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: info@sacovid19mc.co.za.

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 |

I_{A} |
number of asymptomatic individuals (infectious) |

I_{P} |
number of presymptomatic individuals (infectious) |

I_{M} |
number of mildly and moderately ill individuals (infectious) |

I_{S} |
number of individuals who are or will become severely ill but are not yet hospitalised (infectious) |

H_{1} |
number of severely ill individuals who are hospitalized in the general (non-ICU) ward |

H_{2} |
number of individuals who are work will be come critically ill currently in the general (non-ICU) ward |

C_{1} |
number of individuals who are critically ill, will eventually die, and are currently in the ICU |

C_{2} |
number of individuals who are critically ill, will eventually recover, and are currently in the ICU |

H_{3} |
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 |

I_{Md} |
cumulative number of confirmed mild / morderate infections |

I_{Sd} |
cumulative number of confirmed severe and critical infections |

N |
total number of individuals in the population (S + E + I_{A} + I_{P} + I_{M} + I_{S} + H_{1} + H_{2} + C_{1} + C_{2} + H_{3} + 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, *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}.

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 |
---|---|---|---|---|---|---|

γ_{1} |
gamma1 | 1 / duration in exposed class (E) |
91.25 | 60.83 | 182.50 | years^{ − 1} |

γ_{2} |
gamma2 | 1 / duration in presymptomatic class (I_{P}) |
182.50 | 121.67 | 365.00 | years^{ − 1} |

r_{1} |
r1 | 1 / duration of infectiousness for asymptomatic infections (I_{A}) |
52.14 | 45.62 | 60.83 | years^{ − 1} |

r_{2} |
r2 | 1 / duration of infectiousness for mild and moderate cases (I_{M}) |
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 (C_{1}) |
73.00 | 60.83 | 91.25 | years^{ − 1} |

R_{0} |
R0 | basic reproduction number | 2.70 | 2.60 | 2.80 | years^{ − 1} |

τ_{s} |
taus | 1/ time from onset to hospitalisation for severe cases (I_{S}) |
73.00 | 60.83 | 91.25 | years^{ − 1} |

p_{a} |
pa | proportion of infections that will remain asymptomatic | 0.75 | 0.70 | 0.80 | - |

τ_{p} |
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 | - |

r_{3} |
r3 | 1 / duration of hospital stay for severe cases | 30.42 | 26.07 | 45.62 | years^{ − 1} |

r_{4} |
r4 | 1 / duration of stay in ICU for critical cases | 22.81 | 20.28 | 26.07 | years^{ − 1} |

d_{m} |
pdetm | proportion of mild cases detected | 0.25 | 0.25 | 0.25 | - |

d_{s} |
pdets | proportion of severe cases detected | 1.00 | 1.00 | 1.00 | - |

Δ_{m} |
deltam | 1 / time from onset to test result for mild and moderate cases | 45.62 | 36.50 | 91.25 | years^{ − 1} |

Δ_{s} |
deltas | 1 / time from onset to test result for severely ill cases | 91.25 | 73.00 | 182.50 | years^{ − 1} |

r_{5} |
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 |
---|---|---|---|---|---|---|---|---|---|---|---|

p_{mx} |
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 |

p_{sx} |
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 |

d_{sx} |
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 |

d_{cx} |
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 |

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).

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*_{t}*R*_{0} 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.

We consider two scenarios, an optimistic and a pessimistic scenario, as described in detail in our published reports^{1},^{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.

This work is licensed under a Creative Commons Attribution 4.0 International License.

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↩

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↩