# Epidemiological Model Equations and Parameters ## SIR Model ### System of Equations $$ \begin{cases} \dot{S} = -\beta(1 - u)SI \\ \dot{I} = \beta(1 - u)SI - \gamma I & \forall t \in (0, t^*] \\ \dot{R} = \gamma I \end{cases} $$ ### Initial Conditions $$ [S(0), I(0), R(0)]^T = [S_0, I_0, R_0]^T $$ ### Parameters - Basic Reproduction Number: $\mathcal{R}_0 = \frac{\beta}{\gamma}$ - Modified Reproduction Number: $\mathcal{R}_u = \frac{\beta(1 - u)}{\gamma}$ --- ## SIS Model ### System of Equations $$ \begin{cases} \dot{S} = -\beta(1 - u)SI + \gamma I \\ \dot{I} = \beta(1 - u)SI - \gamma I & \forall t \in (0, t^*] \end{cases} $$ ### Initial Conditions $$ [S(0), I(0)]^T = [S_0, I_0]^T $$ ### Parameters - Basic Reproduction Number: $\mathcal{R}_0 = \frac{\beta}{\gamma}$ - Modified Reproduction Number: $\mathcal{R}_u = \frac{\beta(1 - u)}{\gamma}$ --- ## SIRD Model ### System of Equations $$ \begin{cases} \dot{S} = -\beta(1 - u)SI + \delta R \\ \dot{I} = \beta(1 - u)SI - (\gamma + \varepsilon)I & \forall t \in (0, t^*] \\ \dot{R} = \gamma I - \delta R \\ \dot{D} = \varepsilon I \end{cases} $$ ### Initial Conditions $$ [S(0), I(0), R(0), D(0)]^T = [S_0, I_0, R_0, D_0]^T $$ ### Parameters - Basic Reproduction Number: $\mathcal{R}_0 = \frac{\beta}{\gamma + \varepsilon}$ - Modified Reproduction Number: $\mathcal{R}_u = \frac{\beta(1 - u)}{\gamma + \varepsilon}$ --- ## SEIRD Model ### System of Equations $$ \begin{cases} \dot{S} = -\beta(1 - u)SI + \delta R \\ \dot{E} = \beta(1 - u)SI - \phi E \\ \dot{I} = \phi E - (\gamma + \varepsilon)I & \forall t \in (0, t^*] \\ \dot{R} = \gamma I - \delta R \\ \dot{D} = \varepsilon I \end{cases} $$ ### Initial Conditions $$ [S(0), E(0), I(0), R(0), D(0)]^T = [S_0, E_0, I_0, R_0, D_0]^T $$ ### Parameters - Basic Reproduction Number: $\mathcal{R}_0 = \frac{\beta}{\gamma + \varepsilon + \phi}$ - Modified Reproduction Number: $\mathcal{R}_u = \frac{\beta(1 - u)}{\gamma + \varepsilon + \phi}$