Approximation methods for analyzing multiscale stochastic vector-borne epidemic models

Xin Liu, Anuj Mubayi, Dominik Reinhold, Liu Zhu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Stochastic epidemic models, generally more realistic than deterministic counterparts, have often been seen too complex for rigorous mathematical analysis because of level of details it requires to comprehensively capture the dynamics of diseases. This problem further becomes intense when complexity of diseases increases as in the case of vector-borne diseases (VBD). The VBDs are human illnesses caused by pathogens transmitted among humans by intermediate species, which are primarily arthropods. In this study, a stochastic VBD model is developed and novel mathematical methods are described and evaluated to systematically analyze the model and understand its complex dynamics. The VBD model incorporates some relevant features of the VBD transmission process including demographical, ecological and social mechanisms, and different host and vector dynamic scales. The analysis is based on dimensional reductions and model simplifications via scaling limit theorems. The results suggest that the dynamics of the stochastic VBD depends on a threshold quantity R 0 , the initial size of infectives, and the type of scaling in terms of host population size. The quantity R 0 for deterministic counterpart of the model is interpreted as a threshold condition for infection persistence as is mentioned in the literature for many infectious disease models. Different scalings yield different approximations of the model, and in particular, if vectors have much faster dynamics, the effect of the vector dynamics on the host population averages out, which largely reduces the dimension of the model. Specific scenarios are also studied using simulations for some fixed sets of parameters to draw conclusions on dynamics.

Original languageEnglish (US)
Pages (from-to)42-65
Number of pages24
JournalMathematical Biosciences
StatePublished - Mar 2019


  • Fast and slow dynamics
  • Functional central limit theorem
  • Functional law of large numbers
  • Multiscale analysis
  • Quasi-stationary distributions
  • SIS compartment model
  • Time to extinction
  • Vector-borne disease model

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Biochemistry, Genetics and Molecular Biology(all)
  • Immunology and Microbiology(all)
  • Agricultural and Biological Sciences(all)
  • Applied Mathematics


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