Asymptotic Analysis of Optimal Vaccination Policies

Abstract

Targeted vaccination policies can have a significant impact on the number of infections and deaths in an epidemic. However, optimising such policies is complicated, and the resultant solution may be difficult to explain to policy-makers and to the public. The key novelty of this paper is a derivation of the leading-order optimal vaccination policy under multi-group susceptible–infected–recovered dynamics in two different cases. Firstly, it considers the case of a small vulnerable subgroup in a population and shows that (in the asymptotic limit) it is optimal to vaccinate this group first, regardless of the properties of the other groups. Then, it considers the case of a small vaccine supply and transforms the optimal vaccination problem into a simple knapsack problem by linearising the final size equations. Both of these cases are then explored further through numerical examples, which show that these solutions are also directly useful for realistic parameter values. Moreover, the findings of this paper give some general principles for optimal vaccination policies which will help policy-makers and the public to understand the reasoning behind optimal vaccination programs in more generic cases.

Optimality of Maximal-Effort Vaccination

“…However, choosing the correct groups to prioritise is still of crucial importance and can have a substantial impact of the effectiveness of the vaccination campaign (Fitzpatrick and Galvani 2021 ). Applying similarly rigorous techniques to finding the optimal vaccination policy is beyond the scope of this paper, although we extended the results of this paper to apply asymptotic techniques to understand the behaviour of the optimal solution under certain special cases in Penn and Donnelly ( 2023 ).…”

Learning from the COVID-19 pandemic: a systematic review of mathematical vaccine prioritization models

“…In In Penn and Donnelly (2023), an ODE-based SIR model was used to study the effect of the basic reproduction number R 0 on the optimal vaccination plan. An interesting counter-intuitive result was found: It is better to prioritize 45-49 year olds than 55-59 year olds despite higher case fatality rates in the latter group.…”

“…Used in 90 of the 94 investigated models (Table 1), reduction in infection ( ɛ 1 , implemented either as a leaky or all-or-nothing vaccine) is the most frequently considered vaccine function, followed by reduction in symptoms ( ɛ 2 , used in 28 studies), reduction in severe disease ( ɛ 3 , used in 22 studies), reduction in onward transmission ( α , used in 17 studies), and reduction in death ( ɛ 4 , used in 15 studies). Other vaccine functions considered in only a few models include a shorter period of infectiousness (Makhoul et al, 2020; Penn and Donnelly, 2023), as well as a reduced vaccine efficacy for older individuals (Bubar et al, 2021; Aruffo et al, 2022; Buckner et al, 2021) and children (Han et al, 2021). 46 out of 94 studies (48.9%) considered only one type of vaccine function, while three studies (Liu et al, 2022a; McBryde et al, 2021; Mak et al, 2022) differentiated five types (Table 1).…”

Stochastic dual epidemic hypothesis model with Ornstein-Uhlenbeck process: Analysis and numerical simulations with SARS-CoV-2 variants

“…Epidemiological models play a vital role in studying the transmission of diseases in populations. They are a powerful tool for qualitative and quantitative analysis of the spread and control of infectious diseases [1,2,3,4]. The research results obtained from epidemiological models help in predicting the trends in the development of infectious diseases, identifying the key factors influencing their spread, and finding optimal strategies for prevention and control of infectious diseases.…”