Persistence and dynamics of Persistence and dynamics of disease in a host-pathogen disease in a host-pathogen

model with seasonality in the model with seasonality in the host birth rate.host birth rate.

Rachel Norman and Jill IrelandRachel Norman and Jill Ireland

Motivation: RHDMotivation: RHD

• Rabbit Haemorhaggic Rabbit Haemorhaggic disease kills up to 95% of disease kills up to 95% of those infected within 48 those infected within 48 hours.hours.

• First recorded in China in First recorded in China in 1984 where it killed 140 1984 where it killed 140 million farmed rabbits.million farmed rabbits.

• The disease is now in wild The disease is now in wild rabbits and is a rabbits and is a conservation issue in Europe conservation issue in Europe and a pest control issue in and a pest control issue in Australia/New Zealand.Australia/New Zealand.

• Arid areas: 65-90% Arid areas: 65-90% population reduction.population reduction.

• Temperate areas: less Temperate areas: less impact.impact.

The original hypothesisThe original hypothesis

Since breeding patterns for rabbits will be Since breeding patterns for rabbits will be different in temperate (longer) than arid different in temperate (longer) than arid (shorter) regions then the number of (shorter) regions then the number of susceptibles available will vary and so susceptibles available will vary and so disease persistence is likely to be disease persistence is likely to be effected. It might also be the case that effected. It might also be the case that the time of year that the disease the time of year that the disease appears will have an effect on appears will have an effect on persistence.persistence.

The basic model (no The basic model (no seasonality):seasonality):

YsHXYdt

dY

ZsHbYdt

dZ

YHsHrdt

dH

AnalysisAnalysis

• Three possible biologically relevant Three possible biologically relevant equilibria of the form (H,Y,Z)equilibria of the form (H,Y,Z)

• (0,0,0), nothing there, only stable if the (0,0,0), nothing there, only stable if the growth rate is negative.growth rate is negative.

• (K,0,0) host at carrying capacity, no (K,0,0) host at carrying capacity, no infection, stable iff infection, stable iff

• Coexistence equilibriumCoexistence equilibrium

1)(

)(0

bas

baR

Coexistence equilibriumCoexistence equilibrium

sHb

YZ

10 R

0

2

22

2

223

r

bK

r

Kb

r

Kb

r

bKHK

r

KHH

We can show that there is one root of this cubic lying between 0 and K iff

We assume that this equilibrium is stable or there are stable limit cycles if no other equilibrium is stable.

1)(

)(0

bas

baR

Dynamics of the basic Dynamics of the basic modelmodel Numerical experiments Numerical experiments

have not shown stable have not shown stable limit cycles but under limit cycles but under some parameter some parameter regimes we get regimes we get oscillations to a stable oscillations to a stable equilibrium, in other equilibrium, in other words we have a spiral words we have a spiral point in state space:point in state space:

10 20 30 40 50 60 70

27.5

28

28.5

29

29.5

30

30.5

31

2 4 6 8 10

0.01229

0.01231

0.01232

0.01233

SeasonalitySeasonality

• The basic model assumes a constant birth The basic model assumes a constant birth rate. In many real biological systems this is not rate. In many real biological systems this is not the case e.g rabbits, grouse, foxes, bank voles.the case e.g rabbits, grouse, foxes, bank voles.

• We therefore make the birth rate in the model We therefore make the birth rate in the model seasonal, in the first instance we use a sine seasonal, in the first instance we use a sine wavewave

))2sin(1()( 10 taata

The seasonal model:The seasonal model:

YsHXYdt

dY

ZsHbYdt

dZ

YHsHrdt

dH

Analysis: disease Analysis: disease persistencepersistence• In this case we cannot In this case we cannot

carry out equilibrium carry out equilibrium analysis in the same analysis in the same way.way.

• We therefore took a We therefore took a different approach and different approach and assumed that as a(t) is assumed that as a(t) is bounded then we can bounded then we can use those bounds to use those bounds to determine disease determine disease persistence:persistence:

• Assumption: Ro is of Assumption: Ro is of the same form as in the same form as in the basic model i.e.the basic model i.e.

))((

))((0 btas

btaR

Initial ConjecturesInitial Conjectures

• If RIf R00(t)<1 for the whole season, then (t)<1 for the whole season, then the disease will not persist.the disease will not persist.

• If RIf R00(t)>1 for the whole season then (t)>1 for the whole season then the disease will persist.the disease will persist.

• When neither of the above cases hold When neither of the above cases hold then there is some “average” value of then there is some “average” value of RR00(t) above which the disease persists.(t) above which the disease persists.

RR00<1 for all t.<1 for all t.

0.2 0.4 0.6 0.8 1t

0.2

0.4

0.6

0.8

1Rot

105 110 115 120t

210-34

410-34

610-34

810-34

Y

105 110 115 120t

29.5

30

30.5

31

H

RR00>1 for all t.>1 for all t.

0.2 0.4 0.6 0.8 1t

2

4

6

8

10

12

Rot

105 110 115 120t

5.25

5.35

5.4

5.45

Y

105 110 115 120t

11.4

11.6

11.8

12.2

H

In between:In between:

0.2 0.4 0.6 0.8 1t

0.2

0.4

0.6

0.8

1

1.2

Rot

105 110 115 120t

110-18

210-18

310-18

410-18

510-18

Y

105 110 115 120t

29.5

30

30.5

31

H

ConjectureConjecture

• If then the disease If then the disease

will die out.will die out.

