Lyme disease is a common tick-borne illness caused by a bacterium, which is transmitted to humans through the bite of infected ticks. The transmission dynamics of Lyme disease is dependent on a variety of factors, including the length of the tick's life cycle, availability of hosts, climatic conditions and seasonal influences, which are important to understand for control strategies.
In a paper published last month in the SIAM Journal on Applied Mathematics, authors Yuxiang Zhang and Xiao-Qiang Zhao propose a reaction-diffusion model to study transmission dynamics of Lyme disease while taking into account seasonality.
Ticks live for roughly 2 years, and their life cycle includes three stages: larva, nymph and adult. Ticks climb on to host animals who brush against vegetation from the tips of grasses and shrubs. Once they attach themselves, they feed on blood by inserting their mouthparts into the skin of a host, thus transmitting the disease. After obtaining a blood meal-which can take anywhere between 3 and 5 days-ticks drop off their hosts and prepare for the next stage of the life cycle.
Adult ticks feed and mate on larger mammals such as deer. After obtaining their meal, adult females drop off their hosts and lay eggs on the ground. The eggs hatch into larvae, and larval ticks feed on smaller animals such as mice and birds. After obtaining a blood meal from these smaller hosts, larvae drop off and are inactive until they grow into nymphs. Nymphs feed on small rodents, birds and other small mammals, and then molt into adults. The cycle thus repeats itself.
For Lyme disease to exist in an area, the bacteria that cause it, the ticks that carry the bacteria, and mammals that provide food to ticks in their various life stages must be present. Seasonal variations in temperature, rainfall and resource availability also affect disease transmission and dynamics.
"Ticks develop slowly or become less active in colder temperatures, and rainfall is also critically important for their development, survival, and activities," explains author Xiao-Qiang Zhao. "According to a report from the Public Health Agency of Canada on Lyme disease cases in Ontario between 1999 and 2004, most occurred in late spring and summer, when the young ticks are most active and people are outdoors more often."
A previous model proposed a reaction and diffusion model to study global dynamics of Lyme disease. A reaction-diffusion model takes into account the interaction (reaction) of constituents within the system (in this case pathogens, susceptible hosts and infective hosts) and their change in density over time within their respective populations (diffusion).
The previously-proposed spatial model treats population densities in a continuous two-dimensional space, factoring in birth, death, infection and developmental advancement. However, the model does not account for seasonal patterns.
In this paper, the authors modify this previous model into a reaction and diffusion model in a periodic environment, which models seasonable variables (eg. temperature) as a periodic function. A periodic function is one that repeats its values at regular intervals.