The SEIRS Compartmental Model

The underlying system of the SEIRS model is a system of coupled ordinary differential equations that describe the rate of change in the amount of individuals in each of the compartments. The system is expressed as

\({dS \over dt} = -{\beta SI \over N} + \epsilon R\)
\({dE \over dt} = {\beta IS \over N} - \sigma E\)
\({dI \over dt} = \sigma E - \gamma I\)
\({dR \over dt} = \gamma I - \epsilon R\),

where \(N=S+E+I+R\) is the total population. The functions \(S,E,I,R\) respectively represent

• susceptible individuals,
• exposed individuals,
• infected individuals,
• and removed individuals.

The constants \(\beta,\sigma,\gamma,\epsilon\) respectively represent

• the infectious rate,
• the incubation rate,
• the recovery rate,
• and the rate at which recovered individuals regain susceptibility.

Based on the SEIRS system and the expression for the total population, the initial conditions are

\((S_0,E_0,I_0,R_0)\in\{(S,E,I,R)\in[0,N]^4\,|\,S\leq0,\,E\leq0,\,I\leq0,\,R\leq0,\,N=S+E+I+R\}\).

In order to perform simulations using the SEIRS model, the system of coupled ordinary differential equations is solved using a numerical method. The system is solved using the Runge-Kutta 4th order method.

The Runge-Kutta 4th Order Method

The Runge-Kutta 4th order method is also referred to as RK4. The method is a fourth order method since the global truncation error of the method is \(O(h^4)\).
The incrementing function \(\Phi(t_n,\hat{x}_n,h)\) for RK4 consists of a weighted average of four functions. The functions are

\(k_1=f(t_n,\hat{x}_n)\)
\(k_2=f\Bigl(t_n+\frac{h}{2},\hat{x}_n+\frac{h}{2}k_1\Bigr)\)
\(k_3=f\Bigl(t_n+\frac{h}{2},\hat{x}_n+\frac{h}{2}k_2\Bigr)\)
\(k_4=f(t_n+h,\hat{x}_n+hk_3)\).

The incrementing function is

\(\Phi(t_n,\hat{x}_n,h)=\frac{1}{6}(k_1+2(k_2+k_3)+k_4)\).

By applying the incrementing function, the iterative approximation with step size \(h\) is given by

\(\hat{x}_{n+1}=\hat{x}_n+\frac{h}{6}(k_1+2(k_2+k_3)+k_4)\).

Through the application of the iterative approximation above, the simulations of the SEIRS model are performed.

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