# Constrained minimization problems for the reproduction number in meta-population models

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## Abstract

The basic reproduction number (\(\mathcal {R}_0\)) can be considerably higher in an SIR model with heterogeneous mixing compared to that from a corresponding model with homogeneous mixing. For example, in the case of measles, mumps and rubella in San Diego, CA, Glasser et al. (Lancet Infect Dis 16(5):599–605, 2016. https://doi.org/10.1016/S1473-3099(16)00004-9), reported an increase of 70% in \(\mathcal {R}_0\) when heterogeneity was accounted for. Meta-population models with simple heterogeneous mixing functions, e.g., proportionate mixing, have been employed to identify optimal vaccination strategies using an approach based on the gradient of the effective reproduction number (\(\mathcal {R}_v\)), which consists of partial derivatives of \(\mathcal {R}_v\) with respect to the proportions immune \(p_i\) in sub-groups *i* (Feng et al. in J Theor Biol 386:177–187, 2015. https://doi.org/10.1016/j.jtbi.2015.09.006; Math Biosci 287:93–104, 2017. https://doi.org/10.1016/j.mbs.2016.09.013). These papers consider cases in which an optimal vaccination strategy exists. However, in general, the optimal solution identified using the gradient may not be feasible for some parameter values (i.e., vaccination coverages outside the unit interval). In this paper, we derive the analytic conditions under which the optimal solution is feasible. Explicit expressions for the optimal solutions in the case of \(n=2\) sub-populations are obtained, and the bounds for optimal solutions are derived for \(n>2\) sub-populations. This is done for general mixing functions and examples of proportionate and preferential mixing are presented. Of special significance is the result that for general mixing schemes, both \(\mathcal {R}_0\) and \(\mathcal {R}_v\) are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively.

## Keywords

Meta-population model Convexity of reproduction number Optimization problem Vaccination strategy Epidemiology## Mathematics Subject Classification

37N25 49J15 34H05 92D30## Notes

### Acknowledgements

The findings and conclusions in this report are those of the author(s) and do not necessarily represent the official position of the Centers for Disease Control and Prevention or other institutions with which they are affiliated. We thank the anonymous reviewers for comments and suggestions, which helped improve the presentation of the manuscript.

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