The Impact of Coinfections and Their Simultaneous Transmission on Antigenic Diversity and Epidemic Cycling of Infectious Diseases

Table 3

Four equilibria and their stability conditions of model system (A.1).

Equilibrium	Solution	Stability conditions
^&Disease-free (𝑄^∗0)	𝑆=1,  𝐼𝑎=𝐼𝑏=𝐼𝑑=𝑅=0	𝑅^𝑎0<1, 𝑅^𝑏0<1, and 𝑅^𝑑0<1
^$Strain 𝑎 alone (𝑄^∗𝑎)	1𝑆=𝑅^𝑎0,  𝐼𝑏=𝐼𝑑=0,; 𝐼𝑎=𝜎+𝜇𝜎+𝜇+𝛾𝑎11−𝑅^𝑎0,  𝛾𝑅=^𝑎𝜎+𝜇+𝛾𝑎11−𝑅^𝑎0	𝑅^𝑎0>1, 𝑅^𝑎0>𝑅^𝑏0, 𝑅^𝑎0>𝑅^𝑑0, 𝑅^𝑏0𝑄^∗𝑎<1
Strain 𝑏 alone (𝑄^∗𝑏)	1𝑆=𝑅^𝑏0,  𝐼𝑎=𝐼𝑑=0,; 𝐼𝑏=𝜎+𝜇𝜎+𝜇+𝛾𝑏11−𝑅^𝑏0,  𝛾𝑅=^𝑏𝜎+𝜇+𝛾𝑏11−𝑅^𝑏0	𝑅^𝑏0>1, 𝑅^𝑏0>𝑅^𝑎0, 𝑅^𝑏0>𝑅^𝑑0, 𝑅^𝑎0𝑄^∗𝑏<1
Coexistence (𝑄^∗𝑎𝑏)	Interior equilibrium (𝑆,𝐼𝑎,𝐼𝑏,𝐼𝑑,𝑅) obtained numerically by solving equations (A.1) with the left sides being set to zero	𝑅^𝑎0(𝑄^∗𝑏)>1, 𝑅^𝑏0(𝑄^∗𝑎)>1, and the real part of the dominant eigenvalue of Jacobian matrix of system (A.1) being negative (see Appendix A.2)

^&𝑅^𝑖₀, 𝑖=𝑎,𝑏,𝑑, are the basic reproduction number of strain 𝑎, strain 𝑏, and dual infection, respectively, and are given by 𝑅^𝑖₀=𝛽_𝑖/(𝛾_𝑖+𝜇).
^$𝑅^𝑏₀(𝑄^∗_𝑎) represents the effective reproduction number of strain 𝑏 upon the population in which strain 𝑎 was established (𝑄^∗_𝑎) (see equation (A.2)).