Dynamics of an SIR model with vaccination dependent on past prevalence with high-order distributed delay John C. Eckalbar a, *, Walter L. Eckalbar b a California State University, Chico, CA 95929, United States b University of California, San Francisco, CA 94143, United States A R T I C L E I N F O Article history: Received 22 May 2014 Received in revised form 6 October 2014 Accepted 24 December 2014 Available online 30 December 2014 A B S T R A C T This paper investigates the dynamics of an SIR model of childhood vaccination under the assumption that the vaccination uptake rate depends on past values of disease prevalence. The delay kernel is a high order Erlang function, which allows no instantaneous feedback between prevalence at time t and vaccinations at time t. Multiple types of endemic equilibria are found, as are stable and unstable equilibria, periodic orbits, dependence on initial conditions, and apparent chaos. ã 2014 Elsevier Ireland Ltd. All rights reserved. 1. Introduction When a contagious disease outbreak is recognized, both individuals and public health authorities can take steps to mitigate its impact. For example, public health authorities can impose quarantines to limit contact between the infectious and suscepti- ble, expand treatment opportunities to shorten the duration of an individual's infectious period and thus limit that individual’s future transmission to others, and work to develop and/or distribute vaccinations. Susceptible individuals can limit their contacts, employ more stringent hygiene, and obtain vaccinations, while the infected can seek treatment. These reactions serve to endogenize some of the “fixed” parameters in a simple disease model, making the model more difficult to analyze, but also potentially more useful. Further, since some of the responses mentioned above may take time to materialize, it will often make good sense to incorporate delay. This also adds both realism and complexity to a model. In the present paper, we are interested in the dynamics of an SIR childhood vaccination model, where the fraction being vaccinated at time t, V(t), is determined by a function which attaches a weight to all prior levels of disease prevalence, I(t  t), t e (0, 1), where t is the delay duration. Higher past prevalence leads to greater perceived risk, which then leads to higher vaccination uptake. Study of this general issue (i.e., prevalence induced vaccination demand) has a history, which we briefly review: Geoffard and Philipson (1997), who coined the term “preva- lence demand” for the dependence of V on I, studied this question using a conventional SIR model with the assumption that V(t) depends upon I(t), i.e., with zero delay. Reluga et al. (2006) use a game theoretic approach to explore an individual’s vaccination demand. Each individual considers the average vaccination uptake rate for the whole population, since the average rate effects the individual's risk from not vaccinating. Vaccination by others than individual A reduce risk to A. As earlier researchers had found, this leads to a sub-optimal community outcome. The vaccination demand is then embedded into an SIR model much like that of Geoffard and Philipson, and this offers an interesting individual-choice background to Geoffard and Phili- pson. Our main interest in Reluga et al. (2006) is the incorporation of delay between prevalence, an individual’s perception of risk from not vaccinating, and the vaccine uptake rate, V(t) in our notation. Reluga, Bauch, and Galvani model this problem using distributed delay under the assumption of a “weak kernel,” to use Ruan’s (Ruan, 2006) term. These are “fading memory” models where the current value of I(t) is given maximum weight in determining V(t), and earlier values of I are assigned steadily diminishing influence. Of particular interest is that the model of Reluga, Bauch, and Galvani exhibits periodic orbits under some values for average delay duration. d’Onofrio et al. (2007) have an underlying SIR structure much like that of Geoffard and Philipson, except that V (using our notation) is explicitly linked to past levels of disease prevalence. Again we see a fading memory model, but the analysis is more detailed and complete. Various forms of vaccination demand functions are explored, and stability is studied in more detail than * Corresponding author. Tel.: +1 5303436791. E-mail address: jeckalbar@csuchico.edu (J.C. Eckalbar). http://dx.doi.org/10.1016/j.biosystems.2014.12.004 0303-2647/ ã 2014 Elsevier Ireland Ltd. All rights reserved. BioSystems 129 (2015) 50–65 Contents lists available at ScienceDirect BioSystems journal homepage: www.elsevier.com/locate/biosyst ems