Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases
Introduction
Vaccines have been a central factor in improving the standards of living, and the standards of health (Bloom and Cannings, 2004, Livi Bacci, 2005). Mass vaccination has allowed increasing levels of disease control worldwide, which has recently culminated in the elimination of indigenous measles in Finland (Peltola et al., 1997) and of poliomyelitis in many areas of the world (CDC, 2004). A further positive effect of some mass vaccinations is the reduction of the incidence of virus-related tumours (Chang et al., 1997). For example, the anti-HBV vaccines may be considered a preventive anti-tumour vaccine (Lollini et al., 2006).
Nonetheless the recent experience of developed countries shows instances of declines in vaccination coverage for several diseases. In some cases this is a consequences of rumours and adverse publicity against vaccines. For example, the decline in coverage of the Measles–Mumps–Rubella vaccine (MMR) recently observed in the UK (EURO SURVEILLANCE, 1998; CDR weekly, 1998, CDR weekly, 2002, CDR, 2004) has been explained by the role of adverse publicity about possible links between the vaccine, autism, and Crohn's disease (Wright and Polack, 2005). Similar facts have been found for Scotland (Friederichs et al., 2006). Another example is the decline in HBV coverage due to the “Thimerosal” case (Luman et al., 2004). In the future negative effects on coverage could derive from the argument, often raised by anti-vaccination movements, that vaccines could favour the onset of allergic diseases, a point that is still debated by the scientific literature (Koppen et al., 2004, Berndsen, 2004, Souza da Cunha, 2004, Schattner, 2005).
From a wider perspective, the phenomenon of coverage upswing has been common in the history of modern societies, often as a consequence of the tension between public health targets and individual freedom, for example between compulsory vaccination and conscientious or philosophical exemption (Salmon et al., 2006). Exemption against childhood immunization is a good example of this phenomenon: the tension stemming from the fear of damages due to the vaccine is emphasized when compulsory vaccination diminishes parents’ autonomy as regards the decisions on their children health. Such tension can in many cases prove beneficial for the society as a whole, in that it can push research toward better outcomes and eventually lead to an improvement in the safety of vaccines, which is a major target of public health. In the short-term however the main consequence of coverage decline, or of delayed vaccination, is always an increase in susceptibility (CDR, 2004), and thus in the risk of resurgence of diseases that were perhaps thought to be well controlled.
An increasing number of studies deal with the motivations underlying parents’ choices to vaccinate or not their children (Maayan-Metzger et al., 2005, Wright and Polack, 2005, Wroe et al., 2005, Friederichs et al., 2006). Besides the role played by conscientious exemption, these studies suggest the possibility of an “inverted U” relationship between education and income on one hand and propensity to vaccinate on the other. Two main remarks thus arise: first, the increasing well-being of modern societies could, in prospective terms, lead to increasing difficulties in maintaining high coverages. Second, an ultimate responsible in coverage decline is vaccination itself, i.e. the vaccination success in controlling diseases, which tends to encourage forms of “rational exemption”. The argument underlying rational exemption is simple. Consider for example the cases of poliomyelitis and measles control. In several countries the increasing coverage with MMR within the WHO Plan for global measles elimination has driven circulation of the disease to minimal levels or even zero incidence, i.e. a situation where the few observed cases can be traced back to immigration. As the incidence of the diseases continues to decline thanks to vaccination, families become increasingly concerned with the risks associated with vaccines (WHO, 2006). If families start perceiving that the chance of acquiring infection for their children is lower compared to the risk of experiencing damages from the vaccine (this is actually so for poliomyelitis), they could believe it rational not to vaccinate their children, particularly if they perceive that the rest of the population will, instead, vaccinate. This rationality is of course myopic since the decision to not vaccinate should be forward looking and taking into account also expectations of future resurgence of infection due to declining coverage, and not just the currently observed regime of low incidence and high coverage. Moreover, it is an example of “free riding” (Stiglitz, 2000), as by the way all types of exemptions (Salmon et al., 2006).
The widespread adoption of rational exemption would lead to a situation where at least a part of families relate their decision to vaccinate to the available information on the state of the disease, vaccinating more, and promptly, under circumstances of high social alarm due to the disease, and little (and later) otherwise. Such a behaviour always existed, as pointed out by Salmon et al.: “…vaccination rates fell, although uptake tended to increase when outbreaks occurred” (Salmon et al., 2006, p. 438).
Motivated by the above considerations, in this paper we study the dynamic implications of information dependent vaccination for SIR vaccine preventable childhood diseases. The underlying idea is that the vaccine coverage is the outcome of decisions, to vaccinate or not their children, which are partly based on the publicly available information on the state of the disease. There is a growing body of literature on information-dependent vaccination and vaccination choice, and their implications for the dynamics of SIR models for vaccine preventable diseases. Since the seminal paper by Fine and Clarkson (1986), Brito et al. (1991) have explored the conditions under which the free-rider problem can actually be overcome without compulsory vaccination, through the use of taxes and subsidies. Geoffard and Philipson (1997) use SIR-type models to explore the difficulty of eradicating a disease in presence of rational exemption, even if incentives such as subsidies are included. Bauch and Earn (2004) develop a game theoretical interpretation based on an SIR model of the rational exemption phenomenon, and show that under a purely voluntary policy, rational exemption makes eradication impossible. Reluga et al. (2006) expand Bauch and Earn (2004) by setting the game theoretic approach within the “viability” approach, and study the dynamical consequences of rational exemption under both current and delayed information. Bauch (2005) studies SIR-type differential equations with information dependence and analyze a model similar to the one in the present manuscript. Both the latter studies show the existence of oscillations and the impossibility of eradicating the disease due to rational exemption. A further related study is Auld (2003), who models the issue of vaccination choice within the framework of an agent-based model.
