A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

Figures & data

Figure 1. Flow diagram of the model (1). The model consists of five sub-populations: susceptible S, acute A, chronic C, isolated Q and recovered R individuals.

Table

Variable	Description
N(t)	Total population
S(t)	Population of susceptible individuals
A(t)	Population of individuals with acute Hep-C
C(t)	Population of individuals with chronic Hep-C
Q(t)	Population of isolated individuals
R(t)	Population of recovered individuals

Table

Parameter	Description
Π	Recruitment rate
μ	Natural death rate
δa	Disease-induced death rate of individuals with acute Hep-C
δc	Disease-induced death rate of individuals with chronic Hep-C
δq	Disease-induced death rate of isolated individuals
γ	Recovery rate of isolated individuals
ξ	Progression rate from the acute to the chronic stage of Hepatitis C
α	Isolation rate of chronically infected individuals
κ	Natural recovery rate of acutely infected individuals
ψ	Recovery rate of chronically infected individuals
ω	Rate of loss of infection-acquired immunity
β	Effective contact rate
η	Modification parameter for infectiousness of acute individuals
ζ	Modification parameter for infectiousness of isolated individuals

Figure 2. Numerical solution of the deterministic model (1) at {R₀=0.6453}. Initial population:

Π=0.12; γ=0.18; κ=0.2; ω=0.95;

ξ=0.7; α=0.15; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; β=0.1369.

Figure 3. Numerical solution of the deterministic model (1) at {R₀=2.6889}. Initial population: (

(0),

(0),

(0),

(0),

(0))=(600, 20, 60, 12, 10). Π=1; γ=0.18; κ=0.2; ω=0.95;

ξ=0.7; α=0.15; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; β=0.5703.

Figure 4. The dependence of R₀ on some state variables. (a) The contour plot of R₀ as a function of α and γ, (b) R₀ as a function of α and γ, (c) the contour plot of R₀ as a function of α and β and (d) R₀ as a function of α and β. Π=10; γ=0.1; κ=0.7535; ω=0.95;

ξ=0.8; α=0.2; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; β=0.3.

Figure 5. Comparison of the solution to the deterministic model (1) and the numerical means of each discrete random variable from the stochastic model at R₀=2.6889.

Figure 6. The numerical mean of each discrete random variable calculated using 10,000 sample paths. Numerical mean of the discrete random variables (a)

(t), (b)

(t), (c)

(t) and (d)

(t). Initial population: (

(0),

(0),

(0),

(0),

(0))=(600, 20, 60, 12, 10). Π=1; γ=0.18; κ=0.2; ω=0.95;

ξ=0.7; α=0.15; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; β=0.5703.

Figure 7. The variance of each discrete random variable calculated using 10,000 sample paths: (a)

(t), (b)

(t), (c)

(t) and (d)

(t).

Figure 8. The probability distribution of each discrete random variable calculated using 5000 sample paths. (a) Probability distribution of

(t) at {R₀=2.6889}, (b) probability distribution of

(t) at {R₀=2.6889}, (c) probability distribution of

(t) at {R₀=2.6889} and (d) probability distribution of

(t) at {R₀=2.6889}. Initial population: (

(0),

(0),

(0),

(0),

(0))=(600, 20, 60, 12, 10). Π=1; γ=0.18; κ=0.2; ω=0.95;

ξ=0.7; α=0.15; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; β=0.5703.

Figure 9. Mean time to disease extinction plotted as a histogram for three cases: α=0, α=0.15 and α=0.5. Initial population: (

(0),

(0),

(0),

(0),

(0))=(600, 20, 60, 10, 10). Π=1; γ=0.2; κ=0.2; ω=0.95;

ξ=0.8; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; β=0.1880; Δ t=0.0069.

Figure 10. The mean time to disease extinction for two cases in which the basic reproduction number R₀ is slightly greater than unity computed using 5000 stochastic simulations each. Mean time to disease extinction for the cases (a) β=0.439,α=0.50 and R₀=1.01 and (b) β=0.213,α=0.15 and R₀=1.01. Initial population: (

(0),

(0),

(0),

(0),

(0))=(600, 20, 60, 10, 10). Π=1; γ=0.2; κ=0.2; ω=0.95;

ξ=0.8; ψ=0.05; δ_a=0.000233; δ_c=0.00233; δ_q=0.001667; η=0.5; ζ=0.1; Δ t=0.0069.