Deriving Dead - Application Problem And Data Analysis

Deriving Dead Application Problem and Data Analysis

Deriving Dead - Application Problem and Data Analysis

Please type up the answers and show your work. Include units when necessary.

This assignment involves analyzing data collected from a transmission model in a closed environment with 152 individuals. The data is provided as ordered triples, where each triple corresponds to a stage in the infected population. The first number indicates the stage number, the second indicates the number of newly infected individuals at that stage, and the third indicates the total number of infected individuals up to that point.

Specifically, the data provided is as follows:

  • (0, 1, 1)
  • (1, 1, 2)
  • (2, 2, 4)
  • (3, 3, 7)
  • (4, 5, 12)
  • (5, 11, 23)
  • (6, 18, 25, 66)
  • (8, 33, 99)
  • (9, 25, 124)
  • (10, 18, 142)
  • (11, 4, 146)
  • (12, 3, 149)
  • (13, 1, 150)

The total population in the environment is 152 individuals, which plays a key role in modeling the infected population over time.

Paper For Above instruction

Understanding the progression of infection through a population is essential for developing effective response strategies. This analysis focuses on deriving a mathematical model that describes the spread of infection during an outbreak, based on the provided data and total population constraints.

Analysis of the Data

The data captures the progression of the infected population over discrete stages. The total number of individuals infected at each stage accumulates as shown: starting with 1 infected individual at stage 0, increasing through successive stages to a maximum of 150 infected individuals, within the total population of 152. The incremental infections (newly infected at each stage) help understand the rate of spread at each point.

Calculating the Rate of Infection

To analyze the infection rates, we examine the number of newly infected individuals at each stage. For example, from stage 0 to 1, the new infections are 1; from stage 1 to 2, another 1; from stage 2 to 3, 3 new infections, etc. These increments reveal the acceleration or deceleration of disease spread.

Modeling the Spread

Several models can be used to describe the infection trajectory, including exponential, logistic, or SIR models. Given the data's progression and constraints, the logistic model is a promising candidate because it accounts for the saturation effect as the total infected approaches the total population.

Developing the Logistic Model

The logistic function is commonly expressed as:

I(t) = \frac{K}{1 + e^{-r(t - t_0)}}

where:

  • I(t) is the total infected at time t,
  • K is the carrying capacity (here, total population 152),
  • r is the growth rate,
  • t_0 is the inflection point in time.

Using the data, we estimate the parameters of the logistic model. Notably, the maximum I(t) approaches 150 infected individuals, close to the total population, indicating K ≈ 152.

Estimating Model Parameters

Based on the data at the later stages, where the total infected nears the maximum capacity, we determine t_0 to be approximately when the infections accelerate most rapidly, around stage 5 or 6. The growth rate r can be estimated from the increase between stages, for example, calculating the ratio of increase in infected individuals over the change in stage number, adjusted for the exponential growth phase.

Calculating the Number of Dead

Assuming a certain mortality rate, or mortality proportion, the number of dead individuals can be estimated from the total infected, based on disease-specific fatality rates from relevant epidemiological data. For example, if the case fatality rate (CFR) is 2%, then the number of deaths would be approximately:

Number of deaths = Total infected × CFR

Applying this to the maximum total infected infected (≈150), the expected number of dead individuals in the environment would be approximately:

150 × 0.02 = 3

This estimation assumes the fatality rate remains constant over the course of the outbreak, and that all infected individuals are equally at risk.

Conclusion

Using the provided data, a logistic growth model can effectively describe the infection's progression within the closed environment. Estimation of parameters like growth rate and inflection point allows for predicting future spread and resource needs. The number of dead, based on assumed fatality rates, can be predicted once the total infected count is known. Further refinement of the model would require more precise timing information and additional epidemiological parameters.

References

  • Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
  • Brauer, F. (2017). Mathematical Epidemiology: Past, Present, and Future. Infectious Disease Modelling, 2(2), 113–127.
  • Kermack, W. O., & McKendrick, A. G. (1927). A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A, 115(772), 700–721.
  • Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599–653.
  • Murray, J. D. (2002). Mathematical Biology: I. An Introduction. Springer-Verlag.
  • Li, Q., et al. (2020). Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia. New England Journal of Medicine, 382(13), 1199–1207.
  • Vynnycky, E., & White, R. (2010). An Introduction to Infectious Disease Modelling. Oxford University Press.
  • Keeling, M. J., & Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
  • Rothman, K. J. (2012). Epidemiology: An Introduction. Oxford University Press.
  • Grassly, N. C., & Fraser, C. (2008). Mathematical Models of Infectious Disease Transmission. Nature Reviews Microbiology, 6(6), 477–487.

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