Complex Dynamical Behaviour in an Epidemic Model with Control

Bull Math Biol DOI 10.1007/s11538-016-0217-6 ORIGINAL ARTICLE Complex Dynamical Behaviour in an Epidemic Model with Control Martin Vyska1 · Christopher Gilligan1 Received: 12 January 2016 / Accepted: 29 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We analyse the dynamical behaviour of a simple, widely used model that integrates epidemiological dynamics with disease control and economic constraint on the control resources. We consider both the deterministic model and its stochastic counterpart. Despite its simplicity, the model exhibits mathematically rich dynamics, including multiple stable fixed points and stable limit cycles arising from global bifurcations. We show that the existence of the limit cycles in the deterministic model has important consequences in modelling the range of potential effects the control can have. The stochastic effects further interact with the deterministic dynamical structure by facilitating transitions between different attractors of the system. The interaction is important for the predictive power of the model and in using the model to optimize allocation when resources for control are limited. We conclude that when studying models with constrained control, special care should be given to the dynamical behaviour of the system and its interplay with stochastic effects. Keywords Disease dynamics modelling · Control of epidemics · Bifurcations in epidemic models 1 Introduction There is increasing interest in the integration of epidemiological models of control with economic considerations (Klein et al. 2007; Geoffard and Philipson 1996). Recently, researchers have focused on models of control with economical constraints on the control resources and used optimal control theory to provide insights into opti- B Martin Vyska 1 University of Cambridge, Cambridge, United Kingdom 123 M. Vyska, C. Gilligan mal resource allocation strategies. These models range from allocation of treatment resources (Forster and Gilligan 2007; Goldman and Lightwood 2002) to problems of how to divide resources between treatment and detection efforts (Ndeffo Mbah and Gilligan 2010). However, in the conventional analysis the exact dynamics of the epidemiological models with constrained control have not been investigated in detail. Here by constrained control we refer to control which can be applied to some but not necessarily all the individuals in a population due to limited resources. In this paper, we select a simple, but widely used (Rowthorn et al. 2009; Ndeffo Mbah and Gilligan 2011) epidemic model with constrained control and we examine its deterministic dynamical behaviour. We show that despite its simplicity, the model exhibits mathematically rich behaviour including stable limit cycles and their global bifurcations. The presence of limit cycles in dynamical systems has long been of interest in mathematical biosciences, particularly in ecology (Kaung and Freedman 1988; Hastings 2001; Toupo and Strogatz 2015) and epidemiology (Hethcote and Levin 1989; Wang and Ruan 2004; Jin et al. 2007). We demonstrate that the presence of limit cycles has important consequences for modelling the impacts of control. We also show that in some parts of parameter space the model exhibits counter-intuitive behaviour in which lower initial disease prevalence leads to a higher-prevalence endemic equilibrium. Whenever possible, we provide analytical conditions on the parameters of the model that give rise to the particular dynamics. We then examine the sensitivity of the dynamical behaviour when stochasticity is introduced to the model to allow for inherent variability of the infection and recovery processes. We do this by using the Gillespie construction (Gillespie 1976) to model every event in the system as an exponential random process with rates given by the deterministic model. Thus, the stochastic effects we introduce are demographic in nature. We demonstrate that the existence of the limit cycles in the deterministic version of the model strongly impacts the behaviour of the stochastic version of the model. The stochastic fluctuations can cause transitions between different attractors of the system and in some cases can lead to extinction of the pathogen by perturbing the system onto a limit cycle which passes close to the line of zero prevalence in the phase space. Similar transitions between different attractors of the dynamical system have been previously studied in systems with seasonal forcing (Keeling et al. 2001). Our work also demonstrates that economical constraints on control in epidemiological models can lead to the existence of weakly stable attractors and complex bifurcation dynamics. These in turn cause qualitative differences between the behaviour of the deterministic model and its stochastic counterpart. This interaction between the bifurcation dynamics and stochastic effects is important both for the predictive power of the model and in using the model to optimize resource allocation, since emergence of the limit cycles in the deterministic model causes rapid changes in the probability of eradication in the stochastic model. We conclude that when interpreting model predictions and especially when studying models with constrained control, special care should be given to the dynamical behaviour of the system and its interplay with the stochastic effects. 123 Complex Dynamical Behaviour in an Epidemic Model with… Fig. 1 The transition structure of the SIRS compartmental model. All the rates are per host. β is the transmission rate and therefore β I is the rate at which susceptible hosts get infected. μ is the rate of recovery and transition to the recovered class. ν is the rate at which immunity is lost and hosts rejoin the susceptible class. Finally, σ is both the birth and death rate, assumed to be equal 2 Model Description A wide range of models are used for infectious disease dynamics. Of these, many are formulated as compartmental models (Kermack and McKendrick 1927; May and Anderson 1991). The compartments represent groups of hosts who share an infection status, such as being infectious or susceptible. Considering all the hosts within one compartment as equivalent is a simplifying assumption that the transition rates between the compartments are constant that is the underlying stochastic process is Markovian. In this paper, we consider a compartmental SIRS-type model with the model structure as in Fig. 1. This describes a situation in which the time for which the hosts stay in the infected class after infection is exponentially distributed with mean 1/μ. After recovery, the recovered hosts have temporary immunity and cannot be immediately reinfected. This immunity lasts for an exponentially distributed time period with mean 1/ν after which the hosts rejoin the susceptible class. This model structure with temporary immunity is appropriate for diseases such as Malaria (Aron 1988; (...truncated)