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Modelling the seasonality of respiratory syncytial virus

in young children

A.B. Hogana, G.N. Mercera, K. Glassa, H.C. Mooreb

aNational Centre for Epidemiology and Population Health, Australian National University,

Canberra, Australia

bTelethon Institute for Child Health Research, Centre for Child Health Research,

University of Western Australia, Perth, Australia

Email: Alexandra.Hogan@anu.edu.au

Abstract: Respiratory syncytial virus (RSV) is a major cause of acute lower respiratory tract infections in

infants and young children. The transmission dynamics of RSV infection among young children are still

poorly understood (Hall et al., 2009) and mathematical modelling can be used to better understand the seasonal

behaviour of the virus. However, few mathematical models for RSV have been published to date (Moore et al.,

2013; Weber et al., 2001; Leecaster et al., 2011) and these are relatively simple, in contrast to studies of other

infectious diseases such as measles and inﬂuenza.

A simple SEIRS (Susceptible, Exposed, Infectious, Recovered, Susceptible) type deterministic ordinary dif-

ferential equation model for RSV is constructed and then expanded to capture two separate age classes with

different transmission parameters, to reﬂect the age speciﬁc dynamics known to exist for RSV. Parameters in

the models are based on the available literature.

In temperate climates, RSV dynamics are highly seasonal with mid-winter peaks and very low levels of ac-

tivity during summer months. Often there is an observed biennial seasonal pattern in southern Australia with

alternating peak sizes in winter months. To model this seasonality the transmission parameter β(t)is taken to

vary sinusoidally with higher transmission during winter months, such as in models presented in Keeling and

Rohani (2008) for infections such as measles and pertussis:

β(t) = β0[1 + β1sin(2πt

52 )].(1)

This seasonal forcing reﬂects increases in infectivity and susceptibility thought to be due to multiple factors

including increased rainfall, variation in humidity, and decreased temperature (Cane, 2001; Weber et al., 1998).

Sinusoidally forced SIR-type models are known to support complex multi-periodic and even chaotic solutions.

For realistic parameter values, obtained from the literature, and depending on the values selected for β0and

β1, the model predicts either annual peaks of the same magnitude, or the observed biennial pattern that can be

explained by the interaction of the forcing frequency and the natural frequency of the system. This behaviour

is in keeping with what is observed in different climatic zones.

Keywords:

Mathematical

model,

infectious

disease,

respiratory

syncytial

virus,

seasonality

20th International Congress on Modelling and Simulation, Adelaide, Australia, 1–6 December 2013

www.mssanz.org.au/modsim2013

338

A.B. Hogan et al., Modelling respiratory syncytial virus in young children

1 INTRODUCTION

Respiratory syncytial virus (RSV) is a signiﬁcant health and economic burden in Australia and internationally.

Epidemics are strongly seasonal, occurring each winter in temperate climates (Cane, 2001) and during the

rainy season in tropical climates (Simoes, 1999), usually lasting between two and ﬁve months (Hall, 1981;

Kim et al., 1973).

There are only limited RSV data sets published and often these are only over short time spans. Recent data

sets that span numerous years for Utah in the U.S.A. (Leecaster et al., 2011), southern Germany (Terletskaia-

Ladwig et al., 2005) and Western Australia (Moore et al., 2013), show a distinct biennial seasonal pattern with

higher peaks in alternate winter seasons. These regions all have a temperate climate and experience signiﬁcant

seasonal variation in climate. Other data sets, such as for Singapore (Chew et al., 1998) and the Spanish region

of Valencia (Acedo et al., 2011) show annual seasonal behaviour, with peaks of the same magnitude each year.

While the mortality rate for previously healthy children is low, RSV causes high rates of hospitalisation for

children under two years of age and has also been identiﬁed as a cause of mortality in the elderly (Faskey

et al., 2005; Simoes, 1999; Hardelid et al., 2013). Clinical symptoms may vary from those of a mild infection

to severe bronchiolitis or pneumonia (Hall, 1981).

