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Modelling the seasonality of respiratory syncytial virus in young children

Authors:

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 differential 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 activity 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 �(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: �(t) = �0[1 + �1 sin( 2�t 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 �0 and �1, 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.
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 =µβSI 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< β11), 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 =µβAS1(I1+I2)ηS1+νR1(6)
dE1
dt =βAS1(I1+I2)δE1ηE1(7)
dI1
dt =δE1ηI1γI1(8)
dR1
dt =γI1ηR1νR1(9)
dS2
dt =ηS1βBS2(I1+I2)ηS2+ν R2(10)
dE2
dt =ηE1+βBS2(I1+I2)δE2ηE2(11)
dI2
dt =ηI1+δE2ηI2γI2(12)
dR2
dt =ηR1+γI2ηR2νR2.(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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... , A. B.,Mercer, G. N., Glass, K., and Moore, H. C. (2013). Modelling the seasonality of respiratory syncytial virus in young children. ...
Thesis
Respiratory syncytial virus (RSV) causes respiratory tract infections in infants and young children. Almost all children experience an RSV infection within the first two years of life, and while mortality due to RSV infection is low in developed countries, the virus presents a significant burden in Australia and internationally. In temperate regions, RSV displays strong seasonal patterns. In Perth, Western Australia, RSV detections show a distinct biennial cycle, and similar patterns have been observed in other temperate locations. While there is no licensed vaccine for RSV, there are several candidates in clinical trials. Understanding the seasonal patterns of RSV, and developing mathematical models that capture key transmission characteristics, can assist with planning the future rollout of an RSV vaccine. This thesis focusses on three themes: age structure and immunity; seasonality and climate; and vaccination. For the first theme, I present age-structured compartmental mathematical models with waning immunity and seasonal forcing. I fit these models to RSV data for Perth and explore the parameter space and bifurcation structures. The models help explain the different patterns in RSV detections observed globally. In particular, both the seasonality and immunity parameters must exceed certain thresholds for the model to produce biennial patterns, which aligns with observed data. Further, I identify a ‘window’ of birth rate parameters that produces biennial patterns, showing that RSV seasonality may not be only driven by weather and climatic factors as was previously thought. The second research theme involves a time series analysis of both RSV and bronchiolitis data, as approximately 70\% of bronchiolitis hospitalisations are linked to RSV infection. First, I identify a clear shift in seasonality for both RSV and bronchiolitis, from the temperate to tropical regions of Western Australia. I then apply a mathematical time series analysis method, complex demodulation, to assess the validity of using bronchiolitis hospitalisations as a proxy for RSV cases. I find bronchiolitis and RSV are similar in terms of timing, but that epidemic magnitudes differ. To address the third research theme, I adapt the compartmental model to incorporate a finer age structure, contact patterns and naturally-derived maternal immunity, to assess the potential impact of a maternal vaccination strategy for RSV in Perth. I find that the introduction of a maternal vaccine is unlikely to alter the regular biennial RSV pattern, but that the vaccine would be effective in reducing hospitalisations due to RSV in children younger than six months of age. This thesis adopts both mathematical modelling and data analysis approaches to improve our understanding of RSV dynamics. Developing mathematical models for RSV transmission in the Australian context allows a better understanding of the relative importance of age cohorts, immunity, climatic factors, and demography, in driving different RSV epidemic patterns. Further, data analysis shows the extent to which bronchiolitis hospitalisations are representative of RSV detections, and that different approaches to interventions must be considered in temperate versus tropical Western Australia. These findings will be instrumental in planning an effective vaccine rollout strategy for Western Australia.
... The disease states are presented as proportions of a population, such that S + E + I + R = N = 1. This model was first presented in [24] and the relevant equations are reproduced here as Eqs. 5-9. ...
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Respiratory Syncytial Virus (RSV) has long been recognized as the single most important virus causing acute severe respiratory-tract infections with symptoms ranging from rhinitis to bronchitis in children who may require hospitalization. Outbreaks of RSV occur every year and all children become infected within the first two years of life, and that overloads hospital casualty services. The transmission dynamics of RSV are strongly seasonal. Epidemics occur each winter in temperate climates and often coincide with the seasonal rainfall in tropical climates.In this paper we develop a non-standard numerical scheme for a SIRS seasonal epidemiological model for RSV transmission. This non-standard numerical scheme preserves the positivity of the continuous model and is applied to approximate the solution using different sizes of step.Finally this method is compared with some well-known explicit methods and simulations with data from Gambia and Finland are carried out.
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