Optimal allocation of limited vaccine to control an infectious disease: Simple analytical conditions

Abstract

When allocating limited vaccines to control an infectious disease, policy makers frequently have goals relating to individual health benefits (e.g., reduced morbidity and mortality) as well as population-level health benefits (e.g., reduced transmission and possible disease eradication). We consider the optimal allocation of a limited supply of a preventive vaccine to control an infectious disease, and four different allocation objectives: minimize new infections, deaths, life years lost, or quality-adjusted life years (QALYs) lost due to death. We consider an SIR model with interacting populations, and a single allocation of vaccine at time 0. We approximate the model dynamics to develop simple analytical conditions characterizing the optimal vaccine allocation for each objective. We instantiate the model for an epidemic similar to COVID-19 and consider population groups: one group (individuals under age 65) with high transmission but low mortality and the other group (individuals age 65 or older) with low transmission but high mortality. We find that it is optimal to vaccinate younger individuals to minimize new infections, whereas it is optimal to vaccinate older individuals to minimize deaths, life years lost, or QALYs lost due to death. Numerical simulations show that the allocations resulting from our conditions match those found using much more computationally expensive algorithms such as exhaustive search. Sensitivity analysis on key parameters indicates that the optimal allocation is robust to changes in parameter values. The simple conditions we develop provide a useful means of informing vaccine allocation decisions for communicable diseases.

Optimal Control Applied to Vaccination and Testing Policies for COVID-19

“…Regarding the input controls used in the formulation of the corresponding OCPs, some previous works have considered only non-pharmaceutical interventions against COVID-19, such as isolation of the population [27]; quarantining, hospitalization interventions, and treatment of infected people [15,17]; isolation, quarantine, and public health education [16], sensitization, quarantine, diagnosis, and monitoring and psychological support [19]; public health education, treatment of infected individuals, and health care measures for asymptomatic infectious people [21]; and use of face-masks, hand sanitizer, and social distancing; treatment of patients and active screening with testing; and prevention against recurrence and reinfection of people who have recovered [26]. Fewer works have focused on pharmaceutical measures, such as allocation of the treatment [18] and vaccine administration [20,[23][24][25]. In this paper, both kinds of measures, pharmaceutical and non-pharmaceutical, have been considered as the input controls of the proposed OCPs.…”

“…Regarding the objective functional used in the formulation of the corresponding OCPs, some previous works have considered just a single optimality criterion, such as the minimization of the number of infected people [18], the number of deaths [27], or the years of life lost due to premature mortality and the years lost due to disability [23]. Some other works have considered and compared several optimality criteria, such as minimizing the number of new infections, the number of deaths, the life years lost, and the quality-adjusted life years lost due to death [20], or minimizing the number of symptomatic infections, the number of deaths, the number of cases requiring non-ICU hospitalization, and the number of cases requiring ICU hospitalization [24]. Finally, other works have combined different optimality criteria with the cost of applying the controls, such us the number of infected people [16,21], the numbers of exposed and infected people [26], or the numbers of susceptible, infected, exposed, and asymptomatic people [15,17,19].…”

“…The impacts of delays in applying preventive precautions against the spread of the COVID-19 pandemic are assessed in [19] through a mathematical model with multiple delays. The dynamics of an SIR model of SARS-CoV-2 transmission with interacting populations is approximated in [20], in which the optimal vaccine allocation problem is reduced to a knapsack problem. An OCP for a fractional system of the COVID-19 transmission dynamics is designed in [21], in which several non-pharmaceutical intervention policies are incorporated.…”

Dynamic resource allocation for controlling pathogen spread on a large metapopulation network

“…Even if we consider a sub-population as healthy or infected (only two possible states per sub-population), a network with J sub-populations would have 2 J possible health states, which yields an asymptotically intractable optimal allocation when the network is very large. Hence, authors that study resource allocation on metapopulation networks generally build and evaluate their approach in a small number of sub-populations, usually less than 50 [18][19][20][21][22][23][24][25]. Recently, [26] proposed a framework built upon optimal control theory that is able to deal with the dynamic allocation problem in a network of hundreds of sub-populations thanks to several simplifications, among which is considering only a subset of edges for the optimization.…”

“…Finally, published studies assume the resource to be in general distributed only once, before or at the beginning of an outbreak [ 21 , 29 ]. Therefore, the resource allocation problem is static.…”