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

Table 1

The dependence of the successful invasion by an indistinguishable new strain on simultaneous transmission rates of coinfections of multiplicity 𝑚=2,3,4 and the efficiency of partial transmission from coinfections (𝜙). We consider the condition where a new indistinguishable strain can invade a host population at equilibrium with 𝑛_𝑒 endemic strains. After the challenge with a new identical strain, a new endemic will be established if condition (5) holds; otherwise, the invader is not successful and the old endemic crashes into one with a single strain. When the invasion is successful, the new endemic is established “invisibly” because the total prevalence, approximated by (6), remains unchanged. Note the prevalence of triple and quadruple infections is vanishingly small, implying their actual nonexistence. Other model parameter values are immunity period = 3 years, transmission rate of single infections (𝛽₁) = 1.0, and infectious period = 2 days; hence 𝑅₀=2.0 for single infection.

𝑛𝑒	𝛽2	𝛽3	𝜙	𝑝	𝑝1	𝑝2	𝑝3	𝑝4	𝑛ec^∗
3	0.0	0.0	0.0–0.49	9.502×10^−4	9.502×10^−4	—	—	—	1
			0.51	9.502×10^−4	9.489×10^−4	1.351×10^−6	1.284×10^−9	6.097×10^−13	4
	0.5	0.0	0.0–0.24	9.502×10^−4	9.502×10^−4	—	—	—	1
			0.26	9.503×10^−4	9.476×10^−4	2.695×10^−6	3.834×10^−9	3.031×10^−12	4
		0.4	0.0–0.24	9.502×10^−4	9.502×10^−4	—	—	—	1
			0.26	9.503×10^−4	9.476×10^−4	2.697×10^−6	6.398×10^−9	5.463×10^−12	4
			0.0–0.24	9.502×10^−4	9.502×10^−4	—	—	—	1^&
			0.26	9.503×10^−4	9.476×10^−4	2.697×10^−6	6.400×10^−9	9.109×10^−12	4^&
2	0.0	0.0	0.0–0.49	9.502×10^−4	9.502×10^−4	—	—	—	1
			0.51	9.506×10^−4	9.494×10^−4	1.201×10^−6	7.612×10^−10	—	3
	0.5	0.0	0.0–0.24	9.502×10^−4	9.502×10^−4	—	—	—	1
			0.26	9.503×10^−4	9.479×10^−4	2.397×10^−6	2.275×10^−9	—	3
		0.4	0.0–0.24	9.502×10^−4	9.502×10^−4	—	—	—	1
			0.26	9.503×10^−4	9.479×10^−4	2.399×10^−6	3.794×10^−9	—	3
1	0.0	—	0.0–0.50	9.502×10^−4	9.502×10^−4	—	—	—	1
		—	0.51	9.502×10^−4	9.493×10^−4	9.020×10^−7	—	—	2
		—	2	9.529×10^−4	9.520×10^−4	9.097×10^−7	—	—	2
	0.5	—	0.0–0.25	9.502×10^−4	9.502×10^−4	—	—	—	1
		—	0.26	9.503×10^−4	9.485×10^−4	1.801×10^−7	—	—	2

^∗𝑛_ec is the number of endemic strains after the challenge of the invader.
^&In this situation we set 𝛽₄=0.4 while 𝛽₄=0 for all other situations.