (3.4)–(3.6) and a CTMC model. The models are based on the fact that spatial

overlap of habitats leads to greater numbers of interspecies encounters. From the ODE model,

an explicit expression for the basic reproduction number ℛ0 was calculated based on the

assumption (4.9), as well as reproduction numbers for the preferred habitats,

for crossover or interspecies transmission, . In this case, we showed global stability of the

disease-free equilibrium when ℛ0 < 1 and existence of an enzootic equilibrium when ℛ0 > 1.

Greater numbers of interactions among species allow the pathogen to be transmitted more

frequently from an infectious host to a susceptible host. This, in turn, increases ℛ0 so that it

exceeds the reproduction number in the preferred habitat, ℛ0 > ℛr (Figure 7), which ultimately

results in greater likelihood of outbreaks and disease persistence. As illustrated in Figure 2, the

overlap region is spatially- and temporally-dependent. We did not consider temporal variability

of this overlap region which may depend on seasonal variations. But we did include

demographic variability due to births, deaths, transmission, and movement in the CTMC

model. Seasonal variations, in general, will cause additional variability in the solution behavior

(e.g., Allen et al., 2005). As more data are collected, the effects of seasonal and climatic

variations on the reservoir and spillover species will be studied. In addition, controlled studies

are needed to obtain data on the duration and shape of the rodents’ disease stage distributions.

and , and

Interspecies pathogen transmission, where a known virus “jumps” into a new host, is one of

the primary reasons for the large increase in emerging diseases in wildlife in recent years

(Daszak et al., 2000; Parrish et al., 2008; Richomme et. al., 2006). Our mathematical models

illustrate the first step in this emergence and the role that the spillover species may play in

emerging diseases. Our models were developed for spread of hantavirus in rodents but can be

modified and applied to other species, where spatial spread results in spillover infection.

Acknowledgments

This research was supported by a grant from the Fogarty International Center #R01TW006986-02 under the NIH NSF

Ecology of Infectious Diseases initiative. We thank R. K. McCormack for preliminary discussions on this work, the

Fundación Moises Bertoni for facilitating access to the field sites, and the Vendramini family for allowing us to work

Allen et al.Page 13

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 14

in Estancia Rama III. The Secretaría de Ambiente provided necessary permits for working with wildlife. In addition,

we thank the referees for their helpful suggestions.

References

Abramson G, Kenkre VM. Spatiotemporal patterns in hantavirus infection. Phys Rev E

2002;66:0011912-1–5.

Abramson G, Kenkre VM, Yates TL, Parmenter RR. Traveling waves of infection in the hantavirus

epidemics. Bull Math Biol 2003;65:519–534. [PubMed: 12749537]

Allen, LJS. An Introduction to Stochastic Processes with Applications to Biology. Prentice-Hall, Upper

Saddle River; NJ: 2003.

Allen, LJS.; Allen, EJ.; Jonsson, CB. The impact of environmental variation on hantavirus infection in

rodents. In: Gumel, AB.; Castillo-Chavez, C.; Mickens, RE.; Clemence, DP., editors. Contemporary

Mathematics Series, 410, Proceedings of the Joint Summer Research Conference on Modeling the

Dynamics of Human Diseases: Emerging Paradigms and Challenges. AMS, Providence; RI: 2006a.

p. 1-15.

Allen LJS, Langlais M, Phillips CJ. The dynamics of two viral infections in a single host population with

applications to hantavirus. Math Biosci 2003;186:191–217. [PubMed: 14583172]

Allen LJS, McCormack RK, Jonsson CB. Mathematical models for hantavirus infection in rodents. Bull

Math Biol 2006b;68:511–524. [PubMed: 16794943]

Anderson, RM.; May, RM. Infectious Diseases of Humans, Dynamics and Control. Oxford Univ, Press;

Oxford: 1991.

Bailey, NTJ. The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley

&; Sons; New York: 1990.

Begon M, Hazel SM, Baxby D, Bown K, Cavanagh R, Chantrey J, Jones T, Bennett M. Transmission

dynamics of a zoonotic pathogen within and between wildlife host species. Proc Roy Soc Lond B

1999;266:1939–1945.

