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.

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