On the global stability of a delayed epidemic model with transport-related infection

Abstract

a b s t r a c tWe study the global dynamics of a time delayed epidemic model proposed by Liu et al. (2008) [J. Liu, J. Wu, Y. Zhou, Modeling disease spread via transport-related infection by a delay differential equation, Rocky Mountain J. Math. 38 (5) (2008) 1525-1540] describing disease transmission dynamics among two regions due to transport-related infection. We prove that if an endemic equilibrium exists then it is globally asymptotically stable for any length of time delay by constructing a Lyapunov functional. This suggests that the endemic steady state for both regions is globally asymptotically stable regardless of the length of the travel time when the disease is transferred between two regions by human transport.

Transmission dynamics of infectious diseases on transportation networks

“…The SIS-type epidemic model proposed by Liu et al [28] was further investigated by Nakata [33] and Nakata and Röst [34] by describing the global dynamics for an arbitrary n number of regions with different characteristics and general travel networks.…”

“…The models in [28,33,34] provide a good basis to investigate the spread of an infectious disease in regions which are connected by transportation. As a submodel, an age-structured system can be constructed to incorporate the possibility of disease transmission during travel, where age is the time elapsed since the start of the travel.…”

“…The two systems, describing the dynamics in the regions and during transportation, are interconnected; initial values of the system for disease spread during travel depend on the state of the system in the regions, while the inflow term of arrivals to the regions after being in transportation for a fixed time arises as the solution of the subsystem for travel. If the subsystem can be solved analytically (this was the case for the age-structured SI model used in [28,33,34]), then the system for disease spread in the regions decouple from the subsystem. Recalling that initial data of the system during travel comes from the equations for the regions, the inflow term of travelers completing a trip appears as a delayed feedback term.…”

“…Based on such considerations, Liu et al [28] introduced the time needed to complete the travel into the SIS-type epidemic model and also the possible infections during this time. Nakata [33] described the global dynamics of this system for two identical regions in terms of the basic reproduction number. The model was later generalized by Nakata and Röst [34] to the case of n regions with different characteristics and arbitrary travel networks.…”

“…The SIS model can be reduced to a logistic equation and then can be solved analytically. This property was heavily used in the analysis done in [28,33,34]. However, the lack of closed form solution causes substantial technical difficulties in the analysis of SEAIR-type models, as will be shown in this paper.…”