Global analysis for spread of infectious diseases via transportation networks

Abstract

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number R(0) as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if R(0) ≤ 1 then the disease will be eradicated from each region, while if R(0) > 1 there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, R(0) may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

Global stability of a time-delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure

“…It is, however, not well understood some problems on the mathematical properties (e.g., existence, uniqueness and stability of equilibria) of such models. From this point of view, we are interested in the work of Nakata and Röst [26]. For biological reason and mathematical viewpoint, to clarify such properties is always thought to be an important work.…”

“…In particular, time-delayed models (see e.g., [3,22,28]), multi-group epidemic models (see e.g., [11,17,22,24,28,29,32,33,35,38,39]), patchy models (see e.g, [5,12,26,36]) and models with general nonlinear incidence rates (see e.g., [8,28,29,38]) play important roles in studying the transmission of disease. In this field, determining threshold conditions for the persistence, extinction of a disease, and global stability of equilibria remains one of the most challenging problems in the analysis of models due to the dimension of the model is higher.…”

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.…”