# NCEM v1.0 (Provincial Model): Supplementary information

This document provides a technical overview of the National COVID-19 Epi Model (NCEM) provincial model. The model described in this document is the provincial-level NCEM model, created by the South African COVID-19 Modelling Consortium. There is also a separate document, the NCEM Provincial Model Code Guide, that gives an overview of the structure of the model code. If there are any queries regarding the model or the code, please contact us on: .

## Model equations

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

Variable Definition
S number of susceptible individuals
E number of exposed but not yet infectious individuals
IA number of asymptomatic individuals (infectious)
IP number of presymptomatic individuals (infectious)
IM number of mildly and moderately ill individuals (infectious)
IS number of individuals who are or will become severely ill but are not yet hospitalised (infectious)
H1 number of severely ill individuals who are hospitalized in the general (non-ICU) ward
H2 number of individuals who are work will be come critically ill currently in the general (non-ICU) ward
C1 number of individuals who are critically ill, will eventually die, and are currently in the ICU
C2 number of individuals who are critically ill, will eventually recover, and are currently in the ICU
H3 number of individuals who have been critically ill, will recover, and have been discharged from the ICU but remain in hospital for step-down care
R number of individuals who are no longer infectious / recoverd and/or discharged
D number of individuals who have died
IMd cumulative number of confirmed mild / morderate infections
ISd cumulative number of confirmed severe and critical infections
N total number of individuals in the population (S + E + IA + IP + IM + IS + H1 + H2 + C1 + C2 + H3 + R)
X dummy variable representing mild and moderate cases who will be tested before they are tested
Y dummy variable representing severe and critical cases who will be tested before they are tested

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, Φx, 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 pcx = 1 − pmx − psx.

## 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.

Symbol Code Description Mode Lower Upper Units
γ1 gamma1 1 / duration in exposed class (E) 91.25 60.83 182.50 years − 1
γ2 gamma2 1 / duration in presymptomatic class (IP) 182.50 121.67 365.00 years − 1
r1 r1 1 / duration of infectiousness for asymptomatic infections (IA) 52.14 45.62 60.83 years − 1
r2 r2 1 / duration of infectiousness for mild and moderate cases (IM) 73.00 60.83 91.25 years − 1
μ mu 1 / time to from entry into ICU until death for critical cases who will die (C1) 73.00 60.83 91.25 years − 1
R0 R0 basic reproduction number 2.70 2.60 2.80 years − 1
τs taus 1/ time from onset to hospitalisation for severe cases (IS) 73.00 60.83 91.25 years − 1
pa pa proportion of infections that will remain asymptomatic 0.75 0.70 0.80 -
τp tauprog 1 / time from hospitalisation to ICU admission for critical cases 91.25 73.00 182.50 years − 1
ζ zeta1 relative infectiousness of asymptomatic infections 0.75 0.70 0.80 -
r3 r3 1 / duration of hospital stay for severe cases 30.42 26.07 45.62 years − 1
r4 r4 1 / duration of stay in ICU for critical cases 22.81 20.28 26.07 years − 1
dm pdetm proportion of mild cases detected 0.25 0.25 0.25 -
ds pdets proportion of severe cases detected 1.00 1.00 1.00 -
Δm deltam 1 / time from onset to test result for mild and moderate cases 45.62 36.50 91.25 years − 1
Δs deltas 1 / time from onset to test result for severely ill cases 91.25 73.00 182.50 years − 1
r5 r5 1 / duration in step-down care after discharge from ICU 121.67 91.25 182.50 years − 1
Symbol Code Definition Eastern Cape Free State Gauteng Kwa-Zulu Natal Limpopo Mpumalanga Northern Cape North West Western Cape
pmx pm proportion of symptomatic cases that are mild or moderate 0.95 0.95 0.95 0.96 0.96 0.96 0.95 0.96 0.95
psx ps proportion of symptomatic cases that are severe 0.03 0.04 0.04 0.03 0.03 0.03 0.04 0.03 0.04
dsx pd1 proportion of severely ill (not critical) cases who die 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
dcx pd2 proportion of critically ill cases who die 0.29 0.27 0.24 0.26 0.27 0.26 0.27 0.25 0.26

## 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 (Sx ≈ Nx) and (b) there are no measures or practices in place that reduce the contact rate below baseline (δx, 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: Rc(t) = δtR0 denotes the hypothetical reproduction number at a given point in time that would be observed in the absence of previously-infected individuals, where δt is a proportional reduction from baseline; Re(t) = Rc(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 δt (and therefore Rc(t)) varies through time, as depicted below. A no intervention scenario is also shown for comparison.

## References

1. MASHA, HE2RO, SACEMA, and NICD, on behalf of the South African COVID-19 Modelling Consortium. (2020) “Estimating cases for COVID-19 in South Africa: Long-term national projections” https://www.nicd.ac.za/wp-content/uploads/2020/05/SACovidModellingReport_NationalLongTermProjections_Final.pdf

2. MASHA, HE2RO, SACEMA, and NICD, on behalf of the South African COVID-19 Modelling Consortium. (2020) “Estimating cases for COVID-19 in South Africa: Long-term provincial projections” https://www.nicd.ac.za/wp-content/uploads/2020/05/SACovidModellingReport_ProvincialLongTermProjections_Final.pdf

South African COVID-19 Modelling Consortium