An evolutionary model of influenza A with drift and shift

Figures & data

Table 1. List of dependent variables.

Notation	Meaning
B(t)	Number of susceptible birds at time t
Y(t)	Number of H5N1-infected birds
P(t)	Total population size of birds
S(t)	Number of susceptible individuals at time t
I(t)	Number of individuals infected with seasonal human influenza
r(a, t)	Density of recovered individuals who are a amino acid substitutions away from the strain at time t
R(t)	Number of recovered individuals from seasonal human influenza
I b(t)	Number of humans infected with the bird-to-human transmittable flu strain
J(t)	Number of individuals jointly infected with seasonal human flu and the bird-to-human transmittable flu
Z(t)	Number of individuals infected with the pandemic strain at time t
N(t)	Total human population size at time t

Table 2. List of parameters.

Notation	Meaning
	Birth/recruitment rate into the domestic bird population
μb	Per capita natural death rate for domestic birds
νb	Per capita disease-induced death of domestic birds
βb	Transmission coefficient of avian influenza in domestic birds
Λ	Birth/recruitment rate into the human population
μ	Per capita natural death rate
β	Transmission coefficient of human influenza A strains to susceptibles
γ(a)	Transmission coefficient of human influenza A to recovered individuals
β Z	Transmission coefficient of the pandemic flu strain for susceptible and recovered individuals
k	Rate of amino acid substitutions in the virus population
β Y	Transmission coefficient of bird-to-human transmittable avian flu strain
β J	Transmission coefficient of bird-to-human transmittable avian flu strain for those infected with human influenza A
α	Per capita recovery rate from the class I
ρ	Per capita mutation rate of the bird-to-human transmittable strain into human-to-human transmittable strain
ν	Per capita death/recovery rate of infected with bird-to-human flu strain
ν J	Per capita death/recovery rate of jointly infected individuals in class J
ν Z	Per capita death rate of infected with pandemic flu strain

Figure 1. The left figure shows a graph of the function F(I) in the case ℛ₁>1. The graph clearly intersects the horizontal line y=1 at around I*≈ 3.5. The right figure shows a graph of the function F(I) in the case ℛ₁<1. The graph clearly intersects the horizontal line y=1 at two points I*₁ ≈ 0.1 and I ₂*≈ 3.0. The parameters for the two figures are taken as Λ=13.3 births per year, μ=0.0133 (years)⁻¹, α=52 (years)⁻¹, k=3.6 (years)⁻¹, γ(a)=γ a with γ=0.005 (amino acid substitution)⁻¹ (years)⁻¹ (person)⁻¹. The left figure has β=0.07 cases per person per year, while the right figure has β=0.02 cases per person per year. The reproduction numbers for the two figures are ℛ₁=1.34 for the left figure, and ℛ₁=0.38 for the right figure.

Table 3. List of reproduction and invasion reproduction numbers and their interpretation.

RN/IRN	Explanation
	Reproduction number of human influenza
	Reproduction number of avian influenza
	Reproduction number of pandemic influenza
	IRN of pandemic strain at equilibrium of avian strain
	IRN of human strain at the equilibrium of pandemic strain
	IRN of pandemic strain at the equilibrium of human strain
	IRN of human flu at eq. of bird and pandemic strains
	IRN of human flu at eq. of the avian flu strain (ρ=0)
	IRN of pandemic flu at eq. of bird and human flu (ρ=0)

Table 4. List of equilibria and conditions for existence.

Equilibrium	Notation	Conditions for existence	Comments
Human flu	ℰ1
Bird flu
Pandemic flu	ℰ2
Bird + Pandemic	ℰ3	,
Human + Pandemic	ℰ4	, ,
		,
Bird + Human	ℰ6	, ,	ρ=0
All three	ℰ5	, ,
All three		,
All three	ℰ7	, ,
All three		, , ,	ρ=0

Figure 2. The left figure shows parametric plot in the (γ I*,αγ I*) plane. The right figure shows α as a function of w. The plots are made with w∈[π/32, π/16].

Table 5. List of parameter values for simulation. See 25 for discussion.

Parameter	Value
Λ	1000/365 (×10^5) people/day
α	1/5.068732 days^−1
γ	0.000115785 people^−1days^−1
β Y Y*	0.00000001189 days^−1
μ	1/(65*365) days^−1
A k	365 days
ν	1/20 day^−1

Figure 3. The left figure shows oscillations in I(t). We note that the period of the oscillations is 365 days. The scale of the y-axis gives the number of infected people when multiplied by 10⁵. The right figure shows the resulting oscillations in I _b(t).

Martcheva , M. “ Avian influenza: modeling and implications for control ” . (in preparation)

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