WilliamsThesisTemplate

A NOVEL MODEL FOR WHITE NOSE SYNDROME INLITTLE BROWN BATS

by

DAVID STEVENS

JULIE C. BLACKWOOD, ADVISOR

STEVEN J. MILLER, SECONDARY ADVISOR

A thesis submitted in partial fulfillment

of the requirements for the

Degree of Bachelor of Arts with Honors

in Mathematics

WILLIAMS COLLEGE

Williamstown, Massachusetts

May 20, 2014

1ABSTRACT

Bats are important reservoirs for emerging human and wildlife diseases. Certain pathogens

that are highly virulent to humans are able to persist in healthy bats and little is known

about the mechanisms by which bat immune systems are able to cope with these diseases.

Shedding light on bat immune systems may have important implications for developing

intervention strategies, which in turn allows us to control the spillover of zoonotic diseases

into human and wildlife populations. White nose syndrome (WNS) is a prime example

of a devastating emerging infectious disease. In 2006, the first incidence of bats infected

with WNS was discovered in a cave near Albany, New York. It has since spread rapidly

across eastern North America. WNS is caused by a newly described fungus, Geomyces

destructans, that grows on the exterior of hibernating bats. The infection is thought to

rouse infected bats from hibernation, depleting essential fat stores and resulting in death by

starvation. This disease is forecasted to cause the regional extinction of little brown bats

(Myotis lucifugus) in the northeastern United States, with the potential for serious conse-

quences for ecosystem integrity. In this paper we outline disease control strategies for WNS

with the aim of preventing the regional extinction of Myotis lucifugus. For this purpose,

we develop a mixed-time SEI model for WNS in Myotis lucifugus broken into three stages:

(1) roosting, (2) swarming, and (3) hibernation.

2ACKNOWLEDGEMENTS

I would like to thank my advisor, Julie Blackwood, for her invaluable guidance and her

unwavering support.

3CONTENTS

1. Introduction 5

1.1. Background 6

2. The 3-Phase Model 14

2.1. Density-dependent or Frequency-dependent Transmission? 15

2.2. The Roosting Model (May-September) 15

2.3. The Swarming Model (September-October) 17

2.4. The Hibernation Model (October-May) 18

2.5. Parameter Assumptions 20

3. Analytical Results 20

3.1. Constant Birth Models 20

3.2. Disease-Free Equilibria 21

3.3. Basic Reproductive Numbers 23

3.4. Endemic Equilibria 28

4. Numerical Results 29

4.1. Parameter Estimation Using The Basic Reproductive Number 29

5. Using Deterministic Simulations to Isolate the Effects of Individual Parameters 32

5.1. Effective Transmission: The Trade-off between Force of Infection and

Mortality (The Story of and ) 32

5.2. Incubation of WNS and its Dynamical Consequences (The Story of ) 38

6. Control of White-Nose Syndrome 40

6.1. Heat and Fungicide 42

6.2. Culling 44

6.3. Restricted Access to Caves 44

7. Conclusions 47

8. Future Directions 47

8.1. Spatial Dynamics 47

8.2. Field Work 48

Appendix A.

Code for Deterministic Plots 48

Appendix B.

Code for Stochastic Plots 63

Appendix C. The Gillespie Algorithm 70

Appendix D. The Relationship Between Risks and Rates 70

4References 72

51. INTRODUCTION

The year was 2006. It was a particularly cold February when a group of spelunkers

ventured into Howes Cave, about 52 km to the west of Albany, NY. [16] On that visit, they

snapped a picture of bats with a strange white fungus growing on their muzzles. This was

the first documented case of white-nose syndrome (WNS) in North America. In the years

since that photograph, over 5.5 million bats in the northeastern United States have died due

to WNS, the first sustained epizootic observed in bats in history. [?]The ferocity with which this disease has attacked the local bat population is unprece-

dented. According to some estimates, WNS is responsible for one of the fastest rates of

population decline ever observed in wildlife populations due to disease. [13] At the current

rate of decline, the regional extinction of little brown bats is estimated to be fewer than 15

years away.

If we can accurately model the disease dynamics of WNS, then we can test methods of

control with the goal of preventing the collapse of the little brown bat population. In this

paper, we develop a system of ordinary differential equations (ODEs) to model WNS in

the little brown bat population of a hibernacula site from year to year. The model is broken

into three phases, one of which is discrete-time and two of which are continuous-time. In

addition, we develop methods of parameter estimation in the absence of empirical data.

This is particularly applicable to other situations where empirical is not readily available

for whatever reason. It may be that the necessary data is difficult to observe in the field or

that field tests are too costly. Whatever the reason, parameter estimation without the use

of empirical data is an important ability. Based on the typical basic reproductive number

of wildlife diseases, we provide reasonable ranges for the model parameters. These ranges

make it easy to fit the WNS model to empirical data should it become available.

Given parameter ranges, we vary the parameters one at a time to isolate the effect of each

of the overall disease dynamics, focusing on the hibernation period. Among other findings,

we show that the latent period of the disease controls the interepidemic period of WNS, but

does not affect the basic reproductive number. The transmission rate and disease-induced

mortality rate are the determiners of the basic reproductive number and they control the

peak size of the epidemics. Finally, we show that localized heating strategies within the

hibernacula has the potential to alter the fate of a population headed for extinction based

on our model.

61.1. Background.

1.1.1. The fungus. The causative agent of WNS is the fungus Pseudogymnoascus destru-

cans. The fungus is psychrophillic (cold-loving), which means that active growth and

propagation is limited to low-temperature evironments such as bat hibernacula. There is

no growth observed at 24 C or above. The optimum growth temperature is between 4 C

and 15 C. Conveniently, the skin temperature of bats during hibernation is approximately

5 C. [2]

The fungus is quite adaptable. At higher temperatures, it undergoes morphological

changes, which may account for the fungal propagation at higher temperatures. For ex-

ample, under adverse conditions (high temperatures) P. destructans exhibits increased sep-

tation, thickened hyphae, arthospore production, and chlamydospore-like structure forma-

tion. Arthospores, a type of asexual spore formed by the breakup of the mycelium of

the fungus, can become a primary means of fungal propagation, but not to the extent of

the short-chain, asymmetrically curved microconidia seen at lower temperatures. Chlamy-

dospores, a type of large resting spore, act as desiccation resistant structures not involved

in propagation. [12] It may be that the altered phenotype allows the fungus to persist on the

bats during the warmer periods outside of the hibernation months, but is not the main mode

of transmission.

The same fungal strain which is currently decimating North American bat populations

has been isolated from healthy European bats. Isolates of P. destructans from Germany

have been shown to be lethal to experimentally infected North American little brown bats.

This fungus is harmless to the German bats, however. The reason for this disparity in

mortality between North American and European bats is unknown.

P. destructans was isolated from soil samples from hibernacula in states where WNS is

known to occur. This indicates that the fungus is able to persist in the environment. [7]

1.1.2. Disease progression. WNS is a cutaneous fungal infection characterized by white

growth on the muzzle and skin lesions as seen in Figures 1 and 2 The white material is

an abundance of fungal conidia produced by P. destructans. Though this white growth is

the most recognizable visible sign of WNS infection, the bats skin is the main target of P.

destructans.

Several elements of torpor contribute to the success of P. destrucans in infecting little

brown bats. Metabolism and immune function are severely downregulated during hiber-

nation and body temperature drops dramatically as well, which is convenient for the psy-

chrophillic fungus. Due to the lowered immune system, it is likely that P. destructans does

7FIGURE 1. A little brown bat infected with WNS. Note the

white fungal growth on the muzzle and outer epidermis.

(http://en.wikipedia.org/wiki/White_nose_syndrome)

not encounter an immune response while invading the tissue of torpid bats. In addition, bats

cluster together during hibernation to conserve energy and often select humid areas of the

hibernacula to decrease moisture loss. These behaviors may have the effect of increasing

the dispersal rate of P. destructans across the colony and promoting fungal growth.

The wing membranes of bats infected with WNS are severely compromised. The folds of

the membranes stick together, tear easily, and lose tone and elasticity. This causes the wings

to take on the appearance of crumpled tissue paper. [13]. The fungal colonization causes

8FIGURE 2. A cluster of hibernating little brown

bats showing physical signs of white-nose syndrome

infection.(http://www.natureworldnews.com/articles/6144/20140222/noninvasive-

technique-identifying-bat-populations-tested-government-researcher)

epidermal lesions filled with fungal hyphae. P. destructans is thought to cause infarction

of wing tissue not by invading blood vessels, but rather by effacement of the surrounding

vasculature. This wing damage is clearly detrimental to the bats flying ability, but it also

impairs the bats water balance maintenance. Autopsies of bats that have died from WNS

suggest that dehydration is a contributing factor to disease-induced mortality.

