Simplify the model (the ideas carry over): dS/dt = – beta I(t) S(t), dI/dt = beta I(t) S(t) – gammaH iota I(t) – r (iota) I(t), dH/dt = gammaH iota I(t), where r (iota) = (1 – iota) (gammaI (1 – delta1) + gammaD delta1) accounts for all other exits from I (none returning to S or I).

First, the rate of exit to hospitalization is gammaH iota. So, by the probabilistic interpretation of linear loss rates, the average time until hospitalization is 1/(gammaH iota), not 1/gammaH.

Second, from the rates in the differential equation and the definition of iota, the probability a case is hospitalized is gammaH iota / (gammaH iota + r (iota)) = iota. The parameter values do not satisfy this equation.

These issues can be easily fixed by adjusting the parameters and loss rates.

In addition to the probabilistic interpretation of iota as a ratio of loss rates in the I(t) differential equation, the probability a case is hospitalized can also be derived from the solution of the nonlinear model. From the model, the long-term fraction of cumulative hospitalized cases is also gammaH iota / (gammaH iota + r (iota)), since (gammaH iota times the integral of I(t) from 0 to infinity)/ (S(0) – lim t-> infinity S(t)) = gammaH iota / (gammaH iota + r(iota)).

Glenn Webb and Cameron Browne ]]>

Let’s consider hospitalized individuals H. They can either die or recover. According to Table 1, the parameters for Sierra Leone are: Time from hospitalization to death, 1/gamma_DH = 6.26; time from hospitalization to recovery, 1/gamma_IH = 1/15.88; case fatality rate in hospital, delta_2 = 0.75.

One can easily calculate the average time an individual will stay in the hospital as follows: delta_2*1/gamma_DH + (1-delta_2)*1/gamma_IH = 8.7 days. This is the weighted mean of the time to death and time to recovery.

According to the differential equations in the study by Rivers et al. (2014), the total rate at which individuals will leave the hospital compartment is gamma_DH*delta_2 + gamma_IH*(1-delta_2). The inverse of this rate gives the average time individuals in the model stay in the hospital: 7.4 days. Hence, lower than what it should be.

]]>Original: Transition probabilities for compartmental models are indeed 1/rate, and that is the case in this model. It is customary to omit this notation from transition diagrams for the sake of simplicity. See for example Legrand et al 2006, which is the source of this model structure. Table 1 shows our parameters are 1/rate.

]]>The authors weight the rates by probability, but they should be weighting the average time spent in each compartment (i.e. 1/rate) and then inverting this again.

For a simple R example, let us take a compartment spawning two options, one with rate 1/4 and one with rate 1/1, each with 50% probability:

1/mean(c(rexp(1000000,rate=1/4),rexp(1000000,rate=1/1)))

This gives a value of 0.4. Which is equal to 1/(4*0.5+1*0.5). This is NOT equal to 1/4*0.5+1/1*0.5=0.625

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