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A VERY IMPORTANT CONCEPT • Disease epidemiology is first and foremost a population biology problem • Important proponents: Anderson, May, Ewald, Day, Grenfell S.I.R. MODELS • Susceptible, Infected, Recovered • Assign each individual to one of the mutually exclusive categories and then apply equations to describe rates of transition to the different categories TWO KINDS OF TRANSMISSION • Direct - through physical contact or through production of free-living stages (e.g. measles) • Indirect - through one or more obligatory intermediate hosts (e.g. malaria) DIRECT TRANSMISSION BETWEEN HOSTS • • • • ASSUMPTIONS: Hosts do NOT recover Hosts do NOT die Host population is stable (static) DIRECT TRANSMISSION BETWEEN HOSTS • X = density of susceptibles • Y = density of infecteds • N=X+Y = transmission coefficient TRANSMISSION COEFFICIENT • Product of 2 constants: – C = contact rate – T = probability of transmission when a susceptible contacts an infected (0,1) DIRECT TRANSMISSION BETWEEN HOSTS • X = density of susceptibles • Y = density of infecteds • N=X+Y = transmission coefficient dX = - X bY dt dY Y = Xb N dt N = X bY DISEASE DYNAMICS TWO STABLE EQUILIBRIA • When Y = 0 • When Y = N BASIC RATE OF INFECTION • R0 Def’n - the rate of acquisition of susceptible individuals that get converted to infecteds (2º infections) • If R0 > 1 disease is growing MEASLES • Virus-caused disease • Transmission is primarily by large respiratory droplets. • The disease typically consists of a high fever, cough, runny nose and a generalized maculopapular rash. • Infants under one year of age have the highest case fatality rates reaching as high as 20% in epidemic situations. INCIDENCE TRENDS MEASLES ASSUMPTIONS • Human population is stable. Deaths immediately lead to birth of a healthy susceptible individual. • No age structure. • Death can arise from natural causes (b) or from measles ( ) • Immunity once gained is never lost • No incubation time BASIC MEASLES MODEL • X = number of susceptibles • Y = number of infecteds • Z = number of immunes • N=X+Y+Z β = transmission coefficient b = death due to natural causes α= death due to disease = immunity gain rate BASIC MEASLES SIR MODEL d X /dt = b N + Y - β XY -bX dY /dt = β XY -( + b + ) Y dZ /dt = Y - bZ WHAT IS R0 R0 = β N/ ( + b + ) TWO STABLE EQUILIBRIA If β N/ ( + b + ) > 1 the Y* adopts non-zero values If β N/ ( + b + ) < 1 then X* = 1, Y* = 0 and Z* = 0 MINIMUM HOST POP? Nt > ( + b + )/ β for R > 1 ADD REALISM • Add an incubation class Q that includes individuals that are infected but not yet infectious (12 day period) • Let vary according to season PHASE PLANE INFECTEDS WITH AGE STRUCTURE INFECTEDS THE KEYS • The susceptible class is exhausted by infection and replenished by birth of new susceptibles. • Latency period accentuates oscillatory behavior. • Seasonally changing β values accentuates oscillatory behavior. DYNAMICS A CRITICAL ASSUMPTION • We assumed that decisions to vaccinate are tactical i.e. a rationale decision was made based upon costs and benefits independent of the actions of others in the population WHY THAT MIGHT BE A PROBLEM • The payoffs from vaccination could be dependent upon the vaccination decisions of others e.g. if everyone else vaccinated then the risk to you from not vaccinating should be small and vice versa WHAT IS GAME THEORY? • Def’n – the study of strategic decision making •In Biology (Evolutionary Game Theory), the principle utility is fitness WHAT IS GAME THEORY? • Example from economics: The Prisoners’ Dilemma: • Two prisoners are given the option to betray or remain silent •If A and B both betray, each gets a 2 year sentence •If A betrays and B remains silent, and B gets 3 years and vice versa •If A and B remain silent, each receives 1 year •Result: both betray EXAMPLES IN BIOLOGY • Def’n – the study of strategic decision making •In Biology (Evolutionary Game Theory), the principle utility is fitness •In Biology, we solve for the Evolutionary Stable Strategy (ESS), which is a Nash Equilibrium, wherein no one player can gain fitness by unilaterally changing his/her strategy unilaterally PARAMETERS • P – an individual’s strategy that he/she will vaccinate with probability P • p - proportion of the population that is vaccinated • rv – morbidity risk from vaccination • ri - morbidity risk from infection • πp - risk to an unvaccinated individual of eventual infection given vaccination coverage p AN IMPORTANT TRADEOFF • There are two key risks, (i) the morbidity risk from the infection and (ii) the morbidity risk from the vaccination ri vs rv PAYOFF FROM VACCINATION • This will always depend upon the proportion of the population that is vaccinated, p E(P, p) = P (- rv) + (1 – P) (-ri πp) •Set r (relative risk) to rv/ri •This gives: E(P, p) = -rP - πp (1 – P) TO SOLVE THE ESS • You need to calculate the risk of infection πp •This requires that you build an S.I.R. model dS = m (1- p ) - b SI - mS dt dI = b SI - g I - m I dt dR = m p + g I - mR dt TWO IMPORTANT PARAMETERS •Growth rate of the disease: •Infectious period: R0 = b m +g m f= g THE BIG INSIGHT • It is impossible to eradicate a disease through voluntary vaccination whenever the perceived risk from vaccination rv > 0 •For any perceived risk from vaccination (i.e. rv > 0), p* < pcrit •If rv > πp then P* = 0 otherwise 0 < P* < 1 •Recall, πp depends upon R0 •If r > 1, parents will not vaccinate STABLE VACCINATION ESS coverage as a function of vaccination risk and risk of infection RESPONSE TO A VACCINATION SCARE • The change in vaccine uptake ▵P depends upon the new perceived risk from the vaccine and risk from infection ANOTHER INSIGHT • A vaccination scare can cause vaccination rate to drop (relatively easily) however it will be relatively more difficult to restore vaccination rates to pre-scare levels WHAT’S MISSING? • Seasonality •Transient dynamics •Variance in risk perception •Social dynamics