# Global analysis for spread of infectious diseases via transportation networks

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## Abstract

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number \({\mathcal {R}}_0\) as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if \({\mathcal {R}}_0\le 1\) then the disease will be eradicated from each region, while if \({\mathcal {R}}_0> 1\) there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, \({\mathcal {R}}_0\) may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

## Keywords

Epidemic models Transportation networks Global dynamics Delay differential systems## Mathematics Subject Classification (2010)

34K05 92D30## Notes

### Acknowledgments

The authors are grateful for Professor Eduardo Liz for his kind hospitality at the Universidade de Vigo, October, 2010, where YN and GR started this project. The authors are grateful for Professor Teresa Faria for her kind hospitality at the University of Lisbon, July 2012, where the authors had stimulating discussions about multipatch models. YN and GR were supported by European Research Council StG Nr. 259559. YN was also supported by Spanish Ministry of Science and Innovation (MICINN), MTM2010-18318 and by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP-4.2.4. A/2-11-1-2012-0001 ‘National Excellence Program’. Since April 2014 YN was also supported by JSPS Fellowship, No. 268448. GR was supported by European Union and the European Social Fund through project FuturICT.hu (Grant No. TÁMOP-4.2.2.C-11/1/KONV-2012-0013), and Hungarian Scientific Research Fund OTKA K75517. The authors are particularly grateful for the three referees, whose comments significantly improved the manuscript.

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