Model structure
The stages of COVID-19 included within this model are S—susceptible, E—exposed (not currently infectious but have been exposed to the virus), A—asymptomatic infection (will never develop symptoms), I—symptomatic infection (consisting of presymptomatic or mild to moderate symptoms), H—severe symptoms requiring hospitalisation but not IC, C—very severe symptoms requiring IC, R—recovered and D—death. The total population is N=S+E+A+I+H+C+R+D (figure 1).
Figure 1  Compartmental flow model diagram depicting stages of disease and transitions between states. Asymptomatic infection represents the number of people never showing symptoms while symptomatic infection includes all those who show presymptomatic/mild symptoms to those who show more severe symptoms (prehospitalisation). Those who are hospitalised first occupy a non-IC bed (acute bed) after which they can either move into IC, recover or die. Those in IC can either recover or die at an increased rate compared with those in acute beds. This model does not capture those deaths which occur outside of hospital as a result of COVID-19. IC, intensive care. Each compartment Xg is stratified by age group (0–4, 5–17, 18–29, 30–39, 40–49, 50–59, 60–69, ≥70) where X denotes the stage of COVID-19 (S, E, A, I, H, C, R, D) and g denotes the age group class of individuals. Age groups were chosen to capture key social contact patterns (primary, secondary and tertiary education and employment) and variability in hospitalisation rates and outcomes from COVID-19 especially in older age groups. The total in each age group is informed by recent Office for National Statistics (ONS) estimates.21
Susceptible individuals become exposed to the virus at a rate governed by the force of infection λg, and individuals are non-infectious in the exposed category. A proportion δ move from exposed to symptomatic infection and the remaining to asymptomatic infection, both at the latent rate η. Individuals leave both the asymptomatic and symptomatic compartments at rate μ. All asymptomatic individuals eventually recover and there are no further stages of disease: the rate of leaving the asymptomatic compartment is therefore equivalent to the infectious period, μ. A proportion of symptomatic individuals γg go on to develop severe symptoms which require hospitalisation, but not IC. Once requiring hospitalisation, we assume individuals are no longer infectious to the general population due to self-isolation guidelines restricting further mixing with anyone aside from household members (if unable to be admitted to hospital) or front-line NHS staff (if admitted to hospital). Individuals move out of the acute hospitalised compartment at rate ρ, either through death, being moved to IC at rate ϵ, or through recovery (all remaining individuals). A proportion ωg of patients requiring IC will die at rate ψ, while the rest will recover.
The model (schematic in figure 1) is therefore described by the following differential equations:
Susceptible Sg  (1a)  d S g    d t   =  − λ g   S g
Exposed Eg  (1b)  d E g    d t   =  λ g   S g   − η E g
Asymptomatic Ag  (1c)  d A g    d t   = η ( 1 − δ  )  E g   − μ A g
Infectious Ig  (1d)  d I g    d t   = η δ E g   − μ I g
Hospitalised in acute bed Hg  (1e)  d H g    d t   = μ γ g   I g   − ρ H g
Hospitalised in IC Cg  (1f)  d C g    d t   = ρ ϵ H g   − ψ C g
Recovered Rg  (1g)  d R g    d t    = μ A g   + μ ( 1 − γ g    )  I g   +   ( 1 − ϵ  )  ( 1 − κ  )  ρ H g   +   ( 1 − ω g   ) ψ C g
Death Dg  (1h)  d D g    d t   = ( 1 − ϵ  )  κ ρ H g   + ω g   ψ C g