1 / 18

A stochastic model for epidemics in childhood infections

A stochastic model for epidemics in childhood infections. Samuel Nosal Student at ENSTA, ParisTech, France Visiting student in the IIMS (May-July 2005) Supervised by Ass. Prof. Mick Roberts. An SIR model. A class of models for epidemics with immunity (especially in childhood infections)

walker
Download Presentation

A stochastic model for epidemics in childhood infections

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A stochastic model for epidemics in childhood infections Samuel Nosal Student at ENSTA, ParisTech, France Visiting student in the IIMS (May-July 2005) Supervised by Ass. Prof. Mick Roberts

  2. An SIR model • A class of models for epidemics with immunity (especially in childhood infections) • Population divided into 3 groups: Susceptibles, Infected and Recovered • With or without demography: birth, death, emigration, immigration • In this project: Without demography

  3. Deterministic version • Volterra-Lotka equations • Initial conditions: • Constant population: Pairwise rate of infection [time-1people-1] Removal per capita rate [time-1]

  4. Main properties of the deterministic version • An epidemic occurs only when • Final number of susceptible people • Not possible to have an explicit expression for it • But unique ( ) • Linked to the stochastic model

  5. A stochastic version • A continuous-time Markov with states are in the following set: • Generator: • That leads to a discrete-time Markov chain • Length of each state: exponentially distributed random variable with parameter • Transition probabilities

  6. Matlab simulations Deterministic Markov chain Epidemic No Epidemic Different scale on each row

  7. Epidemic or not? • Criterion to give the name “epidemic”: when • Probability distribution for • Kolmogorov forward equations • Computed in a large matrix • Then sum of a few terms:

  8. Probability distribution (simulations) • The only difference is the initial number of infected people (1 vs 10) • Compute the probability matrix and then plot the first column

  9. Final number of infected people • The evolution of with respect to • The deterministic approximates • Then we obtain the following expression: Depends on too The deterministic

  10. Matlab simulations In red: deterministic final number of susceptible people In red: the approximation, which is given on the previous slide In blue: the exact expected final number of susceptible people

  11. To a PDE and an SDE • An epidemic as a chemical reaction • Master equation

  12. Fokker-Planck equation • Taylor series to replace by terms in • We only keep orders 0, 1 and 2  F-P eq. • + Initial and boundary conditions

  13. From F-P equation to an SDE • There is a theorem that gives us an SDE for a random variable, whose probability distribution obeys the F-P equation • But the theorems, that we have to prove (or not) that there is a unique solution, are not valid: for example, coefficients are not Lipschitz functions

  14. Euler scheme • It normally works only on nice SDEs • But it gives some interesting results, that were computed much faster than previously • Where are Gaussian random variable normally distributed

  15. Euler scheme – Matlab simulation

  16. Conclusion

  17. Thank youAu revoir et bonne continuation à tous

More Related