The effects of human movement on the persistence of vector-borne diseases

Abstract

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R0 for both modeling approaches. We show that for both approaches that if the disease free equilibrium is stable (R0 < 1) then it is globally stable and if the disease free equilibrium is unstable (R0 > 1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

“…Another variant of this model, considered in [30] and reviewed in [3,37,15], includes host movement, but excludes vector movement. More spatial models for malaria transmission have been reviewed in [11,15] where space is treated discretely in the form of distinct patches, and in [15,12] where space is either treated as discrete or as continuous, in the latter case leading to models that take the form of reaction-diffusion equations.…”

“…Models that include mobility need to specify how exactly vectors and humans move, and here we have adopted the so-called Lagrangian approach, see [11] and references therein. A salient feature of the Lagrangian model is that all individuals are declared to be residents of a specific patch, but that they can spend parts of their time in other patches, where they might infect others, or pick up the infection.…”

“…This is in contrast to the more popular Eulerian approach, where individuals are not assigned to a particular patch, but instead simply move around between the various patches at certain prescribed rates. Examples of the Eulerian approach can be found in various contexts related to the spread of infectious diseases such as in [1,2,5,11], and are not restricted to malaria. Our methods can be used to study similar patched Ross-Macdonald models based on the Eulerian approach, but to keep our analysis concise, we restrict ourselves to models based on the less frequently used Lagrangian approach.…”

“…Establish the global dynamics of a patched Ross-Macdonald model, a variant of which was first investigated in [17] and reviewed in [11]. This model assumes an arbitrary number of patches between which both humans and mosquitoes are allowed to move.…”

“…Although our proof is based on techniques that are similar to those used in [11] for a closely related model, we have decided to include a concise and self-contained proof in an Appendix here, for two main reasons. First, there are important differences between the modeling assumptions made in [17], and those considered here.…”