Bernshtein AD, Apekina NS, Mikhailova TV, Myasnikov YA, Khlyap LA, Korotkov YS, Gavrilovskaya

IN. Dynamics of Puumala hantavirus infection in naturally infected bank voles (Clethrinomys

glareolus). Arch Virol 1999;144:2415–2428. [PubMed: 10664394]

Carleton, MD.; Musser, GG. Order Rodentia. In: Wilson, DE.; Reeder, DM., editors. Mammal Species

of the World. Vol. 3. Vol. 2. Johns Hopkins University Press; Baltimore, MD: 2005. p. 745-752.

Castillo-Chavez, C.; Thieme, HR. Asymptotically autonomous epidemic models. In: Arino, O.; Axelrod,

D.; Kimmel, M.; Langlais, M., editors. Mathematical Population Dynamics: Analysis of

Heterogeneity, Theory of Epidemics. Vol. 1. 1995. p. 33-49.

CDC MMWR. Hantavirus pulmonary syndrome – United States: updated recommendations for risk

reduction. 2002 July 26;51(RR09):1–12.

Childs JE, Ksiazek TG, Spiropoulou CF, Krebs JW, Morzunov S, Maupin GO, Gage KL, Rollin PE,

Sarisky J, Enscore RE, Frey JK, Peters CJ, Nichol ST. Serologic and genetic identification of

Peromyscus maniculatus as the primary rodent reservoir for a new hantavirus in the Southwestern

United States. J Infect Dis 1994;169:1271–280. [PubMed: 8195603]

Chu YK, Milligan B, Owen RD, Goodin DG, Jonsson CB. Phylogenetic and geographical relationships

of hantavirus strains in eastern and western Paraguay. Am J Trop Med Hyg 2006;75:1127–1134.

[PubMed: 17172380]

Chua KB. Nipah virus outbreak in Malaysia. J Clin Virol 2003;26:265–275. [PubMed: 12637075]

Daszak P, Cunningham AA, Hyatt AD. Emerging infectious disease of wildlife: threats to biodiversity

and human health. Science 2000;287:443–449. [PubMed: 10642539]

Delfraro A, Tome L, D’Elía G, Clara M, Achával F, Russi JC, Rodonz JR. Juquitiba-like hantavirus from

two nonrelated rodent species, Uruguay. Emerg Infect Dis 2008;14:1447–51. [PubMed: 18760017]

Feng Z, Xu D, Zhao H. Epidemiological models for non-exponentially distributed disease states and

applications to disease control. Bull Math Biol 2007;69:1511–1536. [PubMed: 17237913]

Fernández Soto A, Mata Olmo R. Deforestatión y dinámica vegetal en un area de frontera agrícola del

la Región Oriental del Paraguay. Revista Geonotas (Universidade Estadual de Maringá) 2001;5:1.

Allen et al. Page 14

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 15

Fiorello CV, Noss AJ, Deem SL. Demography, ecology, and pathogen exposure of domestic dogs in the

Izozog of Bolivia. Conserv Biol 2006;20:762–771. [PubMed: 16909569]

Glass GE, Livingston W, Mills JN, Hlady WG, Fine JB, Higgler W, Coke T, Frazier D, Atherley S, Rollin

PE, Ksiazek TG, Peters CJ, Childs JE. Black Creek Canal Virus infection in Sigmodon hispidus in

southern Florida. Am J Trop Med Hyg 1998;59:699–703. [PubMed: 9840584]

Goodin DG, Paige R, Owen RD, Ghimire K, Koch DE, Chu Y-K, Jonsson CB. Microhabitat

characteristics of Akodon montensis, a reservoir for hantavirus, and hantaviral seroprevalence in an

Atlantic forest site in eastern Paraguay. Journal of Vector Ecology. Submitted to

Hethcote HW. The mathematics of infectious diseases. SIAM Review 2000;42:599–653.

Hethcote HW, Thieme HR. Stability of the endemic equilibrium in epidemic models with subpopulations.

Math Biosci 1985;75:205–227.

