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

- 19 Downloads

## 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.

### References

- Adler FR (1992) The effects of averaging on the basic reproduction ratio. Math Biosci 111(1):89–98CrossRefMATHGoogle Scholar
- Andersson H, Britton T (1998) Heterogeneity in epidemic models and its effect on the spread of infection. J Appl Probab 35(3):651–661MathSciNetCrossRefMATHGoogle Scholar
- Brauer F, Castillo-Chavez C (2012) Mathematical models in population biology and epidemiology, texts in applied mathematics, 2nd edn. Springer, New York. https://doi.org/10.1007/978-1-4614-1686-9
- Busenberg S, Castillo-Chavez C (1991) A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS. IMA J Math Appl Med Biol 8(1):1–29MathSciNetCrossRefMATHGoogle Scholar
- Castillo-Chavez C, Feng Z (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. Math Biosci 151(2):135–154CrossRefMATHGoogle Scholar
- Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28(4):365–382. https://doi.org/10.1007/BF00178324 MathSciNetCrossRefMATHGoogle Scholar
- Diekmann O, Heesterbeek H, Britton T (2012) Mathematical tools for understanding infectious disease dynamics. Princeton University Press, PrincetonCrossRefMATHGoogle Scholar
- Feng Z, Hill AN, Smith PJ, Glasser JW (2015) An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing. J Theor Biol 386:177–187. https://doi.org/10.1016/j.jtbi.2015.09.006 MathSciNetCrossRefMATHGoogle Scholar
- Feng Z, Hill AN, Curns AT, Glasser JW (2017) Evaluating targeted interventions via meta-population models with multi-level mixing. Math Biosci 287:93–104. https://doi.org/10.1016/j.mbs.2016.09.013 MathSciNetCrossRefMATHGoogle Scholar
- Friedland S (1980/81) Convex spectral functions. Linear Multilinear Algebra 9(4):299–316. https://doi.org/10.1080/03081088108817381
- Glasser J, Feng Z, Moylan A, Del Valle S, Castillo-Chavez C (2012) Mixing in age-structured population models of infectious diseases. Math Biosci 235(1):1–7. https://doi.org/10.1016/j.mbs.2011.10.001 MathSciNetCrossRefMATHGoogle Scholar
- Glasser JW, Feng Z, Omer SB, Smith PJ, Rodewald LE (2016) The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: a modelling study. Lancet Infect Dis 16(5):599–605. https://doi.org/10.1016/S1473-3099(16)00004-9 CrossRefGoogle Scholar
- Hadeler K, Müller J (1996a) Vaccination in age structured populations I: the reproduction number. Models Infect Human Dis Struct Relat Data 6:90CrossRefMATHGoogle Scholar
- Hadeler K, Müller J (1996b) Vaccination in age structured populations II: Optimal strategies. Models for infectious human diseases: their structure and relation to data pp 102–114Google Scholar
- Hill AN, Longini IM Jr (2003) The critical vaccination fraction for heterogeneous epidemic models. Math Biosci 181(1):85–106. https://doi.org/10.1016/S0025-5564(02)00129-3 MathSciNetCrossRefMATHGoogle Scholar
- Jacquez JA, Simon CP, Koopman J, Sattenspiel L, Perry T (1988) Modeling and analyzing HIV transmission: the effect of contact patterns. Math Biosci 92(2):119–199. https://doi.org/10.1016/0025-5564(88)90031-4 MathSciNetCrossRefMATHGoogle Scholar
- Nold A (1980) Heterogeneity in disease-transmission modeling. Math Biosci 52(3–4):227–240. https://doi.org/10.1016/0025-5564(80)90069-3 MathSciNetCrossRefMATHGoogle Scholar
- Nussbaum RD (1986) Convexity and log convexity for the spectral radius. Linear Algebra Appl 73:59–122. https://doi.org/10.1016/0024-3795(86)90233-8 MathSciNetCrossRefMATHGoogle Scholar
- Seneta E (1973) Non-negative matrices. An introduction to theory and applications. Halsted Press, New YorkGoogle Scholar
- van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Dedicated to the memory of John Jacquez. Math Biosci 180(1–2):29–48. https://doi.org/10.1016/S0025-5564(02)00108-6