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Infectious disease, heterogeneous populations and public healthcare: the role of simple models SIAM CSE 2009 K.A. Jane White Centre for Mathematical Biology University of Bath United Kingdom Presentation overview • • • • • Motivating the use of simple models Infectious disease modelling Case study 1: treatment of infections Case study 2: prevention of infections Concluding remarks Co-workers Vicki Brown, Centre for Mathematical Biology Matt Dorey, Health Protection Agency Dushyant Mital, Milton Keynes General Hospital Steven White, Centre for Ecology & Hydrology What categorises a simple model? • Captures key components of real system • Can be used to address specific questions • Model lends itself to analytical techniques: ODEs, PDEs, integral equations, integrodifference/differential equations; nonlinear, low dimension. • Equivalent to, derived from or motivated by, higher dimensional systems more directly linked to data Aside: equivalence of models Modelling spread of insects into discrete spatial locations e.g. pests in agriculture Coupled map lattice N i ,t 1 1 f i ( N i ,t ) jN N f j ( N j ,t ) Integro-difference system Nt 1 x k x y f ( Nt y )dy k x taken as an indicator function Pattern formation Speed of invasion Infectious disease modelling Public Healthcare Treatment Effective Affordable Available Epidemiology Prevention Intervention Education Social Contact Dealing with social contact structure I: The simple modelling approach Compartmental model Population split according to infection status: Susceptible (S), Infected (I), etc. Mass action assumption Rate of infection (incidence) bilinearly dependent on S & I Generally unrealistic for structured contacts. Nonlinear incidence S , I S p 1I q Dealing with social contact structure II: Linking nonlinear incidence to infection on networks Irregular networks e.g. Scale free Good to represent sexual contact network From Andrea Galeotti University of Essex Infecteds Per capita infection rate Infection on scale free network Time S , I S p=1.05; q=0.71 = 0.00016 p 1 q I Time Data from simulation on scale free network Fitted curve (glm) Case study 1: Treatment of Infections Previous work White et al. (2005) JID Vicious and virtuous circles in the dynamics of infectious disease and the provision of healthcare Modelling included: Age structure Sex Activity classes Model structure: Coupled PDEs involving integrals. Analysed using simulations Model outcome: Regions of containment, outbreak and bistability. The simple version p Susceptible (1-p) Asymptomatic Infected Symptomatic Infected s Treated I A z I A N q Collapse to a 2-D system 1I 2I 1I Tmax s s I ,Tmax Tmax 2I otherwise Infection incidence Hysteresis effect Tmax Maximum healthcare provision Simple model can quantify basins of attraction in bistable region Containment and outbreak requirements N I N=Tmax II Tmax Outbreak Bistability Simple model can quantify transitions between outbreak and containment of infection Containment Common Infections Gonorrhoea Chlamydia Symptoms appear 1 week after infection Treatment effective after 1 day Symptoms appear 2 weeks after infection Treatment effective after 1 week N N N=Tmax N=Tmax Tmax Outbreak Bistability Tmax Containment Case study II: Prevention of Infection HPV (Human papillomaviruses) vaccination HPV-16 and HPV-18 causal factor in cervical cancer 80% of women infected with HPV at some time Recent vaccination strategy in England vaccinate pre-teenage girls (3 doses, £240) catch up for 16-18 year old girls. http://images.parenthood.com/hpv-vaccine.jpg The Simple Modelling Approach • Ignore age and optimal control – Understand behaviour of key parameter groupings • Ignore age, include optimal control – Understand interaction of control with behaviours of first model • Include both age and optimal control – Most realistic system for given problem i ij zi Ij Nj Jj , i j q q ; dq 0 d I. Ignore age and optimal control p=Proportion vaccinated Infection eradicated if R0e R0 1 f p 1 1 f p 1 R0 Waning immunity Onset sexual activity Eradication more likely, for fixed p, if •Vaccination protection is long lasting •Slower rate of becoming sexually active Females p p h Asymmetric vaccination has small impact on infection prevalence between sexes Important to consider impact of sexual debut p Males Optimal Control II. Ignore age, include optimal control Time In cases where constant control gives persistence of infection, optimal control can eradicate infection. III. Still to do! Time Concluding remarks • Simple models equivalent to high dimensional systems provide useful analytical techniques • Simple models parameterised from high dimensional systems can be used to analyse more complex problems • Building up complexity of model allows systematic exploration of interactions