In the young, infection with RSV does not cause long lasting protective immunity, meaning that individual

children may be repeatedly infected. There is currently no licensed vaccine available, nor any antiviral treat-

ments commonly used for RSV infection in Australia.

The aim of this study is to develop RSV models that reproduce the biennial pattern observed in temperate

climates. In later work these models will be ﬁtted to available data. Thus we present a SEIRS model for

a single age class for RSV infection, where the transmission rate is seasonally forced, such as in models

presented in Keeling and Rohani (2008). An investigation into the types of behaviour possible in this model is

undertaken. To better reﬂect the known epidemiology of RSV, we then introduce a second age class into the

model with a second transmission rate and investigate how this affects the transmission dynamics.

2 MODEL FOR A SINGLE AGE CLASS

A deterministic ordinary differential equation model is developed for the transmission of RSV for 0-23 month

old children. This age group was chosen as the literature indicates that almost all children have been infected

by the time they reach this age (Hall, 1981; Sorce, 2009). As it remains unclear to what degree adults carry

and shed the virus, thereby infecting children, the adult population was not included in the model.

The population is divided into four compartments, where Srepresents the proportion of the population that

is susceptible to infection; Erepresents the proportion of the population that is exposed but not yet infected;

Irepresents the proportion that is infected with the virus; and Rrepresents the proportion of the population

that is recovered and temporarily immune to reinfection. The SEIRS-type model was selected for two reasons.

Firstly, the virus is known to have a latency period between an individual being exposed to the infection and

becoming infectious. This period is of the same order of magnitude as the infectious period and hence needs

to be included to accurately represent the disease dynamics. Secondly, infection from the virus does not cause

long-lasting immunity, hence recovered individuals may return to the susceptible class and be reinfected.

S E I R

µµµµ

µ

βδγ

ν

Figure 1. Schematic diagram for single age class SEIRS model for RSV transmission.

A schematic representation of the model is shown in Figure 1. The average recovery period is represented by 1

γ,

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A.B. Hogan et al., Modelling respiratory syncytial virus in young children

the average latency period (the time between contracting the infection and becoming infectious) is represented

by 1

δand the average duration of immunity is represented by 1

ν. The rate of entering the model (the birth rate)

is equal to the rate at which individuals age out of the model, and is represented by µ. The virus is transmitted

between individuals at rate β.

The differential equations, where time in weeks is represented by t, are

dS

dt =µ−βSI −µS +νR (2)

dE

dt =βSI −δE −µE (3)

dI

dt =δE −µI −γI (4)

dR

dt =γI −µR −ν R. (5)

2.1 Parameter values

The birth rate and ageing rate µis chosen as in Moore et al. (2013) and is based on the average number of

births per week in the metropolitan region of Western Australia. This gives an average weekly birth rate of

0.012. Assuming the birth and ageing rates are equal simpliﬁes the calculations as it ensures the population

size remains constant. Here this assumption does not change the overall dynamics of the system.

Based on the literature, the average latency period for RSV is assumed to be four days (Weber et al., 2001;

Moore et al., 2013). Other models, such as those presented by Leecaster et al. (2011), assume a latency period

of ﬁve days. A four day latency period equates to δbeing 1.754 (or 1

0.57 ).

Similarly, the average recovery period is based on estimates in previous models for RSV of 10 days (Weber

et al., 2001; Moore et al., 2013; Acedo et al., 2010; Leecaster et al., 2011), and within the range of one to 21

days identiﬁed by Hall et al. (1976). This gives γequal to 0.714 (or 1

1.4).

The immunity period is the time between recovering from a RSV infection to becoming susceptible to the virus

again. Although not currently well understood, there is some evidence that the immunity period is around 200

days which equates to νbeing 0.035 (or 1

28.57 ). This again is the value used in previous modelling of RSV

(Weber et al., 2001; Moore et al., 2013; Acedo et al., 2010).

