Modeling epidemics caused by respiratory syncytial virus (RSV)
Introduction
Respiratory syncytial virus (RSV) is the most common cause of acute lower respiratory tract infection in children. Clinical features of RSV infection range from those of a common cold to bronchiolitis and severe pneumonia. Cases of RSV disease occur in seasonal epidemics. The majority of hospitalized cases are aged under 6 months, but the infection can also be a cause of mortality in the elderly. In this paper we develop a model of RSV transmission, and apply it to the interpretation of series of monthly case reports of RSV disease from developed and developing countries. The mechanisms underlying the different patterns of seasonal epidemics observed in different countries are not well understood; our aim in this paper is to examine the effect of population dynamics of transmission on the interpretation of the role of climatic and social factors in driving these epidemics. Furthermore, RSV vaccines are under development; if suitable models can be developed they can be used to evaluate the likely impact of vaccination programmes.
In Section 2 we outline the properties of RSV. In Section 3 we describe extensions to the SIRS model which includes features specific to RSV transmission. Suitable values for some of the model parameters could be obtained from results given in the medical literature, but some parameters, especially the mean b0 of the transmission parameter β and its seasonal amplitude, had to be estimated by fitting the models to monthly case reports of RSV disease (Section 4). Separately we investigated the effects of different assumptions about the duration of immunity on model behavior (Section 5), and the influence of stochastic variation in some parameter values. In Section 6 we give the results of several simulations when the seasonal variation of the weather is correlated to the annual variation of the transmission parameter. Threshold conditions for our models are given in Section 7. Some final discussions are given in Section 8.
Section snippets
Properties of RSV
RSV is the most common cause of acute lower respiratory tract infection in children worldwide [1], [2], [3]. A striking feature of RSV infection is its seasonality: in temperate climates, most of the RSV-associated disease episodes occur in the cold season, whereas in the tropics, most appears to occur in the wet season [3]. RSV appears to be a predominantly human pathogen: it has been found in some animals, but these are not believed to play a role in the transmission to humans [4]. The
Models
In our models the population is divided into distinct groups (of susceptible, infected, or immune individuals) and we use deterministic continuous transitions between the states. So the overall setting is the one treated in-depth in the literature, see, e.g., [27]. We assume homogeneous mixing in the population and do not model spatial spread of epidemics. For RSV almost simultaneous outbreaks of RSV epidemics with different strains of the virus have been reported throughout a whole area (see,
Parameter estimates
For the simulations and parameter estimation we used Matlab. The functions describing the vector fields and their Jacobians were generated from a representation of the dynamical system in the computer algebra system Maple by a tool written by Goller [35] under the supervision of the first author. The integrators used were the standard Matlab functions ode23t and ode15s.
The optimal parameter fits were obtained in most cases by non-linear least squares using functions provided in the `Matlab
Results of further simulations
In fitting the models we held the parameters μ,ν,σ,γ, and ξ at fixed values. For some of the parameters, especially γ, present knowledge is only sufficient to determine a certain interval for its values, so we performed additional simulations in order to estimate the sensitivity of the simulations to changes in this parameter (Fig. 5) and to changes in b0 and b1 (Fig. 6, Fig. 7).
As initial values for the simulations each group was given the same size and we used 100 years of simulated time as a
Correlating the transmission parameter with weather data
In the work of several authors, e.g., [3], [38], attempts to correlate weather conditions with the epidemics of RSV have been made. This work is motivated by the fact that cyclic changes in the social behavior of school children (the beginning of school terms) do not seem to be linked to RSV epidemics [22]. For example, in Gambia, outbreaks occurred before the beginning of the school year [37]. It is possible that other social factors such as indoor crowding during the rainy season might play a
Thresholds for the models
The SIRS model with simple vital dynamics (as in Section 3.2, but with the transmission parameter β(t) assumed to be a constant, β) has a unique equilibrium if the basic reproduction rate β/(μ+ν) exceeds one [41]. If the birth and recovery rates μ and ν take the values given in Table 2, then β must exceed 36 for the infection to persist in the community. This value is about half the value estimated from fitting the model to the case reports series (Table 2).
It can be shown (by solving the
Discussions
We found that RSV epidemics can be modeled surprisingly well by relatively simple ODE models. These models, which can reproduce the reported regular annual outbreaks of epidemics seen in many places, can also reproduce the alternating cycles of large and smaller annual epidemics observed in some countries, with a change in the value of a single parameter (the relative seasonal amplitude of the transmission parameter). Fitted values of the relative amplitude were greater in places with
Acknowledgements
We are grateful to K. Dietz for helpful remarks on a draft of this paper.
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Partially supported by Deutsche Forschungsgemeinschaft under grants We 1945/1-1 and Ku 966/6-1. Major parts of research done while working at the Symbolic Computation Group, Department of Computer Science, University of Tübingen, Germany.