Dynamic updating of vaccination rate

[pratikunterwegs]

This issue is to request that vaccination rates should be dynamically updated over the course of an epidemic in all models that support <vaccination>. The general method is $\nu_{it} = \nu_i N_i/(N_i - V_{it})$ for each timestep $t$, where $\nu_{it}$ is the vaccination rate for group $i$ at time $t$.

This is because the vaccination rate nu ($\nu$) in a <vaccination> is defined as a fraction of the total number of individuals in each age group, $N_i$. This may be interpreted as the number of doses available for each age group on each day.

Models currently remove a fraction of all susceptibles as vaccinated, which results in fewer and fewer susceptibles being vaccinated in each timestep, which is not the expected behaviour.

In {epidemics}, we only allow susceptibles in the default model, and susceptibles and $V_1$ in the Vacamole model, to be vaccinated, allowing $N_i - V_{it}$ to be replaced by $S_{it}$ and $V_{1_{it}}$ for:

• $S \rightarrow V$ in the default model; making $\delta S_V = - \nu_iN_i$
• $S \rightarrow V_1$ and $V_1 \rightarrow V_2$ in the Vacamole model; making $\delta S_V1 = - \nu_{1}N$, and $\delta V_{1_{V2}} = - \nu_{2}N$ respectively. Note that second doses in the Vacamole model are also defined as a fraction of the full population.

Consequently, dynamic updating can be replaced by passing vaccination numbers ($\nu_iN_i$) rather than rates ($\nu_i$) to the C++ model code. This issue can thus be resolved by editing the appropriate .checK_prepare_args_*() functions for current and future models.

This issue might need to be revisited if the definition of a <vaccination> object changes.

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See also this feedback from @BlackEdder in comments on #125:

... vaccination calendars are defined as the fraction of individuals in the full cohort vaccinated. That would lead to a rescaling to the effective vaccination rate: $\hat{\nu} = \nu (S_i +E_i + I_i + R_i + V_i) / (S_i + E_i + I_i + R_I)$ * interval, where $i$ represents the cohort in the population. $\hat{\nu}$ is then the rate of vaccinating $S_i$, while $\nu$ is the rate of vaccinating cohort $i$. Note that the above can be rewritten to the slightly simpler to understand form: $\nu N_i/(N_i - V_i)$