The fungus is thought to rouse infected bats from hibernation, depleting their essential

fat stores and causing death. The arousals themselves seem to be a means of maintaining

homeostasis and an opportunity for the bats to drink and restore water balance. The link

between dehydration and WNS-mortality may be a result of more frequent arousals driven

by thrust.

There is no evidence that bats recover from the disease in the wild once infected.

Since the first observation of bats infected with WNS in Albany, NY in 2006, the disease

has spread to many other areas across the United States and Canada. The spatial spread of

the disease is visualized in Figure 3.

1.1.3. Life history of little brown bats. The little brown bat (Myotis lucifugus) is currently

one of the most common bats in North America. They live in separate day and night roosts

during the spring, summer and fall and go into hibernation during the winter. Day roosts

are found in a variety of locations, such as attics, barns, trees, wood piles, or caves. Night

9FIGURE 3. WNS was initially localized to eastern New York and southern

New England, but has since spread rapidly in all directions. It has recently

been confirmed in bat populations as far West as the western border of Mis-

sissipi. This shows how quickly the disease is spreading without interven-

tion and emphasizes the need for preventative measures. [3]

roosts are often in the same general location as day roosts, but in slightly different spots.

These tend to be smaller, as the bats pack together to stay warm. Little brown bat popula-

tions in the northeastern United States enter hibernation between September and October

and emerge sometime between April and June, with females emerging first. According to

the study at Aeolus Cave, the female exodus begins at about the end of the first week in

April and most of them have left the cave by the second week of May. This suggests that

maternity (or nursery) colonies begin forming in early April. Males begin leaving about the

end of the first week in May and are gone by the end of the first week in June. [10] They

10

FIGURE 4. Partially decomposed little brown bat carcasses litter the floor

of a cave. (Photo by Marianne Moore, Boston University)

are not "true" hibernators in the sense that if the temperature rises over about 50 C, then

they will rouse from torpor and hunt insects.

Once the bats have emerged from hibernation, they migrate to their summer colonies.

Bats that were all banded at one hibernation site in Vermont were found in southwestern

New Hampshire, Massachusetts, Rhode Island, and northeastern Connecticut, indicating

that the hibernation colonies break into smaller factions and scatter during the summer

months. They begin in sex-specific colonies and remain that way through the nursing period

period (early Summer).

Adults mate in mid- to late-autumn while swarming near the entrances of their hiberna-

tion sites. This swarming behavior is thought to be associated with the selection of a place

for hibernation and begins a bit earlier, around the end of July. In addition, males arous-

ing during hibernation may mate with torpid females. Females then store the sperm from

autumn and winter matings. Fertilization occurs in the spring when females leave their

hibernacula and form nursery colonies. The gestation period is variable, usually between

fifty and sixty days. Females bear their single young in late June or July. When the young

are weaned and capable of flight (at approximately 21-28 days), the females disperse from

11

FIGURE 5. A little brown bat (Myotis lucifu-

gus).(http://csrspreadscience.wordpress.com/2013/11/02/white-nose-

syndrome-proves-resilient-as-little-brown-bats-face-further-declines/)

their nursery colonies and return to their hibernacula sites, where they proceed to mate and

hibernate. Yearling females may bear young, but males do not breed until the end of their

second summer.

The average life span of the little brown bat is 6 to 7 years with many individuals sur-

viving for over 10 years. However, the oldest little brown bat reported in the wild was 34

years old! Their average sleep time is 19.9 hours per day, so that they are asleep for about

82.9% of theirs lives. [6] Little brown bats are predominantly active at night around dusk

and before dawn.

Colony demographics. Between 1960 and 1963, a very thorough investigation of thelittle brown bat population in New England and eastern New York was conducted. Today,

it remains the source of most of our knowledge of the lifestyle of Myotis lucifugus. Esti-

mates of the size of roosting colonies range from a dozen to 1,200 adults, with most counts

between 300 and 800, but the average estimate across all breeding colonies was 280 bats.

The colony size during hibernation is similarly variable. About 300,000 bats are estimated

to winter at Aeolus cave, but several mines in New York have hibernating populations of

about 15,000. Population estimates closer to 15,000 are more common. [15]

A summary of the yearly habits and a compilation relevant information concerning little

brown bats are found in Table 1 and Table 2, respectively.

1.1.4. Why study diseases in bats? Role as Reservoir Hosts. Bats are important reservoirhosts of zoonotic diseases. The bat family was one of the earliest mammals to evolve

and the ancient origins of certain zoonotic viruses suggest a long history of coexistence.

12

Time of Year Behavior

April-May bats emerge from hibernacula and form sex-specific colonies

June - July females bear single young

end of July bats begin swarming at hibernacula entrances

Auguest - September adult bats mate

September - October bats enter hibernation

TABLE 1. A year in the life of a little brown bat.

Average lifespan 6-7 years

Average number of pups per litter 1

Average number of litters per year 1

Gestation period 50-60 days

Size of roosting colonies about 300 bats

Size of hibernation colonies about 15,000

TABLE 2. Relevant information about little brown bats.

As such, bats can carry viruses for extended periods of time without showing evidence of

infection. The question of how certain pathogens are able to persist in (apparently) healthy

bats while remaining highly pathogenic to humans is an important one to which there is

no accepted answer. It has been suggested that their innate and acquired immune system

responses are significantly different from other mammals, though little is currently known

about bat immune systems. They are said to reservoir hosts for those pathogens, allowing

them to persist and infect other hosts. The bat lifestyle is particularly conducive to disease

transmission, both to other bats and to other species. They are long-lived animals, so if

the pathogen is able to persist on a bat for an extended period of time, there is likely to

be a high number of secondary infections. The roosting behavior of bats involves a lot of

contact as well, so the contact rate drives the transmission rate higher. [4]

An understanding of transmission dynamics in bats is essential for controlling the spillover

of zoonotic diseases into human and wildlife populations. It will be important to understand

the role of torpor and hibernation on pathogenesis and pathogen maintenance.

Ecological consequences. Bats are critical for maintaining terrestrial biosystem in-tegrity. Insectivorous bats act as pest suppressants, they reseed forests, pollinate plants,

and there is a large market for their guano, which is used as fertilizer.

13

1.1.5. Compartmental disease models. Mathematical models play an essential role in mod-

ern epidemiology. The mathematical modeling and simulation of infectious disease is a

powerful tool which allows us to explore the dynamics in silico and test hypotheses about

the disease that are untestable or unfeasible in the wild. We will be building a compart-

mental model for WNS, so we provide a brief introduction here before introducing the

model.

Compartmental models separate a population into compartments based on the disease

state of each individual. Some common compartments are

Susceptible (S): Individuals susceptible to infection. Exposed (E): Individuals are infected but not yet infectious; the time spent in this

class is called the latent period of the disease.

Infected (I): Individuals that are infectious. Recovered (R): individuals that were previously infected, but have either tempo-

rary or permanent immunity.

Now we can look at the rates that individuals travel between compartments and set up a

system of differential equations. We will be focusing on the change through time only,

so our model will be a system of ordinary differential equations (ODEs) which only look

at change with respect to one variable. We will describe the logic behind each term after

introducing the WNS model.

14

2. THE 3-PHASE MODEL

There are many open questions surrounding the transmission dynamics of WNS. We

begin to build a 3-phase model with the intent of investigating the following questions:

When is the main transmission period of WNS? During hibernation? During matingseason?

Do bats raised from torpor by the disease infect other bats in the hibernacula? Is disease progression during hibernation suspended due to cold temperatures and

the metabolic effects associated with torpor (cf. rabies)?

Do bats that contract the disease ever recover (i.e., is there a recovered class) ?Based on the yearly life cycle of the little brown bats, we identified three main trans-

mission phases: the roosting period, the swarming period, and the hibernation period. The

roosting period likely plays a role in the spatial dynamics of the disease since the bats

spread out from their hibernacula to smaller, scattered roosting sites. Adult little brown

bats mate while swarming at the entrance to their hibernacula prior to settling down for

the winter. Although this is a shorter period in comparison to the roosting and hibernation

phases, there is a high contact rate between bats during that time which is likely significant

for WNS transmission. The hibernation period is suspected to play a large role in WNS

dynamics for several reasons: (1) the cold temperatures in the hibernacula provide a fa-

vorable environment for the paychrophillic fungus responsible for WNS, (2) bat carcasses

litter the ground of affected hibernacula, and (3) the bats are closely packed together for

warmth during this time.

