Fatal SIR diseases and rational exemption to vaccination

Abstract

A challenge to disease control in modern societies is the spread of rational exemption to vaccination as a consequence of the rational comparison between the steadily declining risk of infection and the risk of side effects from the vaccine. Here, we consider rational exemption in an susceptible-infectious-removed (SIR) model with information-dependent vaccination where individuals use information on the disease's mortality as their information set. Using suitable assumptions on the dynamics of the population, we show the dynamic implications of the interaction between rational exemption, current and delayed information and the risk of death by the disease. In particular, we illustrate the onset of the long cycles caused by rational exemption when vaccination decisions are based on delayed informations.

Vaccine demand driven by vaccine side effects: Dynamic implications for SIR diseases

“…More precisely, as far as p(C) is not too reactive to changing perceived risks of VSE the equilibrium is stable independently of the age mechanism through which VSE arise after vaccination. This agrees with our findings on different models of vaccinating behaviour [11,12,13]. However, as argued in previous work on prevalence-related vaccine uptake [23,11,13], we expect that in realistic circumstances the index M will embed not only the current but also some information on the past history of VSE.…”

“…This agrees with our findings on different models of vaccinating behaviour [11,12,13]. However, as argued in previous work on prevalence-related vaccine uptake [23,11,13], we expect that in realistic circumstances the index M will embed not only the current but also some information on the past history of VSE. This case was important because, as shown in [23,11,13], the inclusion of delayed information was necessary to generate oscillations in simple SIR models with prevalencerelated vaccination behaviour.…”

“…As regards the vaccination sub-system we have demonstrated, both in general and for selected epidemiologically relevant sub-cases, the conditions under which the dynamics of vaccine uptake lands on a globally stable steady state, rather than in stable oscillations, of vaccine uptake. Unlike previous results on epidemiological models with prevalence-driven vaccination [23,11,13], where information lags were necessary to trigger oscillations, in the present case steady oscillations can arise also when current information only is used to evaluate the perceived risk (of VSE). The explanation is that the appropriate modelling of vaccine side effects required the introduction of an age mechanism, e.g.…”

“…This confirms the destabilising role played by the information delay. It is interesting to note from (27) that, consistently with the findings in [11,12,13], the onset of oscillations also requires, besides the above described delay pattern, a high sensitiveness of the vaccine uptake to changing perceived risk at equilibrium. If this occurs then there is a whole window (D 2 = 1/a 2 , D 1 = 1/a 1 ) of values of the average information delay D = 1/a, yielding steady oscillations.…”

“…Following ideas developed in d'Onofrio et al [11,13] we consider the following family of SIR models for a non-fatal disease in a constant homogeneously mixing population, with state-dependent vaccination coverage:…”