• If then the disease If then the disease

will persist will persist

1)(

)(,0

bas

baR

ave

aveave

1)(

)(,0

bas

baR

ave

aveave

Dynamics and applicationsDynamics and applications

• We will now consider two biological We will now consider two biological applications which will illustrate applications which will illustrate interesting aspects of the dynamics of interesting aspects of the dynamics of this model.this model.

• These systems are cowpox in bank These systems are cowpox in bank voles and rabbit haemorhaggic disease voles and rabbit haemorhaggic disease in rabbits. We have parameterised the in rabbits. We have parameterised the model for both of these systems.model for both of these systems.

Cowpox in bank volesCowpox in bank voles

• This disease is endemic in bank voles and This disease is endemic in bank voles and the system is of interest since ecologists the system is of interest since ecologists have been trying to explain the cycles have been trying to explain the cycles seen in vole populations.seen in vole populations.

• One interesting aspect of this disease is One interesting aspect of this disease is that it does no kill the host, this simplifies that it does no kill the host, this simplifies the model. However, the results which the model. However, the results which follow do not rely, mathematically, on follow do not rely, mathematically, on this.this.

Cowpox without seasonalityCowpox without seasonality

5 10 15 20

48.02

48.04

48.06

48.08

48.12

48.14

H

time

SeasonalitySeasonality

• If we add seasonality to the If we add seasonality to the model we get Rmodel we get R0,min0,min=16.14 and =16.14 and RR0,max0,max=21.24, clearly both of =21.24, clearly both of these values lie above 1 and we these values lie above 1 and we would expect the disease to would expect the disease to persist.persist.

• These simulations are consistent These simulations are consistent with the data on cowpox in bank with the data on cowpox in bank voles since they predict an voles since they predict an annual cycles in vole dynamics.annual cycles in vole dynamics.

• In addition, the peaks and In addition, the peaks and troughs in the infected section of troughs in the infected section of the population come after those the population come after those in the susceptible section, again in the susceptible section, again this is consistent with field this is consistent with field observations.observations.

9.2 9.4 9.6 9.8 10

2.2

2.4

2.6

2.8

3

9.2 9.4 9.6 9.8 10

25

30

35

40

Infecteds

Susceptibles

time

RHD without seasonalityRHD without seasonality

2 4 6 8 10

0.01229

0.01231

0.01232

0.01233

RHD with seasonalityRHD with seasonality• In this case, if we add seasonality RIn this case, if we add seasonality R0,min0,min=0, =0,

RR0,max0,max=353.2 and R=353.2 and R0,ave0,ave=180, again, in this case we =180, again, in this case we predict disease persistence.predict disease persistence.

2 4 6 8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07Y

time

• As we can see from the previous As we can see from the previous simulation, the dynamics of RHD are simulation, the dynamics of RHD are predicted to be extremely complex, in predicted to be extremely complex, in fact what we have is resonance fact what we have is resonance between the underlying oscillations in between the underlying oscillations in the non-seasonal model and the the non-seasonal model and the seasonality. This phenomenon has seasonality. This phenomenon has been studied by Greenman et al (2004) been studied by Greenman et al (2004) for a model with seasonality in the for a model with seasonality in the transmission rate. Their model explains transmission rate. Their model explains many of the characteristics of the many of the characteristics of the dynamics of childhood diseases such as dynamics of childhood diseases such as measles and whooping cough.measles and whooping cough.

Method of analysisMethod of analysis• We scale the equations by a factor 1/p and rescale time such that t’=tpWe scale the equations by a factor 1/p and rescale time such that t’=tp• We now have all of the parameters divided by p and, more importantlyWe now have all of the parameters divided by p and, more importantly

• We then vary pWe then vary p• As p varies we effectively looking at a different member of the family of models defined As p varies we effectively looking at a different member of the family of models defined

by our equations.by our equations. ))2

sin(1()( 10 p

taata

Bifurcation diagram for RHDBifurcation diagram for RHD

Resonance diagram for RHDResonance diagram for RHD

When do we get resonance?When do we get resonance?• We get resonance when We get resonance when

our non-seasonal model our non-seasonal model exhibits oscillations exhibits oscillations towards the equilibrium, towards the equilibrium, which is when the which is when the eigenvalues of the eigenvalues of the Jacobian are complex Jacobian are complex with negative real parts. with negative real parts. We can work out a We can work out a formula for the line formula for the line which separates real and which separates real and complex eigenvalues and complex eigenvalues and plot it in parameter plot it in parameter space.space.

Figure 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05

alpha (per day)

Be

ta (

pe

r d

ay

)

RHD

Real

Complex

Summary and conclusionsSummary and conclusions• Many real wildlife systems have seasonal birth rates.Many real wildlife systems have seasonal birth rates.• We cannot analyse the model presented using equilibrium We cannot analyse the model presented using equilibrium

analysis.analysis.• We can find a simple condition for persistence of the We can find a simple condition for persistence of the

disease.disease.• However, the dynamics of the disease can be much more However, the dynamics of the disease can be much more

complex as we get resonance between the oscillations in complex as we get resonance between the oscillations in the non-seasonal model and the seasonal forcing term.the non-seasonal model and the seasonal forcing term.

• In real biological terms this means the if a disease enters a In real biological terms this means the if a disease enters a seasonally oscillating population then resonance and seasonally oscillating population then resonance and chaotic dynamics could occur.chaotic dynamics could occur.

• Adding seasonal birth rates to models of real systems gives Adding seasonal birth rates to models of real systems gives us dynamics which are at least qualitatively similar to the us dynamics which are at least qualitatively similar to the observed dynamics in RHD, rabies and cowpox in bank observed dynamics in RHD, rabies and cowpox in bank voles.voles.