Based on the above, it is critical to understand the nature of transmission during hiberna-

tion. We build a continuous-time SEI model for the hibernation period in order to observe

the dynamics with a fine temporal resolution. We have chosen not to include a recovered

class because there is no evidence to suggest that bats recover from the disease once in-

fected.1 In the wild, infected bats that make it through hibernation have been found dead or

unable to fly near their hibernation sites, which shows that even if infected bats survive hi-

bernation their recovery is not guaranteed. Many bats die during hibernation as evidenced

by the carcasses littering the floor of infected caves (see Figure 4). The wing damage caused

by the fungus leaves infected bats more susceptible to predators as well. [?]Reichert Kunz2009

1With supportive care in a laboratory setting, however, it is possible for bats to recover from the disease. In

one study, 36 WNS-affected little brown bats were collected and given a special type of supportive care. After

70 days, 25 of the bats were PCR-negative for the DNA of P. destructans and showed significant recovery

from the damage to their wing membranes. This is the first documented case of recovery from WNS. [11]

15

The swarming period comes directly before hibernation and this is when bats from many

different summer roosting colonies come together to mate. It is possible that this is an

important time period for disease transmission exposure before bats settle into their hi-

bernacula. To test this, the swarming period will also be a continuous-time SEI model.

However, to improve computational tractability of our model when we run it over several

years, the submodel for the roosting period will be a discrete-time SEI model.

2.1. Density-dependent or Frequency-dependent Transmission? In formulating the trans-mission term, we need to decide whether WNS exhibits density-dependence or frequency-

dependence. The difference is subtle, but will have big repercussions on our parameter

estimation later on. In density-dependent transmission, the transmission rate depends on

the population density. The density-dependent transmission term looks like

1 I S.On the other hand, frequency-dependent transmission assumes that the transmission rate is

independent of the population density, so that the transmission term is

2 IN S.

The force of infection is the per capita rate at which susceptible individuals become infected

or, in our case, exposed.

Wildlife diseases often exhibit density-dependent transmission and we suspect that this

is the case for bat diseases as well. It makes sense that the per capita contact rate in-

creases with the size of the population because the increased number of bats in roosting

and hibernation sites encourages more frequent contacts. For this reason, we will assume

density-dependent transmission in our WNS model.

Without further ado, we present the model for WNS in little brown bats:

2.2. The Roosting Model (May-September). We now present a discrete-time model forthe roosting period. This can be made into a continuous-time partial differential equation

(PDE) model in order to better capture the spatial dynamics of the disease when the bats

spread out from the hibernacula to their roosting sites. See the Section 8.1 on spatial dy-

namics for a further discussion.

St+1 = btNt + e(+t)St (2.1)

Et+1 = (1 et)eSt + e(+r)Et (2.2)It+1 = e

(1 er)Et + eIt (2.3)

16

We see that

Nt+1 = (bt + e)Nt.

This says that the population at the end of the roosting period is exactly the new bats that

were born during roosting and the bats that did not die due to natural causes during the

roosting period.

The initial conditions are taken from the output of the hibernation submodel ( 2.4):

St = Sh(end)

Et = Eh(end)

It = Ih(end)

Here we are using risks instead of rates. We begin by examining the difference equation

for the susceptible class. We assume that all bats born during the roosting period enter

the susceptible class (though the possibility of vertical transmission is something to be

explored) and that the number of bats born is some fixed proportion (bt) of the total popula-

tion, giving us the btNt term. Next we must account for the number of bats that survive the

roosting period (probability e) and are not infected during that time et . This number

is e(+t)St.

Moving onto the difference equation for the exposed compartment, we add the bats that

became exposed to the disease (probability 1-et) and survived the roosting period (prob-

ability e) for a total of (1 et)eSt bats. We also have the bats that survived and arestill in the latent period of the disease (probability er .

For the infected class, we have the exposed bats that survived the roosting period and

became infectious (probability 1er). We also have the infected bats that did not die dueto natural causes or due to disease-induced complications. There is currently no evidence

in the literature that bats die due to WNS during the roosting phase or the swarming phase,

which makes sense because P. destructans is psychrophillic and so it does not grow during

these warm phases of the year. We set the disease-induced mortality rate s equal to 0 so

that bats are only dying due to non-disease related causes with probability e.

The parameters for this submodel are summarized in Table 3.

See Appendix D on Difference Equations" for a derivation of the relationship between

risks and rates.

17

Parameter Description Estimate

bt proportion of bats that give birth 0.4

natural mortality rate of hibernating bats 0.154

t sIr, force of infection -

s rate at which bats leave class Er -

e probability that an individual bat will not die due to natural causes 0.857

et probability that a susceptible bat will not become exposed to the disease -

es probability of not leaving the exposed class -

1 es probability that an exposed bat becomes infected -TABLE 3. Descriptions and estimates for parameters of the roosting submodel.

2.3. The SwarmingModel (September-October). We switch to a continuous-time modelfor the swarming period as we are more interested in the transmission dynamics around this

time.

dSsdt

= sIsSs Ss (2.4)dEsdt

= sIsSs (s + )Es (2.5)dIsdt

= sEs (s + )Is. (2.6)

The initial values are taken from the output of the roosting model:

Ss(0) = St+1

Es(0) = Et+1

Is(0) = It+1.

Bats are moving out of the susceptible compartment and into the exposed compartment

due to infection at a rate of sIsSs. Bats are being removed from the susceptible class due

to natural mortality at a rate of Ss. Let us deconstruct the parameter s. This is actually

a product of terms that mathematical epidemiologists group together for convenience. The

entire disease transmission term is composed of four basic elements:

(1) contact rate (N),

(2) proportion of contacts that are with susceptibles S/N ,

(3) probability that a contact results in infection p,

(4) number of invectives I .

18


1/ average lifespan of little brown bats 6.5 years

natural mortality rate of hibernating bats .154

s contact rate transmission probability -s sIs, force of infection -

s rate at which bats leave class Es -

1/s mean duration of time in class Es -

s disease-induced mortality rate during swarming 0

1/s mean duration of time in class Is -

TABLE 4. Parameter descriptions and estimates for the swarming model.

In one unit of time, a given bat makes contact with (N) other bats. Only a certain

number of those contacts involve susceptible bats, however. In one unit of time, that makes

contact with (N)S/N susceptible bats. If there are I infected bats, then in a unit of time

there will be (N)SI/N contacts between susceptible and infected bats. The probability

that such a contact results in transmission is p, so the transmission rate is (N)pSI/N . We

have set s = (N)p.

Newly infected bats move into the exposed class at a rate of sIsSs. They move out of

the exposed class and into the infected class at a rate of sEs and they are removed from

the exposed class (and the population) due to natural mortality at the rate Es.

Bats move into the infected class from the exposed class at sEs bats per unit time and

they are removed from the infected class (and the population) due to the combined effects

of disease-induced and natural mortality at a rate of (s + )Is.

For the same reasons outlined in the roosting submodel section, we have initialized the

disease-induced mortality parameter s to 0. This makes sense because P. destrucans ex-

periences optimum growth between 4 C and 15 C. However, we have chosen to include

it in the model for ease of manipulation later.

The parameters for the swarming submodel are summarized in Table 4.

2.4. The Hibernation Model (October-May). The hibernation model is similar to theswarming model, but we also consider the possibility of infection due to an environmental

reservoir of P. destructans in the hibernacula. The rate at which susceptible bats are ex-

posed to the disease due to the environmental reservoir is some function of the pathogen

population P . We denote this rate by F (P ). In addition, we provide a compartment P

19

for the pathogen present in the hibernacula. The infected bats add to the reservoir at some

rate and the pathogen grows at some rate g and dies at some rate . This leads us to the

following system of ODEs:

dShdt

= (hIh + F (P ))Sh ShdEhdt

= (hIh + P )Sh (h + )EhdIhdt

= hEh (h + )IhdP

dt= I ( g)P.

Each parameter has the same meaning as the corresponding parameter in the swarming

model. However, it differs in the environmental reservoir term.

