# SEIR Modeling of the Italian Epidemic of SARS-CoV-2 Using Computational Swarm Intelligence

This SEIR model has been constructed to take into account actions to prevent COVID, such as quarantining (pg 2).

Typically, a SEIR model is formatted as follows (pg 3):

$\frac{dS(t)}{dt}=(-\beta*I(t)*\frac{S(t)}{N})$

$\frac{dE(t)}{dt}=(\beta*I(t)*\frac{S(t)}{N})-(\gamma*E(t))$

$\frac{dI(t)}{dt}=\gamma*E(t)-((\lambda+\kappa)*I(t))$

$\frac{dR(t)}{dt}=(\lambda+\kappa)*I(t)$

$S(t)+E(t)+I(t)+R(t)=N$

These differential equations represent the progression between Susceptible (S), Exposed (E), Infected (I) and Recovered (R). The constants $\beta,\gamma,\lambda,\kappa$ are the rates at which populations go from on group to another. $\beta$ is the infection rate (S→E), $(\gamma)$ is the inverse of the latency period (E→I), and $(\lambda), (\kappa)$ are (respectively) the recovery and death rates, which remove people from the Infected stage. N is the total population.

An expanded model for COVID has been proposed (pg 4):

$\frac{dS(t)}{dt}=-\beta*I(t)\frac{S(t)}{N}-\alpha*S(t)$

$\frac{dP(t)}{dt}=(\alpha*S(t)$

$\frac{dE(t)}{dt}=\beta*I(t)*\frac{S(t)}{N}-(\gamma*E(t))$

$\frac{dI(t)}{dt}=\gamma*E(t)-(\delta*I(t))$

$\frac{dQ(t)}{dt}=(\delta*I(t)-\lambda(t)*Q(t)-\kappa(t)*Q(t)$

$\frac{dR(t)}{dt}=(\lambda(t)*Q(t))$

$\frac{dD(t)}{dt}=\kappa(t)Q(t)$

This model includes the Protected (P), Quarantined (Q) categories. $\alpha$ determines the growth in Protected (S→P) due to prevention measures such as lockdowns and distancing. $\delta$ defines the change from Infected to Quarantined. This model separates the recovery and death rates ($\lambda, \kappa$), and makes them functions of time to account for changes (i.e introduction of a vaccine) (pg 5). They are defined as:

$\lambda(t)=\lambda_0[1-exp(-\lambda_1t)]$

$\kappa(t)=\kappa_0[exp(-\kappa_1t)]$

A summary of the SEIR scheme (pg 6):

The differential equations above were solved using the Runge-Kutta method (a way of approximating solutions that reduces error by h^4, where h is the factor by which the step size is reduced). Q, D, and R are considered as the starting conditions (pg 8).

Some example values from datasets from around the world (pg 16):

Godio, Alberto, et al. “SEIR Modeling of the Italian Epidemic of SARS-COV-2 Using Computational Swarm Intelligence.” International Journal of Environmental Research and Public Health, vol. 17, no. 10, 18 2020. PubMed, doi:10.3390/ijerph17103535.