Holmes K. SARS-associated coronavirus. New Engl J Med 2003;348:1948–1951. [PubMed: 12748314]

Karlin, S.; Taylor, H. A First Course in Stochastic Processes. Vol. 2. Academic Press; NY: 1975.

Klein SL, Bird BH, Glass GE. Sex differences in immune responses and viral shedding following Seoul

virus infection in Norway rats. Am J Trop Med Hyg 2001;65:57–63. [PubMed: 11504409]

Klingstrom J, Heyman P, Escutenaire S, Sjölander KB, De Jaegere F, Henttonen H, Lundkvist A. Rodent

host specificity of European hantaviruses: evidence of Puumala virus interspecific spillover. J Med

Virol 2002;68:581–588. [PubMed: 12376967]

LaSalle, JP. The Stability of Dynamical Systems. SIAM; Philadelphia: 1976.

Lee HW, Lee PW, Baek LJ, Song CK, Seong IW. Intraspecific transmission of Hantaan virus, etiological

agent of Korean hemorrhagic fever, in the rodent Apodemus agrarius. Am J Trop Med Hyg

1981;30:1106–1112. [PubMed: 6116436]

Lloyd A. Realistic distributions of infectious periods in epidemic models. Theor Pop Biol 2001a;60:59–

71. [PubMed: 11589638]

Lloyd A. Destabilization of epidemic models with the inclusion of realistic distributions of infectious

periods. Proc R Soc Lond B 2001b;268:985–993.

McCormack RK, Allen LJS. Disease emergence in multi-host epidemic models. Math Med Biol

2007;24:17–34. [PubMed: 17012365]

McIntyre NE, Chu Y-K, Owen RD, Abuzeineh A, De La Sancha N, Dick CW, Holsomback T, Nisbet

RA, Jonsson C. A longitudinal study of Bayou virus, hosts, and habitat. Amer J Trop Med Hyg

2005;73:1043–1049. [PubMed: 16354810]

Mills JN, Ksiazek TG, Ellis BA, Rollin PE, Nichol ST, Yates TL, Gannon WL, Levy CE, Engelthaler

DM, Davis T, Tanda DT, Frampton JW, Nichols CR, Peters CJ, Childs JE. Patterns of association

with mammals in the major biotic communities of the southwestern United States. Am J Trop Med

Hyg 1997;56:273–284. [PubMed: 9129529]

Mills JN, Yates TL, Ksiazek TG, Peters CJ, Childs JE. Long-term studies of hantavirus reservoir

populations in the southwestern United States: Rationale, potential and methods. Emerg Infect Dis

1999;5:95–101. [PubMed: 10081676]

Nadin-Davis SA, Loza-Rubio E. The molecular epidemiology of rabies associated with chiropteran hosts

in Mexico. Virus Res 2006;117:215–226. [PubMed: 16303200]

Nel L, Jacobs J, Jaftha J, Meredith C. Natural spillover of a distinctly Canidae-associated biotype of

rabies into an expanded wildlife host in southern Africa. Virus Genes 1997;15:79–82. [PubMed:

9354274]

Padula P, Fogueroa R, Navarrette M, Pizarro E, Cadiz R, Bellomo C, Jofre C, Zaror L, Rodriguez E,

Murua R. Transmission study of Andes hantavirus infection in wild sigmodontine rodents. J Virol

2004;78:11972–11979. [PubMed: 15479837]

Palma RE, Polop JJ, Owen RD. Hantavirus-host ecology in the Southern cone of South America:

Argentina, Chile, Paraguay and Uruguay. Vector Borne and Zoonotic Disease. 2009Submitted to

Pardiňas UF, D’Elía G. The genus Akodon (Muroidea: Sigmodontinae) in Misiones, Argentina.

Mammalian Biology 2003;68:129–143.

Parrish CR, Holmes EC, Morens DM, Park EC, Burke DS, Calisher CH, Laughlin CA, Saif LJ, Daszak

P. Cross-species virus transmission and the emergence of new epidemic diseases. Microbiol Mol

Biol Rev 2008;72:457–470. [PubMed: 18772285]

Allen et al.Page 15

J Theor Biol. Author manuscript; available in PMC 2010 October 21.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript