Complexity and critical thresholds in the dynamics of visceral leishmaniasis.

Shakir Bilal,Anuj Mubayi,Rocio Caja Rivera,Edwin Michael

doi:10.1098/rsos.200904

Abstract

We study a general multi-host model of visceral leishmaniasis including both humans and animals, and where host and vector characteristics are captured via host competence along with vector biting preference. Additionally, the model accounts for spatial heterogeneity in human population and heterogeneity in biting behaviour of sandflies. We then use parameters for visceral leishmaniasis in the Indian subcontinent as an example and demonstrate that the model exhibits backward bifurcation, i.e. it has a human infection and a sandfly population threshold, characterized by a bi-stable region. These thresholds shift as a function of host competence, host population size, vector feeding preference, spatial heterogeneity, biting heterogeneity and control efforts. In particular, if control is applied through human treatment a new and lower human infection threshold is created, making elimination difficult to achieve, before eventually the human infection threshold no longer exists, making it impossible to control the disease by only reducing the infection levels below a certain threshold. A better strategy would be to reduce the human infection below a certain threshold potentially by early diagnosis, control animal population levels and keep the vector population under check. Spatial heterogeneity in human populations lowers the overall thresholds as a result of weak migration between patches.

Highlights

The existence of a threshold behaviour in the spread of infectious disease was first established by Kermack & McKendrick [1,2]
For a sufficiently large disease-induced death rate in humans [14,20] backward bifurcation (BB), with the characteristic two system thresholds, is observed: the first threshold is the human infection prevalence threshold (calculated using equations (2.10)–(2.12)) and the second threshold is the threshold vector population related to Rc as obtained from the condition when A21 À 4A2A0 1⁄4 0: Rc 1⁄4 f (Ll,ml,dl,Lv,mv, . . .), l 1⁄4 {H, A, B}
We investigated the dynamics of visceral leishmaniasis via a dynamical system model that may exhibit multiple thresholds due to BB phenomenon in a certain parameter regime

Summary

Introduction

The existence of a threshold behaviour in the spread of infectious disease was first established by Kermack & McKendrick [1,2]. Most cases the threshold can be expressed in terms of a single parameter, namely, the basic reproduction 2 number, which has to be greater than unity for the disease to invade and/or persist in the population. This threshold translates to having a threshold population of susceptible hosts. A class of plant and animal infectious diseases (including human neglected tropical diseases, NTDs) [1,3,4,5,6] exhibit multiple thresholds, namely, a threshold susceptible host density together with a threshold initial infection level in the population [4,5,6,7]. It is further known that the threshold criterion for transmission and extinction of a disease may show inter/intra-location variations [12]

Methods

Results

Conclusion