Model equations

The model describes the temporal evolution of the following state variables:

The following equations describe the dynamics of transmission and disease progression within each province, x:

$$ \frac{dS_x}{dt} = - \Phi_x S_x $$ $$ \frac{dE_x}{dt} = \Phi_x S_x - \gamma_1 E_x $$ $$ \frac{dI_{A_x}}{dt} = p_a \gamma_1 E_x - r_1 I_{A_x} $$ $$ \frac{dI_{P_x}}{dt} = (1-p_a) \gamma_1 E_x - \gamma_2 I_{P_x} $$ $$ \frac{dI_{M_x}}{dt} = p_{m_x} \gamma_2 I_{P_x} - r_2 I_{M_x} $$ $$ \frac{dI_{S_x}}{dt} = (1-p_{m_x}) \gamma_2 I_{P_x} - \tau_s I_{S_x} $$ $$ \frac{dH_{1_x}}{dt} = \left(1-\frac{p_{c_x}}{(1-p_{m_x})}\right) \tau_s I_{S_x} - r_3 H_{1_x} $$ $$ \frac{dH_{2_x}}{dt} = \frac{p_{c_x}}{(1-p_{m_x})} \tau_s I_{S_x} - \tau_p H_{2_x} $$ $$ \frac{dC_{1_x}}{dt} = d_{c_x} \tau_p H_{2_x} - \mu C_{1_x} $$ $$ \frac{dC_{2_x}}{dt} = (1-d_{c_x}) \tau_p H_{2_x} - r_4 C_{2_x} $$ $$ \frac{dH_{3_x}}{dt} = r_4 C_{2_x} - r_5 H_{3_x} $$ $$ \frac{dR_x}{dt} = r_1 I_{A_x} + r_2 I_{M_x} + (1-d_{s_x}) r_3 H_{1_x} + r_5 H_{3_x} $$ $$ \frac{dD_x}{dt} = d_{s_x} r_3 H_{1_x} + \mu C_{1_x} $$ $$ \frac{dX_x}{dt} = d_m p_{m_x} \gamma_2 I_{P_x} - \Delta_m {X_x} $$ $$ \frac{dY_x}{dt} = d_s \tau_s I_{S_x} - \Delta_s {Y_x} $$ $$ \frac{dI_{M_d}}{dt} = \Delta_m {X_x} $$ $$ \frac{dI_{S_d}}{dt} = \Delta_s {Y_x} $$

where the force of infection, Φ_x, is defined as $$ \Phi_x = \frac{\beta_{x} \delta_{x,t} \left(\zeta I_{A_x}+I_{P_x}+I_{M_x}+I_{S_x}\right)}{N_x} $$ and p_cx = 1 − p_mx − p_sx.

Parameter definitions

Model parameters and the values used are defined in the tables below. The first table contains parameters that are the same across provinces; the second contains parameters that vary across provinces. To take into account parametric uncertainty, we varied key parameters, drawing them stochastically from a triangular distribution defined by the lower, upper, and mode values as given in the tables.

Basic reproduction number

The expected number of secondary infections produced by a single infection introduced into a naive population (basic reproduction number) can be caluclated as: $$ R_{0_x} = \beta_{x,0} \left(\frac{p_a \zeta}{r_1} + \frac{(1-p_a)}{\gamma_2} + \frac{(1-p_a) p_{m_x}}{r_2} + \frac{(1-p_a)(1-p_{m_x})}{\tau_s}\right) $$ In this context, a ‘naive’ population is the population at the start of the epidemic when (a) there are no previously-infected individuals (S_x ≈ N_x) and (b) there are no measures or practices in place that reduce the contact rate below baseline (δ_x, 0 = 1).

Time-varying reproduction numbers

The reproduction number is assumed to vary over time, reflecting changes in the contact rate that result from both government-enforced and individually-enacted measures. We refer to two types of time-varying reproduction numbers: R_c(t) = δ_tR₀ denotes the hypothetical reproduction number at a given point in time that would be observed in the absence of previously-infected individuals, where δ_t is a proportional reduction from baseline; R_e(t) = R_c(t)S(t)/N(t) denotes the realized reproduction number at a given point in time, taking into account accumulation of infeciton and immunity in the population.

Scenarios

We consider two scenarios, an optimistic and a pessimistic scenario, as described in detail in our published reports¹,². The scenarios can be summarized based on how the value of δ_t (and therefore R_c(t)) varies through time, as depicted below. A no intervention scenario is also shown for comparison.

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References