Now we examine the function F (P ) in more detail. Initially, we will not be overly con-

cerned with the ways in which the environmental reservoir changes the disease dynamics.

It is likely to be a contributing factor, perhaps even a significant one, but the function itself

is not our primary interest. For this reason, we choose to value simplicity in our computa-

tion of F (P ) rather than exactness. We let = = g = 0 so that the pathogen population

is fixed. That is, P = P0 for all t 0. Since P is constant, whatever function we choosefor F will be constant over the hibernation season as well. For simplicity, we let F be a

linear function of P :

F (P ) = P,

so susceptible bats become exposed due to the environmental reservoir at rate P . Our P.

destructans population is no longer changing, so we omit the differential equation for P .

dShdt

= (hIh + P )Sh Sh (2.7)dEhdt

= (hIh + P )Sh (h + )Eh (2.8)dIhdt

= hEh (h + )Ih (2.9)

Note that this model collapses to the swarming model when is 0 and all other parame-

ters are equal.

The evidence for environmental persistence of P. destructans (formerly known as Ge-

omyces destructans) is taken from [8]:

20


1/ average lifespan of little brown bats 6.5 years

natural mortality rate of hibernating bats .154

h hIh, force of infection -

rate at which susceptible bats are infected due to environmental reservoir -

h rate at which bats leave class Eh -

1/h mean duration of time in class Eh -

h disease-induced mortality rate during hibernation -

1/h mean duration of time in class Is -

TABLE 5. Parameter descriptions and estimates for the hibernation submodel.

Geomyces destructans was cultured from 27 of the 195 sediment samples collected

from bat hibernacula in 2011 to 2012, with viable fungus detected in 11 of the 14 sites

during at least one sampling interval. Seven of the 14 sites were found to harbor viable G.

destructans in late summer, when bats were either absent from the hibernacula or present in

only low numbers. Sequences of the rRNA gene ITS regions of isolates from each site were

100% identical to the ITS region of the type isolate of G. destructans (GenBank accession

no. EU884921 [2])."

2.5. Parameter Assumptions. In order to simplify the analysis of our model, we makethe following biologically reasonable assumptions

h s: We expect the contact rate between bats to be significantly higher duringthe swarming period than during hibernation. Since is calculated as contact rate

times transmission probability, we would expect that h s. (Unless transmissionprobability is much higher during hibernation, which is a possibility.)

s < h: The psychrophilic nature of P. destructans and the absence of bat carcassesoutside of the hibernacula suggests that bats are dying at higher rate due to WNS

during hibernation than during the swarming or roosting periods.

3. ANALYTICAL RESULTS

3.1. Constant Birth Models. The final model, to be used in our simulations, will use abirth pulse during the roosting phase in order to more closely model the seasonal birthing

pattern of little brown bats. However, the absence of birth terms in the swarming and

hibernation phases causes difficulties in equilibria analysis based on the current theory. To

21

overcome this, we will consider versions of the swarming and hibernation models with

constant birth terms. These will also be used in calculating R0 for each sub model. In

performing simulations, however, the constant birth terms will be dropped and we will

only use a birth pulse during the roosting phase.

Here is the 3-phase model modified to include constant births during the swarming and

hibernation phases.

Roosting:

St+1 = btNt + e(+t)St

Et+1 = (1 et)eSt + e(+s)EtIt+1 = e

(1 es)Et + eIt.

(3.1a)

(3.1b)

(3.1c)

Swarming:

dSsdt

= bsNs sIsSs sSsdEsdt

= sIsSs (s + )EsdIsdt

= sEs (s + )Is.

(3.2a)

(3.2b)

(3.2c)

Hibernation:

dShdt

= bhNh (hIh + P + )ShdEhdt

= (hIh + P )Sh (h + )EhdIhdt

= hEh (h + )Ih.

(3.3a)

(3.3b)

(3.3c)

In the next couple of sections, we will calculate the disease-free and endemic equilibria

for each submodel.

3.2. Disease-Free Equilibria. We show that each submodel has a disease-free equilibrium(DFE), which means that there are no exposed or infected individuals.

3.2.1. The Roosting Model. We show (S, E, I) = (Nt, 0, 0) is a disease-free equilib-

rium for the roosting sub-model simply by demonstrating that plugging these values into

the model returns an unchanged value for each compartment, i.e., St+1 = St. Note that at

(Nt, 0, 0), the force of infection t = rIt = 0 since It = 0. Thus

22

St+1 = brNt + e(+t)St = brNt + e(+0) Nt = (br + e)Nt.

This returns St = Nt when bt + e = 1 or bt = 1 e.Plugging (Nt, 0, 0) into the difference equations for the exposed and infected compart-

ments, we see that Et+1(Nt, 0, 0) = Et(Nt, 0, 0) and It+1(Nt, 0, 0) = It(Nt, 0, 0):

Et+1 = (1 et)eSt + e(+s)Et = (1 e0)eNt + e(+s) 0 = 0 + 0 = 0 = Et,

It+1 = e(1 es)Et + eIt = 0 + 0 = 0 = It.

Therefore (Nt, 0, 0) is a disease-free equilibrium for the roosting sub-model when bt =

1 e.

3.2.2. The Swarming Model. In the same way, we show that (Ns, 0, 0) is a disease-free

equilibrium for the swarming sub-model, assuming the birth rate is equal to the natural

mortality rate (bs = ).

Plugging Ss = Ns, Es = 0, and I

s = 0 into the equations for the swarming model, we

obtain

dSsdt

= bsNs sIsSs sSs = bsNs sNs,which will be 0 if and only if bs = .

dEsdt

= sIsSs (s + )Es = 0,

dIsdt

= sEs (s + )Is = 0.Hence we have shown that (Ns, 0, 0) is a disease-free equilibrium for the swarming

model if and only if bs = .

3.2.3. The Hibernation Model. We show that (Sh, Eh, I

h, P

) = (Nh, 0, 0, 0) is a disease-

free equilibrium when bh = . That is, in the absence of an environmental reservoir, the

disease-free state is an equilibrium point.

dShdt

= bhNh (hIh + P + )Sh = bhNh Nh,which will be 0 if and only if bh = .

23

dEhdt

= (hIh + P )Sh (h + )Eh = (0 + 0)Nh 0 = 0,dIhdt

= hEh (h + )Ih = 0 0 = 0.

Therefore (Nh, 0, 0, 0) is a disease-free equilibrium for the hibernation sub-model when

bh = .

3.3. Basic Reproductive Numbers. In the study of a new disease, a natural questionarises: What is the ability of the disease to invade a population? A more precise way

to phrase this question is: If an infected individual enters an entirely susceptible popula-

tion, will the disease invade or die out? The basic reproductive number R0 captures this

notion. Conceptually, R0 is defined to be the average number of secondary infections re-

sulting from the introduction of one infected individual to a population consisting entirely

of susceptible individuals.

It can be shown that ifR0 < 1, then the disease-free equilibrium is locally asymptotically

stable. Small perturbations from the disease-free state return to a disease-free state. On the

other hand, ifR0 > 1, then the disease-free equilibrium is unstable. (Reproduction numbers

and sub-threshold endemic equilibria for compartmental models of disease transmission, P.

van den Driessche, James Watmough) Small perturbations from the disease-free state may

result in an epidemic. In practice, of course, this is not so clear cut. There may be an

endemic equilibrium near the the disease-equilibrium even if R0 < 1.

We now calculateR0 for each of the sub-models using the next generation matrix method.

This requires that the population can be divided into homogeneous compartments within

which individuals are indistinguishable. In our case, we have broken the population into

susceptible, exposed, and infected subpopulations. This method also assumes that param-

eters do not depend on the length of time an individual has spent in a compartment, which

the parameters for our model satisfy. For a detailed description of the next-generation

matrix approach, see The Basic Reproduction Number in Some Discrete-Time Epidemic

Models" (Allen, van den Driessche).

3.3.1. R0 for the Roosting Model. Here, we analytically find R0 for the roosting model.

First, we reorder the compartments so that the infected compartments come first and the

uninfected compartments come after. In this case, we have (Er, Ir, Sr). Formally, we

24

consider our system to be a function F : R3 R3 whereF (Er, Ir, Sr) = (F1(Er, Ir, Sr), F2(Er, Ir, Sr), F3(Er, Ir, Sr))

=

((1 et)eSt + e(+s)Et, e(1 es)Et + eIt, btNt + e(+t)St

).

Now we determine the Jacobian matrix for the system. Recall that t = rIt.

