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Published byFrances Ferring Modified over 2 years ago

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Epidemics Modeling them with math

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History of epidemics Plague in 1300’s killed in excess of 25 million people Plague in London in 1665 killed 75,000 in one year Polio, measles SARS outbreak in 2003 –15% mortality rate –Potential for pandemic

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Simple model S(t) the number of people susceptible with S(0) = N I(t) is the number of people infected B is some constant of proportionality

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Solving the simple model S(t) = N – I(t) k = BN > 0 and known as the transmission rate

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Solving using the initial condition of I(0) =1, k =.16858 predicted values can be found

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Modeling with simple model

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Problems with simple model The model doesn’t fit well in the middle –because it assumes dS/dt varies linearly with the number infected

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Simple model with p factor p can be adjusted to account for the relationship between dS/dt

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Results using p

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Problems with simple model These models don’t take into account: Incubation rate Recovery rate I(0) = 1 is not realistic S(0) = N is not realistic

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SIR Model β is transmission rate λ is mean recovery rate Mean infectious period is 1/ λ R(0) = βN/ λ is basic reproduction number.

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Things to consider S’(t) < 0 which means that the susceptible population decreases over time –People either develop immunity or they die

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Outbreak If we integrate S(t) + I(t) 0 as t-> infinity I’(t) = [B(0)S(t) - ]I(t) –If BS(0) 0 –If BS(0) > then I’(t) > 0 for t < t* –This is the threshold for a real epidemic outbreak

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Model without recovery

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Eigen Values Solving for eigen values Second shows a change in value depending on BS > it’s positive if < it’s negative = bifurcation

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SIS model

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Solving SIS model We can reduce the two equations since S(t) + I(t) =N

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SIS model solution if R <= 1 then I’(t) < 0 and lim I(t) = 0 If R > 1 then lim = n( 1- 1/R) Bifurcation exists at R=1 with it stable below 1 and unstable above 1

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More complicated models Models get more complicated by adding in birth and death rates B(t) = B(0)(1+ aCos2πt) which explains the periodic nature of real epidemics where B(0) is the mean transmission rate, a is the amplitude of seasonal variation and t is measured in years

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