Analysis of a reaction–diffusion dengue model with vector bias on a growing domain

Abstract

In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number ${\mathrm{\Re }}_0^{\rho }$. The results show that (i) if ${\mathrm{\Re }}_0^{\rho }>1$, the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when ${\mathrm{\Re }}_0^{\rho }<1$. Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure.

Keywords:

• Dengue model
• growing domain
• basic reproduction number
• vanishing and spreading
• Lyapunov function

Mathematic Subject classifications:

• 34K30
• 35K57
• 35Q80
• 92D25

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China (No. 12071115) and Heilongjiang Natural Science Funds for Distinguished Younger Scholar (No. JQ2023A005).