Why do we model infectious diseases? Following Heesterbeek & Roberts (1995) 1. Gain insight into mechanisms influencing disease spread, and link Introduction to infectious disease modelling individual scale ‘clinical’ knowledge with population-scale patterns. 2. Focus thinking: model formulation forces clear statement of assumptions, hypotheses. 3. Derive new insights and hypotheses from mathematical analysis or simulation. Jamie Lloyd-Smith 4. Establish relative importance of different processes and parameters, to focus research or management effort. Center for Infectious Disease Dynamics 5. Thought experiments and “what if” questions, since real experiments Pennsylvania State University are often logistically or ethically impossible. 6. Explore management options. with thanks to Ottar Bjornstad for sharing some slides… Note the absence of predicting future trends. Models are highly simplified representations of very complex systems, and parameter values are difficult to estimate. � quantitative predictions are virtually impossible. Epidemic models: the role of data The SEIR framework for microparasite dynamics Why work with data? Basic aim is to describe real patterns, solve real problems. Test assumptions! S E I R Get more attention for your work � jobs, fame, fortune, etc � influence public health policy Susceptible: naïve individuals, susceptible to disease Challenges of working with data Exposed: infected by parasite but not yet infectious Hard to get good data sets. The real world is messy! And sometimes hard to understand. Statistical methods for non-linear models can be complicated. Infectious: able to transmit parasite to others What about pure theory? Removed: immune (or dead) individuals that don’t contribute to Valuable for clarifying concepts, developing methods, integrating ideas. further transmission (My opinion) The world (and Africa) needs a few brilliant theorists, and many strong applied modellers. The SEIR framework for microparasite dynamics The SEIR framework for microparasite dynamics λ ν γ λ ν γ S E I R S E I R β λ dS SI “Force of infection” = − = β I under density-dependent transmission dt N = β I/N under frequency-dependent transmission β dE SI Ordinary differential equations = − ν E ν Rate of progression to infectious state dt N are just one approach to = 1/latent period dI modelling SEIR systems. = ν − γ E I γ Rate of recovery dt = 1/infectious period dR = γ I dt 1
Disease with environmental reservoir (e.g. anthrax) S E I S I R S I S I R SEI SIRS SIS Adapt model framework to disease biology and to your problem! X No need to restrict to SEIR categories, if biology suggests otherwise. Death of pathogen in e.g. leptospirosis has chronic shedding state � SICR environment S I R Vector-borne disease C Humans S H I H R H Depending on time-scale of disease process (and your questions), add host demographic processes. birth Vectors S V I V births S I R deaths death TB treatment model TB treatment model TB treatment model TB treatment model Residence times RN ss − DOTS Susc. Susc. Susc. Latent Latent Latent Active TB Active TB Active TB ν E ss − ss − ss − ss − ss − non-DOTS ss − ss − ss − ss − Slow Slow Slow Slow How long does an individual spend in the E compartment? det det det det “Detectable” Ignoring further input from new infections: Susc Susc Susc Susc cases ss+ ss+ ss+ ss+ dE ss+ DOTS Fast Fast Fast Fast ss+ ss+ ss+ ss+ = − ν ⇒ = − ν t Under treatment Under treatment E E ( t ) E ( 0 ) e det det det det dt For a constant per capita rate of leaving compartment, the ss+ non-DOTS ss+ non-DOTS ss+ non-DOTS residence time in the compartment is exponentially distributed. Tx Defaulters Part. Part. Part. Completers Rec Rec Rec Rec ODE model Data from rec rec rec SARS Recovered Recovered Partially recovered t t Data from Residence times Basic reproductive number, R 0 SARS How to make the model fit the data better? Expected number of cases caused by a typical infectious individual • “Box-car model” is one modelling trick t in a susceptible population. λ ν /n ν /n ν /n … S E 1 E 2 E n I R 0 ≤ 1 R 0 > 1 disease dies out disease can invade Divide compartment into n sub-compartments, each with constant leaving rate of ν / n . n =40 Outbreak dynamics Disease control Residence time is now gamma- distributed, with same mean and • probability of fade-out • threshold targets flexible variance depending on the • epidemic growth rate • vaccination levels n =10 number of sub-compartments. n =3 n =1 t See Wearing et al (2005) PLoS Med 2: e174 2
Calculating R 0 – Intuitive approach Effective reproductive number Expected number of cases caused by a typical infectious individual in a population that is not wholly susceptible. R 0 = Per capita rate × Duration of R effective = R 0 × S/N of infecting others infectiousness … in a completely susceptible population. Endemic disease: At equilibrium R eff = 1, so that S */ N = 1/ R 0 Under frequency-dependent transmission: Epidemic disease: R eff changes as epidemic progresses, as susceptible pool is depleted. Rate of infecting others = β S/N No. new cases = β in wholly susceptible pop’n Note: Sometimes “effective reproductive number” is Duration of infectiousness = 1/recovery rate R eff > 1 R eff < 1 used to describe = 1/ γ transmission in the presence of disease � R 0 = β / γ Time control measures. This is also called R control . What does R 0 tell you? R effective and herd immunity • Epidemic threshold R effective = R 0 × S/N NOTE: not every epidemic threshold parameter is R 0 ! • Probability of successful invasion If a sufficiently high proportion of the population is immune, then • Initial rate of epidemic growth R effective will be below 1 and the disease cannot circulate. • Prevalence at peak of epidemic • Final size of epidemic (or the proportion of susceptibles The remaining susceptibles are protected by herd immunity . remaining after a simple epidemic) • Mean age of infection for endemic infection The critical proportion of the population that needs to be immune is • Critical vaccination threshold for eradication determined by a simple calculation: • Threshold values for other control measures • For R eff < 1, we need S / N < 1/ R 0 • Therefore we need a proportion 1-1/ R 0 to be immune. The basic framework for macroparasite dynamics The basic framework for macroparasite dynamics dM = β − τ − μ + μ For macroparasites the intensity of infection matters! d L ( t ) ( ) M 1 1 1 dt Basic model for a directly-transmitted macroparasite: dL = λ − τ − μ − β s d NM ( t ) L NL 2 2 2 dt M L β infection rate μ death rate of hosts μ 1 death death death rate of adult worms within hosts μ 2 death rate of larvae in environment State variables proportion of ingested larvae that survive to adulthood d 1 N ( t ) = Size of host population proportion of eggs shed that survive to become infective larvae d 2 τ 1 time delay for maturation to reproductive maturity M ( t ) = Mean number of sexually mature worms in host population τ 2 time delay for maturation from egg to infective larva L ( t ) = Number of infective larvae in the habitat proportion of offspring that are female s Further complexities: parasite aggregation within hosts and density-dependent effects on parasite reproduction. 3
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