Table 1. Definitions and descriptions of the variables, parameters and rates that influence the dynamics of the two-host, two-vector system and the parameter values used to estimate R ₀.

Variable, Parameter or Rate	Construction	Definition or Description	Point Estimate and/or Feasible Range	Comments and Formula if Temperature-dependent[vector species]
xi	Xi/Hi	proportion of host type i that are susceptible		i can be C (cattle) or S (sheep)
yi	Yi/Hi	proportion of host type i that are infectious
zi	Zi/Hi	proportion of host type i that have recovered
Hi	Xi+Yi+Zi	total number of host type i
λHi		rate at which susceptible hosts of type i become infectious through being bitten by infectious vectors		j can be 1 (C. imicola) or 2 (C. bolitinos)
bj		probability of transmission from vector type j to a host given an effective contact	0.8–1.0	[C. sonorensis]
aj		biting rate of vector type j	0–0.5	[C. sonorensis]
		proportion of vectors of type j attracted to hosts of type i		As Gubbins et al. [6], for clarity we replace and with and respectively.
		host preference of vector type j <1 indicates a preference for cattle>1 indicates a preference for sheep	0–1	C. imicola feeds predominantly on cattle and sheep [16], [17], but prefers cattle [23]. C. bolitinos feeds on cattle and horses [16], [17] and breeds in cattle dung [14].
mij	Nj/Hi	ratio of vectors of type j to hosts of type i	Many areas:mC 1 = mS 1 = 500 (0–5000)mC 2 = mS 2 = 50 (0–500)Colder high-lying areas:mC 1 = mS 1 = 50 (0–100)mC 2 = mS 2 = 500 (0–5000)	In general, C. imicola is approx. 10 times more abundant than C. bolitinos [15]. In colder, high-lying areas, C. imicola is approx. 10 times less abundant than C. bolitinos [14].
Nj	Sj+Lj+Ij	total number of vectors of type j
ri		recovery rate of host type i	rC = 1/20.6rS = 1/16.4
di		pathogen-induced mortality rate of host type i	dC = 0dS = 0.001–0.01
Sj		number of vectors of type j that are susceptible
Lj		number of vectors of type j that are latent
Ij		number of vectors of type j that are infectious
λVj		rate at which susceptible vectors of type j become latent through biting infectious hosts
βj		probability of transmission from a host to vector type j given an effective contact	β 1 = 0.0021–0.0654β 2 = 0.0268–0.6444	[C. imicola][C. bolitinos]Both from data in [15].
νj		rate at which latent vectors of type j become infectious ( = 1/EIP, where EIP = extrinsic incubation period)	1/4–1/26	[C. sonorensis]
μj		natural mortality rate of vector type j	0.1–0.5	[C. sonorensis]
ρj		replacement rate of vector type j

Unless otherwise stated, values were taken from Gubbins et al. [6]. Subscripts 1 and 2 denote C. imicola and C. bolitinos respectively.