Table 1 Notation key

\(N\)

Population size

\({S}_{N}\)

Number of non-resistant hosts in the susceptible state

\({S}_{R}\)

Number of resistant hosts in the susceptible state

\({I}_{NW}\)

Number of non-resistant hosts infected with the wild-type pathogen

\({I}_{RW}\)

Number of resistant hosts infected with the wild-type pathogen

\({I}_{NE}\)

Number of non-resistant hosts infected with the escape mutant pathogen

\({I}_{RE}\)

Number of resistant hosts infected with the escape mutant pathogen

\({I}_{NWE}\)

Number of non-resistant hosts infected with both pathogen types

\({I}_{RWE}\)

Number of resistant hosts infected with both pathogen types

\({I}_{W}\)

Total number of hosts infected with the wild-type pathogen (\({I}_{NW}+{I}_{RW}+{I}_{NWE}+{I}_{RWE}\))

\({I}_{E}\)

Total number of hosts infected with the escape mutant pathogen (\({I}_{NE}+{I}_{RE}+{I}_{NWE}+{I}_{RWE}\))

\({f}_{R}\)

Frequency of resistant hosts

\(\beta\)

Transmission rate parameter

\(\alpha\)

Recovery rate parameter (\(\alpha =1\) is assumed throughout)

\(r\)

Cross resistance parameter (\(r=1\) for full cross resistance, \(r=0\) for no cross resistance)

\({P}_{W}^{*}\)

Equilibrium prevalence of wild-type pathogen (\(({I}_{NW}^{*}+{I}_{RW}^{*})/N\))

\({P}_{NW}^{*}\)

Equilibrium prevalence of wild-type pathogen in non-resistant hosts (\({I}_{NW}^{*}/N\))

\({P}_{RW}^{*}\)

Equilibrium prevalence of wild-type pathogen in resistant hosts (\({I}_{RW}^{*}/N\))

\({R}_{NW}\)

Basic reproduction ratio for wild-type pathogen in non-resistant hosts

\({R}_{RW}\)

Basic reproduction ratio for wild-type pathogen in resistant hosts

\({R}_{NE}\)

Basic reproduction ratio for escape mutant pathogen in non-resistant hosts

\({R}_{RE}\)

Basic reproduction ratio for escape mutant pathogen in resistant hosts

\({R}_{INV}\)

Reproduction ratio of invasion of escape mutant in population endemically infected with wild-type pathogen