Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1810–1848 | Cite as

Multi-stage Vector-Borne Zoonoses Models: A Global Analysis

  • Derdei Bichara
  • Abderrahman Iggidr
  • Laura Smith
Original Article


A class of models that describes the interactions between multiple host species and an arthropod vector is formulated and its dynamics investigated. A host-vector disease model where the host’s infection is structured into n stages is formulated and a complete global dynamics analysis is provided. The basic reproduction number acts as a sharp threshold, that is, the disease-free equilibrium is globally asymptotically stable (GAS) whenever \({\mathcal {R}}_0^2\le 1\) and that a unique interior endemic equilibrium exists and is GAS if \({\mathcal {R}}_0^2>1\). We proceed to extend this model with m host species, capturing a class of zoonoses where the cross-species bridge is an arthropod vector. The basic reproduction number of the multi-host-vector, \({\mathcal {R}}_0^2(m)\), is derived and shown to be the sum of basic reproduction numbers of the model when each host is isolated with an arthropod vector. It is shown that the disease will persist in all hosts as long as it persists in one host. Moreover, the overall basic reproduction number increases with respect to the host and that bringing the basic reproduction number of each isolated host below unity in each host is not sufficient to eradicate the disease in all hosts. This is a type of “amplification effect,” that is, for the considered vector-borne zoonoses, the increase in host diversity increases the basic reproduction number and therefore the disease burden.


Vector-borne zoonoses Stage progression Multi-host One health Amplification effect Global stability Nonlinear dynamical systems 

Mathematics Subject Classification

92D30 34D23 34D20 34D40 34A34 



We are grateful to two anonymous referees for valuable comments and suggestions that led to an improvement in this paper. A. Iggidr acknowledges the partial support of Inria in the framework of the program STIC AmSud (Project MOSTICAW).


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Computational and Applied MathematicsCalifornia State UniversityFullertonUSA
  2. 2.IECL (UMR 7502)Inria, Université de Lorraine, CNRSMetzFrance

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