1. Epidemics Modeling them with math

2. 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

3. 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

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

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

6. Modeling with simple model

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

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

9. Results using p

10. 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

11. 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.

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

13. 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

14. Model without recovery

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

16. SIS model

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

18. 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

19. 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