Analysis of a reaction–diffusion SVIR model with a fixed latent period and non-local infections

Abstract

This paper concerns with a reaction–diffusion susceptible-vaccinated-infectious-recovered model with a fixed latent period. The model is formulated as a non-local and time-delayed reaction–diffusion model due to the fact that an individual infected by the disease in one place may not stay at the same space in the domain due to the movement of human during the incubation period. We then derive the basic reproduction number ${\mathrm{\Re }}_0$ as the spectral radius of the next infection operator and show that it serves as a threshold role in predicting whether the disease will spread. Further, the explicit formula of basic reproduction number is obtained when all model parameters to be positive constants and the domain to the one-dimensional case. Moreover, we demonstrate the differences in the form of basic reproduction number between the standard incidence and bilinear incidence rate.

COMMUNICATED BY:

• X. Zou

Keywords:

• SVIR epidemiological model
• basic reproduction number
• spatial heterogeneity
• uniform persistence

2010 Mathematics Subject Classifications:

• 34K30
• 35K57
• 35Q80
• 92D25

Acknowledgments

The authors would like to thank the anonymous referees and the editor for their helpful suggestions and comments which led to the improvement of our original manuscript. The authors would like to thank the editors and the referees for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

J. Gao was supported by the National Natural Science Foundation of China [grant number 61761002] and the First-Class Disciplines Foundation of Ningxia, China [grant number NXYLXK2017B09]. C. Zhang was supported by the Graduate Innovation Project of North Minzu University, China [grant number YCX19129]. J. Wang was supported by the National Natural Science Foundation of China [grant numbers 11871179], the Natural Science Foundation of Heilongjiang Province [grant number LC2018002 and LH2019A021] and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, China.