J =

F1/x1 F1/x2 F1/x3F2/x1 F2/x2 F2/x3F3/x1 F3/x2 F3/x3

= e

(+s) re(t+)St (1 et)e)e(1 es) e 0

bt bt re(+t)St bt + e(+t)

.Next, we evaluate the Jacobian at the disease-free equilibrium (Nt, 0, 0). Note that t =

0 since t = rIt and bt = 1 e.

JDFE =

e(+s) reNt 0

e(1 es) e 0bt bt reNt bt + e

.The 2 2 submatrix in the upper left corner of the Jacobian matrix may be written as

F + T , where F is the probability matrix that new infections survive the time interval and

T is the probability matrix of all of transitions between infected compartments. The matrix

F is known as the fertility matrix and the matrix T is known as the transition matrix.

Splitting the terms in this way and evaluating at the disease-free equilibrium, we get

F =

[0 re

Nt0 0

]

T =

[e(+s) 0

e(1 es) e].

The next generation matrix K is F (I T )1, so we must calculate (I T )1, where

I T =[

1 e(+s) 0e(1 es) 1 e

].

Using that the inverse of a 2 2 matrix[a b

c d

]is 1

adbc

[d bc a

],

(I T )1 = 1(1 e(+s))(1 e)

[1 e 0

e(1 es) 1 e(+s)].

We may now compute the next generation matrix K:

25

K = F (I T )1 = 1(1 e(+s))(1 e)

[0 re

Nt0 0

][1 e 0

e(1 es) 1 e(+s)]

=

[re2(1es )

(1e(+s))(1e)Ntre(1e(+s))

(1e(+s))(1e)Nt

0 0

].

Since K is upper triangular, the eigenvalues are the diagonal entries. The only positive

nonzero, and hence largest, diagonal entry is R0 =re2(1es )

(1e(+s))(1e)Nt.

Therefore R0 for the roosting model is

re2(1 es)

(1 e(+s))(1 e)Nt

.

Noting that is very close to 0, we know e is close to 1. We may obtain a reasonable

estimate for the value of R0 as:

R0 rNt

3.3.2. R0 for the Swarming Model. From the previous section, we see that the infected

classes are the only ones required so we take only Es and Is into consideration. We con-

struct the matrix F such that the ij th entry is the rate of new infections entering class j

from infected class i. We take new infections to be those individuals entering an infected

class from an uninfected class, so in particular any movement between infected classes is

not considered a new infection. The ij th entry of the matrix V is the rate at which new

individuals enter class j from class i.

F =

[0 sSs

0 0

]

V =

[s + 0

s s +

].

From this, we can calculate V 1:

V 1 =

[1

s+0

s(s+)(s+)

1s+

].

Now we are in position to compute the next generation matrix K.

26

K = FV 1 =

[0 sSs

0 0

][1

s+0

s(s+)(s+)

1s+

]=

[sSss

(s+)(s+)sSs

(s+)

0 0

].

Since K is upper triangular, the eigenvalues are the diagonal entries. The only positive

nonzero, and hence largest, diagonal entry is sSsN ss+s

1s+s

. Thus R0 for the swarming

model is

sSs ss + s

1s + s

.

Noting that s is close to 0, we can simplify R0 to

R0 sSs 1s.

In calculating R0, we are assuming that the system is at the disease-free equilibrium, S =

S(0), E = 0, and I = 0. Ignoring seasonality in births, we may assume S(0) = N . This

gives us

R0 sNss

.

Intuitively, this is the rate at which susceptible individuals become infected times the av-

erage amount of time spent in the infected class before disease-induced death times the

number of bats in the population.

3.3.3. R0 for the HibernationModel. The next generation matrix approach is well-established

for epidemiological models using the standard S,E,I, and R compartments. However, our

free-living pathogen compartment P requires some additional thought. In the next genera-

tion matrix approach, we must decompose the Jacobian matrix J of our system, evaluated at

the DFE, into F V . Recall that F is the transmission matrix, describing the rate at whichnew individuals are infected, and V is the transition matrix, describing all other changes

between classes. This step raises some questions about the role of P . Should we include

movement into P in F or V ? Typically, the transmission matrix only needs to consider

the generation of secondary infectious hosts (new infections). Free-living pathogens are

not hosts, but their presence in the environment clearly plays a role in transmission of the

disease. "[D]ecompostion of JDFE is greatly dependent on how the role of the environment

is interpreted in transition and transmission of secondary infectious hosts and [free-living

pathogens]." [9]

We must first choose how to interpret the role of the environment in this capacity, then

we proceed with the next generation matrix approach. Bani-Yaghoub et al. consider three

possible interpretations:

27

(I) transition: The environment is considered an extended state of host infectiousness.

Movement into the P class, whether by pathogen shedding or growth, is taken to

be a transition within the initial infectious state. In this interpretation, the shedding

and growth rates are placed in the transition matrix V .

(II) transition-reservoir: The environment is considered to be both an extended state

of host infectiousness as well a reservoir of infection. Pathogen shedding into the

enviroment is a transition within the initial infectious state and so it is placed in the

transition matrix V . Growth of new pathogen in the environment is thought to be

vertical transmission of infection and so it is placed in the transmission matrix F .

(III) reservoir: The environment is treated as a reservoir for infection. Here, pathogen

shedding and growth are considered generation of secondary free-living pathogen

and so they are both placed in the transmission matrix F .

We will use the reservoir interpretation, so that the shedding rate and the growth rate

g are added to the transmission matrix F .

R0 =

hS0h(+)(+h)

g

+

(g hS0h

(+h)(+h)

)2 4

(S0hghS0h(+h)(+h)

)2(S0hghS0h(+)(+h)

) (3.4)We may now consider a simplified version of R0, noting that is close to 0 and ignoring

the dynamics of the free-living pathogen class (g = 0, = 1, = 1).

R0 hNhh

+

(hNhh

)2 4Nh

h

2Nh.

Recall that we are assuming that the free-living pathogen population remains constant by

setting the shedding rate, the growth rate and the death rate of the pathogen equal to 0. This

doesnt go over well in our current formulation ofR0 because we cannot divide by 0. Rather

than making this section an exercise in algebraic manipulation, we will instead recalculate

R0 assuming the shedding rate, the growth rate and the death rate of the pathogen are equal

to 0 so that dPdt

= 0. We leave the previous formulation of R0 intact, in case we wish to

investigate how the dynamics of the environmental pathogen affect the progression of WNS

in the future.

We set up the transition and transmission matrices for the infectious classes E, I, and P ,

F =

0 hSh Sh0 0 00 0 0

,

28

V =

h + 0 0h h + 00 0 0

.We calculate the inverse of the transition matrix V to get

V 1 =

1

h+0 0

h(h+)(h+)

1h+

0

0 0 0

.The next generation matrix is

K = FV 1 =

0 hSh Sh0 0 00 0 0

1h+

0 0h

(h+)(h+)1

h+0

0 0 0

=

hhSh(h+)(h+)

hShh+

0

0 0 0

0 0 0

.The only positive, and hence largest, eigenvalue is hhSh

(h+)(h+).

Evaluated at the disease-free equilibrium, we get

hh(h + )(h + )

Nh.

Therefore the basic reproductive number we will be working with during the hibernation

period is

R0 =hh

(h + )(h + )Nh.

We can simplify this further by noting that 0 so that

R0 hhNh.

Note that in the absence of the environmental pathogen dynamics, the basic reproductive

number is calculated in the same way as during the swarming period. This is indicative of

the fact that the basic reproductive number captures the intrinsic ability of the disease to

invade a susceptible population regardless of its static presence in the environment.

3.4. Endemic Equilibria. An endemic equilibrium (EE) point has a positive number ofindividuals in either the exposed class or the infected class (or both!). These are equilibria

in which the disease persists in the population rather than dying out or driving the popu-

lation to extinction. However, the individual EE will be different across the submodels.

When it comes to the model as a whole, even if we reach the EE for one of the submodels,

after a period of time we will jump to the next submodel. When that happens, the model

29

will no longer be at an equilibrium point. This is meant to point out that the individual EE

for the submodels will not serve as an EE for the 3-phase model as a whole (except in the

unlikely event that the EE is the same for each submodel). Instead, we will use simulation

studies to understand the long-term dynamics.