The transmission rate β(t), given at Equation 1, was chosen to reﬂect the observed annual seasonality of RSV

in temperate climates. Similar seasonal forcing has been applied in other models for RSV transmission (Weber

et al., 2001; Moore et al., 2013; Acedo et al., 2010; Arenas et al., 2008; Leecaster et al., 2011). The sinusoidal

function, with a period of 52 weeks, accounts for the observed higher transmission between children during

the winter months. The term β0represents the average transmission rate and β1represents the amplitude of

the seasonal ﬂuctuation (Keeling and Rohani, 2008).

For the purpose of investigating the overall dynamics of this model, a range of values was considered for β0.

For β1, a value of 0.6 was assumed (noting that 0< β1≤1), in order to replicate the conditions in temperate

climates where strong seasonality is observed. For some seasonally forced models for RSV, values as high

as 1 (Leecaster et al., 2011) have been assumed. Other models assume much lower values for β1, such as

between 0.10 and 0.36 (Arenas et al., 2008). In future work we will estimate these parameters β0and β1using

longitudinal data from Western Australia.

2.2 Numerical solution

The system of differential equations was solved and plotted using MATLAB’s ode45 routine. A burn in time of

80 years was used to allow the system to stabilise and thereby remove the dependence on the initial conditions.

When there is no seasonality in the transmission rate (when βis constant, β1=0), the natural oscillations in

the system die out and the system reaches a steady state. With seasonality, there is either a distinct biennial

pattern, with higher peaks in alternate winter seasons, or peaks of the same magnitude each year, depending

on the values selected for β0and β1. Figure 2(a) depicts a plot of a biennial pattern solution for the infected

population, with β0=1.1 and β1=0.6.

As there are values of β0and β1where there is no biennial pattern but instead where the seasonal peak reaches

the same maximum each year, adjacent seasonal peaks versus the parameter β0were plotted in order to better

340

A.B. Hogan et al., Modelling respiratory syncytial virus in young children

0 50 100 150 200 250 300 350

0

0.02

0.04

0.06

0.08

0.1

0.12

Weeks

Proportion Infected

β0=1.1, β1=0.6

(a) A numerical solution for the infected population of a single age

class SEIRS model for RSV transmission with a sinusoidally forced

transmission parameter.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

0

0.02

0.04

0.06

0.08

0.1

0.12

β0

Proportion Infected

(b) Maximum proportion of infectives at adjacent seasonal peaks

versus the bifurcation parameter β0, for the single age class system.

The parameter determining the amplitude of the forcing, β1, is 0.6

Figure 2.

understand the bifurcation patterns of the system. Figure 2(b) gives an impression of the bifurcation structure

of the model, showing seasonal peaks over two adjacent years. Where the seasonal pattern is annual, only a

single peak is shown, whereas biennial dynamics gives two distinct peaks. There is a speciﬁc range of possible

values for β0for which the system will feature the biennial seasonal pattern. Outside this range, the system

reverts to a regular annual seasonal pattern. These results are in keeping with what is observed in different

climatic zones.

3 MODEL FOR TWO AGE CLASSES

Studies show that the transmission dynamics of RSV change as children age. That is, incidence is higher

for children aged less than 12 months than those in the 12-23 month age class (Moore et al., 2010). It is

still unclear why older children are less affected. Possible reasons are reduced susceptibility, or less severe

symptoms (so less infections are reported), as a result of prior infection with the virus; or due to having better

developed immune systems than younger children. Thus, we present a second set of differential equations to

account for two age classes and two transmission parameters, βAand βB.

