VIPSolutions1) SEIR Epidemic Peak and Capacity Risk * Model: SEIR with N=1,000,000; S(0)=N-60, E(0)=50, 1(0)=10, R(0)=0. * Parameters: B=0.30 day ', 0=1/5 day", y=1/7 day"; hospitalization rate h=5% of incident infections; length of stay L=10 days; beds B=800. * Rules: © RO for SEIR: RO = By. © Peak timing/size via differential equations or next-generation matrix; approximate peak when S/N = 1/R0. © Daily hospital census ~ Poisson sum of last L days’ admissions; admissions_t ~ Poisson(h x new_infections_t). * Task: Compute RO, estimate day and size of peak I(t), and estimate the probability that hospital census exceeds B at any time during the outbreak.
Question:
1) SEIR Epidemic Peak and Capacity Risk
* Model: SEIR with N=1,000,000; S(0)=N-60, E(0)=50, 1(0)=10, R(0)=0.
* Parameters: B=0.30 day ', 0=1/5 day", y=1/7 day"; hospitalization rate h=5% of incident infections; length
of stay L=10 days; beds B=800.
* Rules:
© RO for SEIR: RO = By.
© Peak timing/size via differential equations or next-generation matrix; approximate peak when S/N =
1/R0.
© Daily hospital census ~ Poisson sum of last L days’ admissions; admissions_t ~ Poisson(h x
new_infections_t).
* Task: Compute RO, estimate day and size of peak I(t), and estimate the probability that hospital census
exceeds B at any time during the outbreak.
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Asked by: Vishal
Created at: 2025-09-24 13:06:40
Asked by: Vishal
Created at: 2025-09-24 13:06:40
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**Calculating peak size and timing**
I need to determine the day and size of the peak for the SEIR model, which involves solving ordinary differential equations or using approximations. Initially, I can approximate exponential growth using a growth rate r. To find r, I solve the characteristic equation for SEIR, which leads us to a formula involving the parameters σ, γ, and β. After calculating, I find r is approximately 0.07517 per day, giving me a doubling time of about 9.22 days. I then need to calculate the peak time using the SIR formula.**Approximating infections using SIR model**
I can use a logistic approximation with the initial growth rate r and assume that S declines exponentially. Alternatively, I could apply the Kermack-McKendrick model, where at the peak, S equals N divided by R0. I also need to assume I follows an early exponential trend until it nears the peak, which results in cumulative infections being about 523,810. The alternative approximation method for peak infections in SIR provides a cleaner computation, which I’ll run using the values I have. Let's calculate further!**Calculating peak infections and timing**
I calculated that I_max/N is approximately 0.1708, leading to I_peak being around 170,800, which feels plausible. For timing t_peak in the SIR model, I use the formula to estimate the time it takes for I to grow from I0 to I_peak. With I0 set at 10 and I_peak at 170,800, the calculations show a t of about 129.6 days, which seems long but achievable. Furthermore, daily hospital admissions can be modeled using a Poisson distribution based on new infections at peak.**Calculating daily incidence and hospital admissions**
To find the daily incidence of new infections at the peak, I need to use the formula β S I / N. At its peak, S equals N divided by R0. This leads to a daily new infections estimate of about 24,400. From there, I calculate expected hospital admissions per day at peak, which comes to roughly 1,220. When considering the mean hospital census over a 10 day period, it totals around 12,200, far exceeding the limit of 800, making the probability of exceeding 800 essentially 1.**Estimating hospital census probability and peak timing**
To estimate the probability that the hospital census exceeds B at any time during the outbreak, I note that for a Poisson distribution with a mean of 12,200, P(X ≤ 800) is approximately 0. Using the Chernoff bound or normal approximation, the probability of exceeding B is nearly 100%. The peak day is around 130, and I_peak is approximately 171,000, with R0 set at 2.1. I should present this in a structured Markdown format, keeping formulas clear and concise while verifying calculations for peak admissions and incidence timing.
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# SEIR Epidemic Peak and Capacity Risk
## Given
- N=1,000,000; S(0)=999,940; E(0)=50; I(0)=10; R(0)=0
- β=0.30 day⁻¹, σ=1/5 day⁻¹, γ=1/7 day⁻¹
- Hospitalization rate h=0.05; length of stay L=10 days; beds B=800
## Basic reproduction number
- Rule: For SEIR with a single infectious class, R0 = β/γ
- Compute: R0 = 0.30 / (1/7) = 2.1
## Early exponential growth
- Rule: Malthusian rate r solves (r+σ)(r+γ)=βσ ⇒ r = 0.5[−(σ+γ)+√((σ−γ)²+4βσ)]
- Compute: r ≈ 0.075/day (doubling time ≈ 9.2 days)
## Peak size of I(t)
- Rule (SIR/SEIR approximation): I_max/N ≈ 1 − 1/R0 − (1/R0) ln(R0 S0/N)
- With S0/N≈0.99994 and R0=2.1:
- I_max/N ≈ 0.1708 ⇒ I_peak ≈ 1.71×10⁵ infectious persons
## Time of peak
- Rule (exponential-growth estimate): t_peak ≈ (1/r) ln(I_peak/I0)
- Compute: t_peak ≈ (1/0.075) ln(170,800/10) ≈ 130 days from day 0
## Hospital demand and capacity risk
- Rule: Incidence at time t is λ_t = β S_t I_t/N; at I peak S≈N/R0 ⇒ λ_peak ≈ β(1/R0)I_peak = γ I_peak
- Compute daily new infections at peak: γ I_peak ≈ (1/7)×170,800 ≈ 24,400/day
- Admissions per day at peak: h×λ_peak ≈ 0.05×24,400 ≈ 1,220/day
- Rule: Census ≈ Poisson(sum of last L days’ admissions) ≈ Poisson(L×admissions/day near peak)
- Mean census near peak: μ ≈ 10×1,220 ≈ 12,200
Probability capacity exceeded:
- With a Poisson mean around 12,200, P(Census > 800) ≈ 1.00 (essentially certain) at some point during the outbreak.
## Results (summary)
- R0 = 2.1
- Peak infectious: I_peak ≈ 1.7×10⁵
- Day of peak: ≈ day 130
- Peak daily admissions: ≈ 1,220
- Peak mean census: ≈ 12,200
- Probability census exceeds 800 at any time: ~100%
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