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, integro- difference/differential equations; nonlinear, low dimension. Equivalent to, derived from or motivated by, higher dimensional systems more directly linked to data

Aside: equivalence of models Coupled map lattice Modelling spread of insects into discrete spatial locations e.g. pests in agriculture Integro-difference system taken as an indicator function Pattern formation Speed of invasion

Infectious disease modelling Epidemiology Social Contact Public Healthcare Prevention Intervention Education Treatment Effective Affordable Available

Dealing with social contact structure I: The simple modelling approach Mass action assumption Rate of infection (incidence) bilinearly dependent on S & I Generally unrealistic for structured contacts. Compartmental model Population split according to infection status: Susceptible (S), Infected (I), etc. Nonlinear incidence

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

Infection on scale free network Data from simulation on scale free network Fitted curve (glm) p=1.05; q=0.71  = Per capita infection rate Time Infecteds

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 Susceptible Asymptomatic Infected Symptomatic Infected Treated  p (1-p)    Collapse to a 2-D system

Hysteresis effect Simple model can quantify basins of attraction in bistable region Maximum healthcare provision Infection incidence T max

Containment and outbreak requirements N=T max I II N T max OutbreakBistabilityContainment Simple model can quantify transitions between outbreak and containment of infection

Common Infections Gonorrhoea Symptoms appear 1 week after infection Treatment effective after 1 day Chlamydia Symptoms appear 2 weeks after infection Treatment effective after 1 week N T max N=T max OutbreakBistabilityContainment T max N N=T max

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 year old girls.

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. Ignore age and optimal control Infection eradicated if Eradication more likely, for fixed p, if Vaccination protection is long lasting Slower rate of becoming sexually active Waning immunity Onset sexual activity p=Proportion vaccinated

Females Males   p p p  Asymmetric vaccination has small impact on infection prevalence between sexes Important to consider impact of sexual debut

II. Ignore age, include optimal control In cases where constant control gives persistence of infection, optimal control can eradicate infection. Time Optimal Control III. Still to do!

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