Modeling Lyme disease transmission

Abstract

Lyme disease, a typical tick-borne disease, imposes increasing global public health challenges. A growing body of theoretical models have been proposed to better understand various factors determining the disease risk, which not only enrich our understanding on the ecological cycle of disease transmission but also promote new theoretical developments on model formulation, analysis and simulation. In this paper, we provide a review about the models and results we have obtained recently on modeling and analyzing Lyme disease transmission, with the purpose to highlight various aspects in the ecological cycle of disease transmission to be incorporated, including the growth of ticks with different stages in the life cycle, the seasonality, host diversity, spatial disease pattern due to host short distance movement and bird migration, co-infection with other tick-borne pathogens, and climate change impact.

Keywords: Basic reproduction number; Biodiversity; Bird migration; Co-infection; Lyme disease; Mathematical model; Seasonality; Spatial model; Tick-borne disease.

Fig. 1

A schematic diagram for the Lyme disease transmission (reproduced from (Lou, Wu, & Wu, 2014)). To describe the tick development and biting activities, the tick population is divided into 7 stages, stratified further as the uninfected or infected epidemiological classes for postegg stages. Immature ticks can feed on two host species, the mouse ($H_1$) and an alternative host ($H_2$), while adult ticks are assumed to feed only on deer in the study.

Fig. 2

Reproduced from figures in (Lou & Wu, 2014) but with different parameters. (a) The relationship between a given index and the rodent population size with frequency-dependent seeking rate (a) and density-dependent seeking rate (b) respectively. In the case (a), the number of infectious nymphs is a linearly decreasing function of the rodent population.

Fig. 3

Reproduced from figures in (Lou & Wu, 2014) but with different parameters. The relationship between a given index and the rodent population size with Holling-type seeking rate. In figures (a) and (b), different parameters are used.

Fig. 4

30 year normal mean monthly temperature under two settings near Long Point (Lou et al., 2014, Environment Canada). The blue solid and red dashed curves represent the monthly temperature for the periods $1961-1990$ period and $1981-2010$, respectively. The monthly temperature was set to be $0^{\circ }\text{C}$ if it is lower than $0^{\circ }\text{C}$. Both are collected from Environment Canada website (Environment Canada).

Fig. 5

Both figures are from (Heffernan et al., 2014). (a) Diagram for the Lyme disease transmission. (b) The spring and fall bird migration rates.

Fig. 6

Figures are from (Heffernan et al., 2014). (a) Ticks establish in the poor habitat ($H=100$ and $D=10$) with the seasonal inputs by migratory birds. (b) Ticks can expand into a new region (where $H=100$ and $D=10$) with the help of bird migration. (c) Ticks establish at a higher density in the rich habitat ($H=150$ and $D=15$) with seasonal inputs by migratory birds. (d) Lyme disease can invade into the rich habitat with seasonal tick inputs by migratory birds.

Fig. 7

A schematic diagram, in (Lou et al., 2017), of co-infection in the tick and mice M populations with subscripts denoting the infectious status for each pathogen. Here E (eggs), $LQ$ (questing larvae), $LF$ (feeding larvae), $NQ$ (questing nymphs), $NF$ (feeding nymphs) and A (adults) represent the stages of tick population with subscripts denoting the infectious status for each pathogen. Subscript 0: no pathogen in ticks; 1: Borrelia only; 2: Babesia only; 3: both pathogens.

Fig. 8

Solution simulations in (Lou et al., 2017) through different initial values converge to the constant level for ticks (a), constant infected ticks for Borrelia infection only (b) and Babesia transmission cycle can not establish without the co-infection (c). However, on the scenario of coinfection, both pathogens can get established ((d), (e) and (f)). More interestingly, some ticks becomes infected with only Babesia or Borrelia while some others get infected with both pathogens.