# Infection Dynamic Model This tutorial is also available for Matlab in: echodemo TutorialInfectionDynamicModel Tutorial for the graphical user interface File -> New -> Tutorial -> Tutorial 3. Infection Dynamic Model (see also Import Tutorial)

# Problem definition

This example shows how to perform the global sensitivity analysis of a mathematical model representing an infective process at its early state, where we assume that the infection is propagated through some kind of contact between individuals who do not take any precaution to avoid contagion.

The equations describe the dynamics of I and S, representing the model of the infection process are: $\frac{dI}{dt}=\gamma \kappa I S -rI -\delta I$

and $\frac{dS}{dt}=-\gamma \kappa I S +bS + m$

where:

• I is the number of Infected idividuals at time t
• S number of individual susceptible to infection
• $\kappa < 1$ contact coefficient
• $\gamma < 1$ infection coefficient
• r recovery rate
• $\delta$ death rate
• m migration rate
• b birth rate

At the early stage (t ~ 0) S(t) >> I(t) and then S(t)~S0. Therefore the first equation becomes linear and the solution is: $I=I_0 e^{\gamma \kappa S_0 -r -\delta}$. If the coefficient of the exponential is greater then 0 the infection spreads, otherwise the infection dies out.

This tutorial is base on the example provided in the Saltelli's book.

## Aim of the Tutorial

The aim of the tutorial is to show how to perform global sensitivity analysis in COSSAN-X. The first order and the total effect sensitivity indices of the input factors of the mathematical model are estimated,