PubMed · 8133673
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T J Romano. 1994. Tuberculosis.. https://pubmed.ncbi.nlm.nih.gov/8133673/
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The effectiveness of a vaccination program depends on how the vaccinations are spread over the households of the community. Here we formulate the optimal allocation of vaccinations as a linear programming problem, when the objective is to prevent epidemics with the minimum vaccination coverage. A vaccine efficacy of less than 100%, as is usual in practice, is allowed for. Optimal vaccine allocations attempt to leave the same number of susceptibles in every household if the disease has a very high transmission rate within households. This means that proportionately more individuals need to be vaccinated in larger households if the vaccine efficacy is < 100%. The linear programming formulation can accommodate heterogeneity among individuals of the proportionate mixing form and can also minimize the initial reproduction number for a given achievable vaccination coverage.
A SIRS model with vaccination is considered. The vaccination is assumed to have side effects (for simplicity, these side effects are modeled as a probability of becoming ill because of vaccination). It is the interest of the total population to minimize the prevalence of disease; hence, the vaccination rate that minimizes the prevalence will be determined. In Section 2, the individual is considered: an individual tries to minimize his or her own risk. This angle of approach results in a vaccination rate dependent on the prevalence of the disease. The bifurcations of this system are analyzed, and the optimal vaccination coverage for the individual is computed. This coverage is then compared with the optimal vaccination coverage for the total population: it is found that they disagree for some parameter sets.