Epidemics Modeling them with math

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

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

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

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

Modeling with simple model

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

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

Results using p

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

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.

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

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

Model without recovery

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

SIS model

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

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

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