4. NUMERICAL RESULTS

4.1. Parameter Estimation Using The Basic Reproductive Number. The basic repro-ductive number is a useful tool for estimating the order of magnitude of our parameter

values. These values will approximate the actual disease parameters, but we dont yet

have the means to make them exact. In order to get more accurate values, we need some

population data to compare to the model output.

However, we can decide on a reasonable range for the basic reproductive number of

white-nose syndrome and find parameter values which place the basic reproductive number

in that range.

We can get an upper bound for the basic reproductive number of WNS by comparing it

to well-known infectious diseases. Airborne human diseases such as measles and pertussis

have R0 between 12 and 18. This is possible due to the ease of transmission afforded

by airborne infections and the high population density found in cities. Wildlife diseases

typically have lower R0 values than human diseases, so we may designate an upper bound

for the basic reproductive number of WNS at about 10. It is also clear that WNS has

successfully invaded the susceptible little brown bat population of the northeastern United

States so we may set a lower bound of 1 for the basic reproductive number. [1]

We will look at each submodel individually, beginning with the roosting phase.

4.1.1. The Roosting Phase. We would like to estimate the transmission risk er and the

risk of leaving the exposed class er .

The basic reproductive number for the roosting period is

re2(1 es)

(1 e(+s))(1 e)Nt.

However, since we know that is very close to 0, e is very close to 1. Substituting 1 for

e gives a simplified, but reasonable estimate of R0,

R0 rNr.

We see that since the basic reproductive number is not significantly affected by changes

to r we cannot use reasonable values for R0 in order to estimate r. On the other hand,

30

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 107

0

2

4

6

8

10

12

14

16

18

20

Transmission rate (r)

Basic

rep

rodu

ctiv

e nu

mber

(R 0)

FIGURE 6. Plot of the basic reproductive number during the swarming pe-

riod as a function of the transmission rate (r). The total population N is

set to 15,000.

we have the opportunity to glean information about appropriate values of r based on the

basic reproductive number.

Using the unsimplified formulation of R0, we plot the basic reproductive number as a

function of r using a fixed bat population size of 15,000. Figure 6 shows the range of rvalues that produce a basic reproductive number of less than 20 for the swarming period.

In order for 1 < R0 < 10, we must have 2.5 108 < r < 2.5 107.2.5 108 < r < 2.5 107

4.1.2. The Swarming Phase. We are looking to estimate the disease-induced death rate s,

the transmission rate s, the rate at which individual leave the exposed class sBased on our simplification of the formula for R0, we suspect that s has little effect on

the basic reproductive number for the swarming phase. This is confirmed by the experi-

ments shown in Figures 7, 8, and 9. In Figures 7 and 8 which show s on the y-axis, the

basic reproductive number does show any visual change in the vertical direction, indicating

that s is not changing the value of R0 substantially. In Figure 9, we see that (a), (b), and

31

(c) show identical distributions of the basic reproductive number despite having different

values of s.

We first focus on estimating the disease-induced mortality rate s. As shown by Figure 7,

the range of permissible s (those which give us a basic reproductive number between 1

and 10) is influenced by the value of s. A very large value for s results in a low basic

reproductive number because then infected bats would die almost instantly and never have

the chance to infect susceptibles. Therefore we are looking for the lower permissible bound

of s as a function of s. When s is low, the disease-induced mortality rate may be lower

than when s is high. As a general trend, we see that as we raise s, the lower bound

of the range of permissible values for s increases as well. Biologically speaking, if we

increase the transmission rate of the disease, then we have to increase the rate at which bats

are dying due to the disease in order to maintain the same average number of secondary

infections (R0).

Looking at our simplified formula for R0, we can get s as a function of s:

sR0Ns.

Now using that the average colony size is 15,000 bats and substituting the extremal R0values of 1 and 10, we obtain a range of permissible s values depending on s:

150 s s 15000 s.This relationship is shown in the color maps in Figure 9.

We will now look into estimating the transmission rate of WNS during the swarming

period. The plots in Figure 8 provide a qualitative look at how the permissible values of

s change as we vary s. As we increase s, the transmission rate s can safely increase

as well while maintaining a basic reproductive number within the appropriate range. As

the biological converse to our previous example, if the bats are dying rapidly due to the

disease, then we need a higher transmission rate to maintain the same number of secondary

infections.

We may obtain a reasonable range for s using the same analytical methods that we used

to estimate s:

1

15000s s 1

150s.

Having found bounds for the transmission rate based on the disease-induced mortality

rate, we may use some intuition to get more concrete estimates. Once a bat becomes

infected, what is the average number of days it has left to live (1/s)? It is unlikely that

32

infected bats die in under a day, so we may assume 1/s > 1, which in turn implies that

s < 1. Therefore, the maximum possible value for s is about 1150 .0067. In alllikelihood, bats spend an even longer average amount of time in the infected class so the

actual transmission rate during the swarming period is probably smaller than this upper

bound by an order of magnitude or two.

We summarize this in an updated parameter table for the swarming period.

Parameter Description Theoretical Range Estimate

1/ average lifespan of little brown bats fixed 6.5 years

natural mortality rate of hibernating bats fixed .00042

s contact rate transmission probability 115000s s 1150s < .0067s rate at which bats leave class Es - -

1/s mean duration of time in class Es - -

s disease-induced mortality rate during swarming 150 s s 15000 s 01/s mean duration of time in class Is s15000 s s150 -

4.1.3. The Hibernation Phase. The basic reproductive number for the swarming period

is the same as the intrinsic basic reproductive for the hibernation period. Therefore the

analysis that was applied to the disease-induced mortality rate and the transmission rate of

the swarming period applies to those rates during hibernation as well.

5. USING DETERMINISTIC SIMULATIONS TO ISOLATE THE EFFECTS OF INDIVIDUAL

PARAMETERS

5.1. Effective Transmission: The Trade-off between Force of Infection and Mortality(The Story of and ). Since our model is divided into three distinct submodels, eachwith their own set of parameters, it can be tricky to parse the individual effects of each

parameter on the disease dynamics as a whole. We take the approach of "freezing" the

dynamics of two of the submodels in order to get a better sense of how one season, whether

that is roosting, swarming, or hibernation, affects the disease dynamics. We can then ex-

amine the long-term effects of varying the parameters for that season. Keep in mind that

this will be a qualitative analysis. We are more concerned with describing the effects of

parameter variation than with the actual number of bats in each class. When we secure

data on the little brown bat population size in the years since WNS was introduced, we can

adjust the parameters to fit that data. It will be important to know how to properly change

the dynamics and that is the purpose of this analysis.

We begin this analysis by focusing on the hibernation period. We reduce the effects

due to the dynamics of the roosting and swarming periods by assuming that there is no

transmission during either of these periods (r, s = 0). In addition, let us assume that

the average about of time spent in the exposed class remains constant throughout the year,

33

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Diseaseinduced mortality rate (s)

Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(s)

0

2

4

6

8

10

12

14

16

18

20

(A) s = .00001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(s)

0

2

4

6

8

10

12

14

16

18

20

(B) s = .00015

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(s)

0

2

4

6

8

10

12

14

16

18

20

(C) s = .0003

FIGURE 7. Colormaps showing values for the basic reproductive number

during the swarming season as the disease-induced mortality rate (s) and

the rate that exposed bats move into the infected class (s) are varied. The

basic plot has been reproduced for low, medium, and high values of a third

disease-parameter, s.

so that r = s = h. We will look at the dynamics in the absence of environmental

transmission since the P term, as it currently stands, simply adds to the transmission rate.

Therefore, we can achieve the same effects of altering by raising the transmission rate h.

It is important to keep in mind that this would be the case if we chose a different function

F (P ) to model environmental transmission or if the pathogen class were not assumed to

remain constant.

We fix an intermediate R0 value of 3 for the hibernation period and see how the dy-

namics change as we vary the parameters while maintaining the same basic reproductive

34

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Transmission rate (s)

Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(s)

0

2

4

6

8

10

12

14

16

18

20

(A) s = .1

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(s)

0

2

4

6

8

10

12

14

16

18

20

(B) s = .5

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(s)

0

2

4

6

8

10

12

14

16

18

20

(C) s = 1


during the swarming season as the transmission rate (s) and the rate that

exposed bats move into the infected class (s) are varied. The basic plot

has been reproduced for low, medium, and high values of a third disease-

parameter, s.

number. We ran the deterministic simulation with a low h, an intermediate h, and a high

h, paired with the appropriate values of h to ensure that R0 remains three. The plots of

these simulations are shown in Figure 13.

