The role of deer in facilitating the spatial spread of the pathogen Borrelia burgdorferi

Abstract

Borrelia burgdorferi is a vector-bourne zoonosis which propagates in wild populations of rodents and deer. The latter are incompetent for the pathogen but are required for the life cycle of hard-backed ticks which act as a vector for the pathogen. Increasing the diversity of hosts has previously suggested the presence of a ‘dilution effect’ in which such an increase reduces successful pathogen transmission as it increases the chance that a tick will encounter an incompetent host. This paper will produce a model which shows that whilst a dilution effect is possible for a system in which deer are the only incompetent host, this effect is not likely to be strong. Extending the population dynamics to include movement of deer into regions previously only inhabited by competent hosts, we find that, although ticks come in with the deer, there is a significant time lag before Borrelia appears.

Appendix: Nomenclature

The following gives a list of all parameters used with their description (including units), as well as estimated value.

r::

Density dependent growth rate of deer (day^− 1); 1/150 (Prior 1995).

α _j ;:

Encounter rate of juvenile ticks with competent hosts (e.g. deer) (day^− 1); 0.031 (Pugliese and Rosà 2008).

α _a ;:

Encounter rate of adult ticks with incompetent hosts (e.g. rodents) (day^− 1); 0.25 (Pugliese and Rosà 2008).

t _j ;:

Proportion of ticks which are larvae or nymphs (juveniles); 0.7.

t _a ;:

Proportion of ticks which are adults; 0.3.

a₁ = t _a α _a ;:

Overall encounter rate of adult ticks; 0.075

a₂ = t _j α _j ;:

Overall encounter rate of juvenile ticks; 0.0217

B;:

Number of eggs laid per fed adult tick; 500 (Norman et al. 1999).

K _T ;:

Carrying capacity of ticks (ha^− 1); 50 (equal to B K _{H 2}).

H₁;:

Density of competent hosts (ha^− 1); 15 (Pugliese and Rosà 2008).

$K_{H_2}$;:

Carrying capacity of incompetent hosts (ha^− 1); 0.1 (Gilbert et al. 2001).

$b_{H_1}$;:

Birth rate of competent hosts (day^− 1); 0.005.

$\delta_{H_1}$;:

Death rate of competent hosts (day^− 1); 0.005 (Pugliese and Rosà 2008).

δ _T ;:

Death rate of ticks (day^− 1); 0.05775 (Pugliese and Rosà 2008).

D;:

Diffusion rate of deer (dist.² day^− 1); N/A.

E;:

Diffusion rate of ticks per deer (dist.² day^− 1 deer^− 1); N/A.

K;:

Ratio of competent hosts with tick carrying capacity; 0.3.

H;:

Ratio of competent hosts with incompetent hosts; 150.

ε;:

Dimensionless diffusion rate of ticks; 0.5–2.5.