The present work aims to contribute to this literature by (a) incorporating information dependence not only on current disease levels but also on the history of disease in the population, (b) including the possibility of catch-up vaccination as a strategy for those who decided to not vaccinate during epochs of low perceived risk, (c) providing more general mathematical result: for instance all our stability results on the model with vaccination dependent on current information are shown to hold globally.
More specifically, we consider some SIR models in which the vaccination coverage of newborn is the sum of two components: a steady one, given by the fraction of parents who, while taking the decision to immunize their children are not affected by the state of information on the disease, and an “information-dependent” one, which is taken to be an increasing function of the perceived risk (or the social alarm) due to the disease, as summarized by some information variable depending on the current and past state of the disease. We feel that our assumptions on coverage capture well the idea of rational exemption.
Our results are as follows. First, if the information function only summarizes the current state of the disease, then unless the steady component is above the elimination threshold, a unique endemic state will exist and is globally asymptotically stable (GAS). This result continues to hold even when we allow in the model delayed catch-up vaccination of older individuals as a “recuperation strategy” for families that did not vaccinate their children during epochs of low perceived risk. Second, if the information function also summarizes the past history of the disease according to an exponentially fading memory then we can also observe the emergence of stable oscillations through Hopf bifurcation of the endemic state, i.e. delayed state-dependent vaccination can be a source of steady oscillations for common childhood diseases. Analysis of selected subcases and numerical simulations gives insight on the conditions under which stable oscillations are more likely, and on the amplitude of the inter-epidemic period that would result as a consequence of the interaction between the factors traditionally included in SIR models, average age at infection, vaccination, and demographics, on the one hand, and those due to social behaviour by individuals on the other hand. Numerical simulations also suggest that the involved limit cycles are globally stable.
The paper is organized as follows. In Section 2 we introduce a general model encompassing the various special models considered. Section 3 reports some results on equilibria and local stability of the general model. Sections 4 and 5 report, respectively, the stability analysis of the undelayed and of the delayed case. Examples, numerical results, and a discussion of the implications of information dependent vaccination for the period of oscillations, are reported in Sections 6 and 7. Concluding remarks follow.
Section snippets
A family of models for information-related vaccinating behaviour
We consider the following family of SIR models for a nonfatal disease in a constant homogeneously mixing population, with state-dependent vaccination coverage:where X, Y, Z, V are functions of time, respectively denoting the number of susceptibles, infectious (and infectives), immune and vaccinated individuals at time t. Moreover, denotes the birth and death rate, which are assumed identical, the rate of recovery from
Properties of the general model
Let us consider initially the unlagged model (8) with constant transmission rate. This model differs from the standard text-book SIR model with vaccination at birth (Capasso, 1993) by the appearance of the state-dependent component of coverage . Obviously if the maximal coverage is below the critical coverage (or elimination threshold) where is the corresponding basic reproduction number, we cannot expect outcomes different from the standard one, i.e. the
Stability analysis of the endemic equilibrium in the unlagged case
We now focus on the stability of the endemic state in the no delay case (8) under the assumption of a constant transmission rate . We assume that is differentiable. The following general result holds (proof postponed to the Appendix A.2): Proposition 8 Let be constant, and . Then the unique endemic state EE of system (8): ifis LAS; if, in particular,is GAS in the positively invariant set:
In the undelayed system limit
Onset of stable oscillations under exponentially fading memories
In this section we prove that when the actual coverage also depends on past information then stable oscillations may appear even under the simplest pattern of delay, i.e. the exponentially fading memory . To make computations simpler we assume in (3), obtaining from (9), under the assumption of constant , the following three-dimensional system:As we have seen in Proposition 4, under condition , system (29) has the
Examples and numerical simulations
In this section we report some numerical simulation of system (29) under three noteworthy functional forms of the function . We focus on the relation between patterns of information delay and the reactivity of information-dependent vaccination in determining the onset of oscillations. Subsequently we shall discuss in greater detail the implications for the period of inter-epidemic oscillations.
Most our computations will be based on the following benchmark parameter constellation roughly
The period of oscillations
The previous simulation results suggest the possibility that information delays in vaccination behaviour might yield a wide range of outcomes in terms of the period of the ensuing long-term sustained oscillation, compared to the basic (i.e. nonperiodically forced) SIR model. In the SIR model without vaccination only damped oscillations occur and are generated by the inter-play of the epidemic mechanism, i.e. exhaustion of susceptibles, and the demographic one, i.e. regeneration of susceptibles
Discussion
This work has investigated the implications of information-dependent vaccination for the dynamics and control of SIR childhood vaccine preventable infectious diseases. Here information-dependent vaccination is used to model the phenomena of rational exemption to vaccination and social alarm as a consequence of the spread of public information on the disease, by assuming that a component of the overall vaccination coverage is positively correlated with the available information on the disease.
Acknowledgment
The authors thank Marta Ciofi degli Atti for useful suggestions, and two anonymous referees of this Journal whose suggestions greatly improved the quality and the exposition of the paper. Usual disclaimers apply.
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