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

Table 4

Impact of coinfection on invasion conditions.

Contribution from coinfection to transmission	Strain 𝑏 can invade only if
&𝛽𝑑=0 and 𝜙𝑏=0	𝑅^𝑏0>𝛾1+^𝑎+𝜇𝐼𝑎𝑅^𝑎0𝛾𝑏𝑅+𝜇^𝑎0>𝑅^𝑎0
^$𝛽𝑑>0 and 𝜙𝑏=0	𝑅^𝑑0>𝑅^𝑎0
𝛽𝑑=0 and 𝜙𝑏>0	^%when 𝑅^𝑏0=𝑅^𝑎0, 𝜙𝑏>12𝛾𝑑+𝜇𝛾𝑏+𝜇
	^#when  𝑅^𝑏0<𝑅^𝑎0, 𝜙𝑏>𝛾𝑑+𝜇𝑎𝛽𝑎𝐼𝑎1−Γ1+Γ
𝛽𝑑>0 and 𝜙𝑏>0	^%when 𝑅^𝑏0=𝑅^𝑎0, 𝑅^𝑑0𝑅^𝑎0+2𝜙𝑏𝛾𝑏+𝜇𝛾𝑑+𝜇>1
	^#when  𝑅^𝑏0<𝑅^𝑎0, 𝑅^𝑑0𝑅^𝑎0+𝜙𝑏𝛽𝑎𝐼𝑎𝛾𝑑+𝜇1+Γ1−Γ>1

We consider the successful invasion conditions by investigating the scenario where the host population in which strain 𝑎 was already established is challenged by strain 𝑏. In principle, it is simply that the invasion reproductive number is larger than unit (i.e., 𝑅^𝑏₀(𝑄^∗_𝑎)>1). In Table 4 we explore the influence of the coinfection on the successful invasion by illustrating different special situations.
^&𝐼_𝑎 is the fraction of infected individuals at equilibrium 𝑄^∗_𝑎.
^$Strain 𝑎 will be replaced by the coinfection.
^#Γ≡(𝑅^𝑏₀/𝑅^𝑎₀)[1+((𝛾_𝑎+𝜇)/(𝛾_𝑏+𝜇))𝐼_𝑎𝑅^𝑎₀]⁻¹.
^%The approximation under the condition 𝛾_𝑏+𝜇≫𝛽_𝑎𝐼_𝑎.