Seasonal population dynamics of ticks, and its influence on infection transmission: a semi-discrete approach

Abstract

In this paper, a simple semi-discrete (ticks' feeding is assumed to occur only during summers of each year) model for tick population dynamics is presented. Conditions for existence, uniqueness, and stability of a positive equilibrium are found; the system is then studied numerically using parameter estimates calibrated for the tick Ixodes ricinus in Trentino, Italy, and the sensitivity to parameters is examined. Then, this model is extended to consider a tick-transmitted infection of one species of hosts, while other hosts are incompetent to the infection. Assuming, for simplicity, that the infection is not affecting the total number either of hosts or ticks, a threshold condition for infection persistence is obtained. The dependence of the equilibrium infection prevalence on parameters is studied numerically; in particular, we considered how infection prevalence depends on host densities. This analysis reveals that a 'dilution effect' occurs both for competent and for incompetent hosts; this means that, besides a lower threshold for host densities for infection to persist, there exists also an upper threshold: if host densities were higher than the upper threshold, the infection would go to extinction. Numerically, it is found that, for realistic parameter values, the upper threshold is not much higher than observed densities.

The importance of the aggregation of ticks on small mammal hosts for the establishment and persistence of tick-borne pathogens: an investigation using the R 0 model

“…Heterogeneities in the efficiency of different transmission routes of tick-borne pathogens will affect R 0 and, therefore, the likelihood of a pathogen becoming established within a system (Woolhouse et al 1997). As expected, due to the applicability of the basic reproduction number, many attempts have been made to estimate R 0 for tickborne pathogens (Randolph, 1998;Norman et al 1999;Randolph et al 1999;Caraco et al 2002;Rosa et al 2003;Ghosh and Pugliese, 2004;Rosa and Pugliese, 2007). Indeed, Rosa et al (2003) previously investigated the role of tick aggregation on the persistence of tick-borne pathogens; however, many of these approaches were lacking in their relevance to the tick life cycle, assuming bite rates akin to insect vector models, or used derived expressions that could not be strictly interpreted as a biologically meaningful value of R 0 (Hartemink et al 2008).…”

Investigating the persistence of tick-borne pathogens via the R0model

“…This problem is often compounded by the presence of multiple host species and transmission routes (Hartemink et al 2008). Given the importance of R 0 in the epidemiology of infectious diseases there have been many attempts to define R 0 for tick-borne infections (Randolph, 1998;Norman et al 1999;Caraco et al 2002;Rosa et al 2003;Ghosh and Pugliese, 2004;Rosa and Pugliese, 2007). More recently, next generation matrix methods have been employed to address the complexities of infections in natural systems (Hartemink et al 2008) which has resulted in the most comprehensive and biologically correct estimation of R 0 for tick-borne infections.…”

Dynamics of a Time-Delayed Lyme Disease Model with Seasonality

“…Since the third equation of system (10) is decoupled from the first two equations of system (10), it suffices to use the following system to define the basic reproduction ratio:…”