The contact process with fitness on random trees

Natalia Cardona-Tobon; Marcel Ortgiese

doi:10.1017/apr.2026.10066

The contact process with fitness on random trees

Natalia Cardona-Tobon
, Marcel Ortgiese

Universidad Nacional de Colombia, Bogotá

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Abstract

The contact process is a simple model for the spread of an infection in a structured population. We consider a variant of this process on Bienaymé-Galton-Watson trees, where vertices are equipped with a random fitness representing inhomogeneous transmission rates among individuals. In this paper, we establish conditions under which this inhomogeneous contact process exhibits a phase transition. We first prove that if certain mixed moments of the joint offspring and fitness distribution are finite, then the survival threshold is strictly positive. Further, we show that, if slightly different mixed moments are infinite, then this implies that there is no phase transition and the process survives with positive probability for any choice of the infection parameter. A similar dichotomy is known for the contact process on a Bienaymé-Galton-Watson tree. However, we show that the introduction of fitness means that we have to take into account the combined effect of fitness and offspring distribution to decide which scenario occurs.

Original language	English
Pages (from-to)	1-37
Journal	Advances in Applied Probability
Early online date	13 Jul 2026
DOIs	https://doi.org/10.1017/apr.2026.10066
Publication status	E-pub ahead of print - 13 Jul 2026

Acknowledgements

The authors thank the anonymous referee and the editor for their detailed comments and suggestions, which helped to improve this article. N.C.-T. acknowledges support from CONACyT-MEXICO (grant no. 636133). This manuscript was partially prepared while N.C.-T. was visiting the Department of Mathematical Sciences at the University of Bath, and she is grateful for the hospitality and collaboration. N.C.-T. acknowledges support from SAMBa (Statistical Applied Mathematics at Bath) and the Dorothea Schlözer-Programm at Georg-August-Universität Göttingen.

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There are no funding bodies to thank relating to the creation of this article.

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10.1017/apr.2026.10066Licence: CC BY

https://arxiv.org/pdf/2110.14537.pdf

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title = "The contact process with fitness on random trees",

abstract = "The contact process is a simple model for the spread of an infection in a structured population. We consider a variant of this process on Bienaym{\'e}-Galton-Watson trees, where vertices are equipped with a random fitness representing inhomogeneous transmission rates among individuals. In this paper, we establish conditions under which this inhomogeneous contact process exhibits a phase transition. We first prove that if certain mixed moments of the joint offspring and fitness distribution are finite, then the survival threshold is strictly positive. Further, we show that, if slightly different mixed moments are infinite, then this implies that there is no phase transition and the process survives with positive probability for any choice of the infection parameter. A similar dichotomy is known for the contact process on a Bienaym{\'e}-Galton-Watson tree. However, we show that the introduction of fitness means that we have to take into account the combined effect of fitness and offspring distribution to decide which scenario occurs. ",

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AB - The contact process is a simple model for the spread of an infection in a structured population. We consider a variant of this process on Bienaymé-Galton-Watson trees, where vertices are equipped with a random fitness representing inhomogeneous transmission rates among individuals. In this paper, we establish conditions under which this inhomogeneous contact process exhibits a phase transition. We first prove that if certain mixed moments of the joint offspring and fitness distribution are finite, then the survival threshold is strictly positive. Further, we show that, if slightly different mixed moments are infinite, then this implies that there is no phase transition and the process survives with positive probability for any choice of the infection parameter. A similar dichotomy is known for the contact process on a Bienaymé-Galton-Watson tree. However, we show that the introduction of fitness means that we have to take into account the combined effect of fitness and offspring distribution to decide which scenario occurs.

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The contact process with fitness on random trees

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