Hospital and critical-care models
Hospital and critical-care models translate epidemic incidence into healthcare demand: admissions, occupied beds, critical-care use, peak demand and capacity exceedance.
From infections to admissions
Let \(J(t)\) be new infections per unit time and \(p_H\) the probability that an infection eventually requires hospital admission. With no delay,
\[A(t)=p_HJ(t).\]A simple fixed-delay version is
\[A(t)=p_HJ(t-d_H),\]where \(d_H\) is the infection-to-admission delay. More realistic models can use a distribution of delays.
A single-exit hospital compartment
First consider a hospital compartment with only one aggregate exit process:
\[\boxed{\frac{dH}{dt}=A(t)-\delta_HH}.\]If \(\delta_H\) is the only per-capita exit rate from \(H\), then the waiting time in the compartment is exponential with mean
\[\boxed{\frac{1}{\delta_H}}.\]Adding a separate transfer to critical care
Now suppose hospital patients can either leave the ward through discharge/other exit at rate \(\delta_H\) or transfer to critical care at rate \(\alpha\). Then
\[\boxed{\frac{dH}{dt}=A(t)-\alpha H-\delta_HH},\]\[\boxed{\frac{dC}{dt}=\alpha H-\delta_CC}.\]The total per-capita rate of leaving the hospital compartment is \(\alpha+\delta_H\). Therefore the mean time spent in \(H\), under these constant-rate assumptions, is
\[\boxed{\frac{1}{\alpha+\delta_H}}.\]Competing exits
The probability that the next exit from \(H\) is transfer to critical care is
\[\frac{\alpha}{\alpha+\delta_H},\]and the probability of discharge or another non-critical-care exit is
\[\frac{\delta_H}{\alpha+\delta_H}.\]Critical-care occupancy
If critical-care patients leave at rate \(\delta_C\), their mean stay under the same constant-rate assumption is \(1/\delta_C\), and
\[\frac{dC}{dt}=\alpha H-\delta_CC.\]Admissions versus occupancy
| Quantity | Meaning |
|---|---|
| admissions | new patients entering hospital per unit time |
| occupancy | patients currently using beds |
| length of stay | time a patient remains in a compartment before an exit |
Capacity exceedance
If \(K_H\) and \(K_C\) are the available hospital and critical-care capacities, exceedance occurs when
\[H(t)>K_H\qquad\text{or}\qquad C(t)>K_C.\]Peak demands are
\[H_{\max}=\max_tH(t),\qquad C_{\max}=\max_tC(t).\]Age-specific risk
If hospitalisation risk differs between groups, a simple delayed formulation is
\[A(t)=\sum_i p_{H,i}J_i(t-d_{H,i}).\]The same total number of infections can therefore produce different healthcare demand depending on which groups are infected.
Stochastic capacity risk
In a stochastic model, peak demand is random. A useful planning quantity is
\[\boxed{P(C_{\max}>K_C)},\]the probability that peak critical-care demand exceeds capacity.