The second model is the set of differential equations

dS1

dt =µ−βAS1(I1+I2)−ηS1+νR1(6)

dE1

dt =βAS1(I1+I2)−δE1−ηE1(7)

dI1

dt =δE1−ηI1−γI1(8)

dR1

dt =γI1−ηR1−νR1(9)

dS2

dt =ηS1−βBS2(I1+I2)−ηS2+ν R2(10)

dE2

dt =ηE1+βBS2(I1+I2)−δE2−ηE2(11)

dI2

dt =ηI1+δE2−ηI2−γI2(12)

dR2

dt =ηR1+γI2−ηR2−νR2.(13)

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A.B. Hogan et al., Modelling respiratory syncytial virus in young children

S1E1I1R1

ηηηη

µ

βAδγ

S2E2I2R2

βBδγ

ηηηη

ν

ν

Figure 3. Schematic description of a SEIRS model for RSV transmission that takes into account two separate

age classes: children aged <12 months, where the virus is transmitted at rate βA; and 12-23 month old

children, where the virus is transmitted at rate βB.

The parameters µ,γ,δand νare as presented for the single age class model in Equations (2)-(5). An additional

parameter ηis introduced to reﬂect the ageing from the <12 month age class into the 12-23 month age class,

and also to reﬂect the ageing out of the 12-23 month class. This rate is assumed to be equal and distributed

evenly over time, therefore ηis taken to be 1

52 . A schematic diagram of the two age class system is given in

Figure 3.

The transmission rates are βA, for the <12 month age class, and βB, for the 12-23 month age class. Both

are of the sinusoidal form presented for the single age class model at Equation (1). Here the parameter β0in

βBwas selected to produce a reduced average transmission rate for the 12-23 month old age class, to better

reﬂect the different transmission dynamics for older children. The β1parameter is the same for βAand βB, to

represent the same climatic region.

3.1 Numerical solution

The system of differential equations was solved using MATLAB’s inbuilt ode45 routine, with a burn intime of

80 years. The model accurately mimics the expected lower number of infectives in the 12-23 month age class

and again produces the biennial pattern for both age classes, as observed in the single age class system. Figure

4(a) shows a solution for the infected population for each age class. Adjacent seasonal peaks for increasing

values of β0were again examined, showing that, as for the single age class system, there is a speciﬁc range of

possible values for β0that produce the biennial seasonal pattern 4(b).

4 DISCUSSION

By sinusoidally forcing the transmission parameter, both models depict either a distinct biennial seasonality, or

annual seasonal peaks of the same magnitude, for realistic parameter values depending on the values selected

for β0and β1. These results are in keeping with what is observed in different climatic zones. We showed that

a simple single age class model, with demography, is sufﬁcient to achieve these seasonal patterns. Both the

single age class model, and the expanded model with two age classes, now provide a base on which to add

complexities.

Future work will investigate varying the recovery, latency and immunity parameters for different age classes, as

well as a more detailed bifurcation analysis of both systems. There is a possibility of more complex behaviour

than the two-cycle pattern observed here being present. We will also investigate whether prior exposure is

the reason for reduced susceptibility, through expanding the model and ﬁtting with population-based linked

laboratory data for the metropolitan region of Western Australia.

There is currently no licensed vaccine for RSV available in Australia. Vaccine development to date has been

problematic due to lack of an ideal animal model for the the disease, and the challenges of immunising infants

342

A.B. Hogan et al., Modelling respiratory syncytial virus in young children

0 50 100 150 200 250 300

0

0.02

0.04

0.06

0.08

0.1

0.12

Weeks

Proportion Infected

I1: β0=3.2 β1=0.6

I2: β0=2.4 β1=0.6

(a) A numerical solution for the infected populations for the <12

month age class and the 12-23 month age class of a SEIRS model

for RSV transmission, where the transmission parameter for each

age class is sinusoidally forced.

2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

β0

Proportion Infected

(b) Maximum proportion of infectives at adjacent seasonal peaks

versus β0(with βB= 0.75βA), demonstrating the biennial be-

haviour over a range of values. In this case, the parameter β1is

0.6.

Figure 4.

who are immunologically immature (Crowe Jr, 2002). However, with a new vaccine currently undergoing

phase two trials (Anderson et al., 2013), we will also look at the optimal timing in the transmission cycle for

the roll out of a vaccination program.

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