First, we note that in addition to the yearly dynamics, the disease shows interepidemic

periods that span across several years. The disease appears to be eradicated for a few years,

with very few individuals in the exposed and infected classes, but then another outbreak

occurs. It also appears that the population remains around the initial population size of

15, 000 on average. There are fluctuations above and below the initial population size

35

0 3e05 6e05 9e05 0.00012 0.00015 0.00018 0.00021 0.00024 0.00027 0.00030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Dis

ease

ind

uced

mor

tali

ty r

ate

(s)

0

2

4

6

8

10

12

14

16

18

20

(A) s = .1

0 3e05 6e05 9e05 0.00012 0.00015 0.00018 0.00021 0.00024 0.00027 0.00030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Dis

ease

ind

uced

mor

tali

ty r

ate

(s)

0

2

4

6

8

10

12

14

16

18

20

(B) s = .5

0 3e05 6e05 9e05 0.00012 0.00015 0.00018 0.00021 0.00024 0.00027 0.00030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Dis

ease

ind

uced

mor

tali

ty r

ate

(s)

0

2

4

6

8

10

12

14

16

18

20

(C) s = 1


during the swarming season as the disease-induced mortality rate (s) and

the transmission rate (s) are varied. The basic plot has been reproduced for

low, medium, and high values of a third disease-parameter, s.

before and after an epidemic, respectively, but the mean population size is close to the

initial population size.

This experiment illustrates several features of the ways in which disease dynamics can

change while maintaining a constant basic reproductive number. Most obviously, we see

that the number of epidemics seen in a forty year period changes with h. The peak size of

the exposed class during an outbreak remains roughly constant because we are not changing

h across these simulations. The peak sizes of the infected class and the total population,

however, vary as we change h and h. When the disease induced death rate and the

transmission rate are low (h = .01, h = 2 106), as shown in Figure 13 in green, we

36

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Diseaseinduced mortality rate (h)

Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(h)

0

2

4

6

8

10

12

14

16

18

20

(A) h = .00001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(h)

0

2

4

6

8

10

12

14

16

18

20

(B) h = .00015

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(h)

0

2

4

6

8

10

12

14

16

18

20

(C) h = .0003


during the hibernation season as the disease-induced mortality rate (h) and

the rate that exposed bats move into the infected class (h) are varied. The

basic plot has been reproduced for low, medium, and high values of a third

disease-parameter, h.

see only two outbreaks with an interepidemic period of about 15 years. The peak population

size is close to 30,000 and the peak number of infected bats is about 7,000. On the opposite

end of the spectrum, when the disease-induced death rate and the transmission rate are

high (h = 1, h = 2 104), as shown in Figure 13 in red, there are 4 outbreaks with aninterepidemic period of about 10 years. The peak population size is about 20,000 and the

peak number of infected bats is about 1,500. Running the simulation with the intermediate

values of h and h yields results that lie in somewhere in between, as shown in Figure 13

in black.

37

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Transmission rate (h)

Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(h)

0

2

4

6

8

10

12

14

16

18

20

(A) h = .1

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(h)

0

2

4

6

8

10

12

14

16

18

20

(B) h = .5

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Rate at whi

ch e

xpos

ed b

ats

beco

me i

nfec

ted

(h)

0

2

4

6

8

10

12

14

16

18

20

(C) h = 1


during the swarming season as the transmission rate (h) and the rate that

exposed bats move into the infected class (h) are varied. The basic plot

has been reproduced for low, medium, and high values of a third disease-

parameter, h.

In general, we see that as we increase h and h (always keeping the basic reproduc-

tive number constant), the interepidemic period decreases and so we see epidemics more

frequently. Intuitively, since we are increasing h, outbreaks are able to occur with fewer

susceptibles so as the population climbs during non-outbreak years, we hit the critical epi-

demic threshold earlier. We also notice that increasing h and h decreases the peak number

of susceptibles and invectives. This is because the higher h moves bats out of the suscepti-

ble class more quickly and the higher h removes bats from the infected class more quickly

as well.

38

0 3e05 6e05 9e05 0.00012 0.00015 0.00018 0.00021 0.00024 0.00027 0.00030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Dis

ease

ind

uced

mor

tali

ty r

ate

(h)

0

2

4

6

8

10

12

14

16

18

20

(A) h = .1

0 3e05 6e05 9e05 0.00012 0.00015 0.00018 0.00021 0.00024 0.00027 0.00030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Dis

ease

ind

uced

mor

tali

ty r

ate

(h)

0

2

4

6

8

10

12

14

16

18

20

(B) h = .5

0 3e05 6e05 9e05 0.00012 0.00015 0.00018 0.00021 0.00024 0.00027 0.00030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Dis

ease

ind

uced

mor

tali

ty r

ate

(h)

0

2

4

6

8

10

12

14

16

18

20

(C) h = 1


during the swarming season as the disease-induced mortality rate (h) and

the transmission rate (h) are varied. The basic plot has been reproduced

for low, medium, and high values of a third disease-parameter, h.

To summarize, while maintaining a constant basic reproductive number, it is possible to

change the interepidemic period, the peak population size, the peak number of susceptible

bats, and the peak number of infected bats during an epidemic by varying h and h.

5.2. Incubation of WNS and its Dynamical Consequences (The Story of ). In thenumerical analysis section, we saw that the average amount of time that bats spend in

the exposed class does not have a significant effect on the basic reproductive number of

WNS during the roosting, swarming, or hibernation periods. Does this mean that there is

no exposure period for WNS at all? This seems to call into question the inclusion of an

"exposed" class in our model.

39

10 15 20 25 30 35 40 45 500

1

2

3x 104

Time (years)

Susc

eptib

les

10 15 20 25 30 35 40 45 500

5000

10000

Time (years)

Expo

sed

10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

Time (years)

Infe

cted

10 15 20 25 30 35 40 45 500

1

2

3x 104

Time (years)

Popu

latio

n

FIGURE 13. Deterministic simulations with R0 = 3. For each simulation,

r = s = h = 1/100. Different values of h and h are used for each

color: h = 0.01, h = 2 106 (green), h = 0.1, h = 2 105 (black),h = 1, h = 2 104 (red).

We will see that the average length of time that bats spend in the exposed class has a very

large effect on the overall dynamics of the population, despite changing R0 very little.

Having seen the effects of varying h and h on the long-term disease dynamics, we turn

our attention to h. We will fix the intermediate values of h = .1 and h = 2 105 for

40

the previous example and re-run the simulations for high, intermediate and low values of

h. The results are shown in Figure 14.

Interestingly, we dont see the same monotonic changes that we did while changing hand h. Instead, there appears to be diminishing returns in some sense as the bats spend

longer in the exposed class. For example, the peak number of exposed bats during an

epidemic is highest when = 1/100, shown in Figure 14 in black. It makes sense that as

gets smaller, the size of the exposed class would get larger because bats are spending a

longer time in the exposed class on average. However, at some point, the peak number of

exposed bats begins to decrease, as demonstrated by the transition between black and red in

Figure 14. These are mainly quantitative considerations that will become more important in

the face of empirical data. For predominantly intuitive reasons, we have chosen to bound

from below by 1/365. This value of would mean that bats spend an average of one year in

the exposed class. Any longer would be unreasonable because it is clear that the fungus acts

as an active disease agent during the cold hibernation months from its demonstrated ability

to kill the bats. Therefore we expect that any bats in the exposed class before hibernation

do not remain in the exposed class after hibernation.

We now turn our attention back to Figure 14. There is an important qualitative change

caused by that we have not talked about yet. As decreases, the interepidemic period

decreases as well. We are seeing more frequent epidemics as bats spend a longer time in

the exposed class. Moving away from the comfort of monotonicity, we will discuss the

relative strength of the epidemics. The epidemics appear to be getting stronger, in terms

of the peak number of invectives, as we decrease . However, once we reach = 1/365,

the interepidemic period is very smaller and the strength of the average epidemic is much

smaller. The strength of the epidemics appears to be cyclic, ebbing and flowing over a

period of almost a decade. These are important considerations to take into account when

attempting to fit this model to the empirical data. The key point to keep in mind is that

changes the interepidemic period in a predictable way, but there are some diminishing

returns with respect to the amount of time spent in the exposed class, so be mindful.

6. CONTROL OF WHITE-NOSE SYNDROME

The future of the little brown bat in the northeastern United States is in danger. One anal-

ysis of the situation places the probability of extinction within the next 16 years at 99%. We

will push the limits of the parameters of our model and evaluate the extinction probabilities.

Doing so will elucidate the most appropriate methods of disease control. Each method of

41

10 15 20 25 30 35 40 45 500

1

2

3x 104

Time (years)

Susc

eptib

les

10 15 20 25 30 35 40 45 505000

0

5000

10000

Time (years)

Expo

sed

10 15 20 25 30 35 40 45 502000

0

2000

4000

Time (years)

Infe

cted

10 15 20 25 30 35 40 45 500

1

2

3x 104

Time (years)

Popu

latio

n

FIGURE 14. Deterministic simulations with R0 = 3. For each simulation,

h = 2 105 and h = .1. Each color uses a different value of , which isused for r, s, and h: = 1 (green), = 1/10 (blue), = 1/100 (black),

and = 1/365 (red).

42

disease control works by affecting a particular parameter and pushing it back into a non-

extinction zone of parameter space. We have to move away from deterministic simulations

and into stochastic simulations so we can effectively discuss extinction probability.

We define the probability of extinction to be the proportion of 1000 simulations that the

total population drops below a quasi-extinction threshold. We will let the quasi-extinction

threshold be about 3% of the initial population size, which in our case is 500 bats. The rea-

son for this threshold is two-fold. First, it takes any possible Allee effects into account.

If the little brown population exhibits strong Allee effects, then the population would

be driven to extinction below a certain population density. Second, the quasi-extinction

threshold accounts for demographic stochasticity, which is just the variability in population

growth rates as a result of random differences in individual bats. Demographic stochasticity

generally only plays an important role when the population is small enough to be affected

by these differences.

6.1. Heat and Fungicide. The causative agent of WNS, the fungal pathogen P. destructans,is particularly hardy. It has demonstrated an ability to persist in the environment in the ab-

sence of bat hosts for long periods of time. The fungus undergoes complex morphological

changes in order to persist and can survive on a wide variety of carbon and nitrogen based

sources (http://www.mnn.com/earth-matters/animals/blogs/bat-killing-fungus-all-but-invincible-

study-finds). The fungus does have one glaring weakness: its psychrophilic nature. P.

destructans cannot grow above 24C and its growth is significantly hindered above 15C.

This points us to a method of disease control involving heat.

The placement of heat lamps in targeted areas of bat hibernacula has the potential to

deplete the environmental reservoir and, in doing so, lower the environmental transmission

term P . The number of free-living WNS pathogen P would decrease due to heat-induced

pathogen death and if the growth rate of the fungus is slowed or brought to a halt, then

the rate at which hibernating bats are infected due to the environmental reservoir () will

decrease as well.

In Figure ??, we demonstrate how the application of heat lamps can save a populationfrom potential extinction. In Figure ??(A), the population is driven to extinction within a50 year window. By applying the heat lamp control method, we lower by two orders of

magnitude (from 2 106 to 2 104), pushing the population back up to the sustainablelevels shown in Figure ??(B). This is meant to serve as a demonstration of how this controlmethod has the potential to improve the little brown bats prospect of survival in the long

run by lowering the environmental transmission term.

43

10 15 20 25 30 35 40 45 500

50

100

150

200

250

300

350

400

450

Time (years)

Num

ber o

f Bat

s)

SusceptiblesExposedInfectivesTotal

(A) Before

10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

8000

Time (years)

Num

ber o

f Bat

s)


(B) After

FIGURE 15. Reducing the environmental reservoir has the potential to save

bat populations from extinction.

We may model the heat lamp control method mathematically by incorporating a heat-

induced pathogen weakening term (h) into the hibernation submodel. The new environ-

mental transmission term becomes hP , so that rate at which bats are infected due to the

environmental reservoir is inversely proportional to the pathogen weakening term h.

44

dShdt

= [hIh +

hP

]Sh Sh (6.1)

dEhdt

=

[hIh +

hP

]Sh (h + )Eh (6.2)

dIhdt

= hEh (h + )Ih (6.3)

In our example, h was 100 so that it reduced the environmental transmission term by two

orders of magnitude. Using a fungicide would have the same effect, but it acts by wiping

out large populations of the fungus instead of inhibiting its growth.

We can use heat in another capacity: to slow the rate of disease-induced deaths during

hibernation. Boyles and Willis present evidence to suggest that artificial warming of lo-

calized areas within hibernacula may increase survival of bats infected with WNS. They

propose using wooden boxes and heating coils to provide a localized warm areas for hiber-

nating bats. These "thermal refugia" are designed to lessen the energetic costs due to heat

loss associated with the periodic arousals caused by the fungus. [3]

This heating strategy, in theory, lowers the disease-induced mortality rate h. Having

tested this disease-control using our WNS model, we conclude that its efficacy depends on

the initial value of h. That is, if h is on the high end of the acceptable range (which we

determined to be simply less than 1), then the thermal refugia strategy has the potential

to worsen the long-term prospects of the bat population associated with that hibernacula.

However, for lower values of h, this strategy could save a population from potential ex-

tinction. These possibilities are demonstrated in Figures 16 and 17, respectively.

6.2. Culling. Culling bats in hibernacula has been proposed as a way to inhibit the spreadof WNS. One study performed by Hallam and McCracken concludes that in general,

culling is ineffective in the control of animal diseases in the wild," based on their simu-

lations of WNS. [5] This is a sweeping generalization based on simulations which are too

narrow in scope to warrant such a statement. In addition, the temporal resolution of their

model is one year, which completely ignores the seasonal dynamics of the bat populations.

Unfortunately, since we are currently modeling a single hibernacula, we are unable to look

at the effects of destroying an entire colony on the spread of WNS. When the model is

adjusted to look at spatial dynamics, we will examine the culling of infected bats in hiber-

nacula within our 3-phase model to see if a finer temporal resolution changes the verdict.

6.3. Restricted Access to Caves. Restrict human access to bat hibernacula has proposedin an effort to minimize the anthropogenic spread of the WNS fungus to other caves in

45

10 15 20 25 30 35 40 45 500

200

400

600

800

1000

1200

1400

1600

1800

Time (years)

Num

ber o

f Bat

s)


(A) Before

10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5x 104

Time (years)

Num

ber o

f Bat

s)


(B) After

FIGURE 16. Localized heating strategy saves bat population from extinc-

tion. The initial value of h is 1/300 and after the heat boxes are added hdrops to 1/800.

the area. Our model is not spatially-coupled since we focused on the dynamics of the

population of a single hibernacula site, so it is not yet amenable to testing this strategy.

Several strategies of this type are analyzed in a WNS management report organized by

the U.S. Fish and Wildlife Service and State Natural Resource Agencies. The authors

46

10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

3500

4000

Time (years)

Num

ber o

f Bat

s)


(A) Before

10 15 20 25 30 35 40 45 500

200

400

600

800

1000

1200

1400

1600

1800

Time (years)

Num

ber o

f Bat

s)


(B) After

FIGURE 17. Localized heating strategy drives sustainable bat population

to extinction. The initial value of h is 1/100 and after the heat boxes are

added it drops to 1/300

recommend that infected sites be closed to all access and that sites within 75 miles of an

infected site be closed to all access except research. [14]

47

7. CONCLUSIONS

We determined the individual effects of each parameter on the qualitative behavior of the

disease dynamics.

The latent period of the disease ( ) during hibernation plays a role in the the peak number

of exposed bats during an epidemic. The effect is not monotonic in the sense that an

intermediate value of in our range attains the highest peak. The time spent in the exposed

class is also responsible for the length of the interepidemic period. The interepidemic

period is proportional to (and hence inversely proportional to the latent period of the

disease).

The dynamics caused by the transmission rate and disease-induced mortality rate are

intimately tied together because and determine the basic reproductive of WNS. We

have to look at how they simultaneously alter the dynamics in order to maintain the same

basic reproductive number. The interepidemic period is inversely proportional to h and

h. The peak numbers of susceptible and infected bats between and during epidemics,

respectively, are inversely proportional to h and h as well.

After exploring disease-control strategies involving heat and fungicide, we demonstrated

that reducing the environmental reservoir has the potential to reverse the fate of a little

brown bat population destined for extinction. Localized heating strategies with the intent to

reduce the number of infected bats dying due to the disease have varying effects depending

on the initial disease-induced mortality rate. In general, if the rate of death is fairly high in

our range, then reducing it may cause the population to crash because infected bats survive

hibernation and spread the disease to other

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