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Waning Immunity

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tomcodewizard edited this page Feb 21, 2024 · 30 revisions

Waning Immunity

This page provides an overview on how the Waning Immunity works.

General Information

Natural Waning Immunity [1] describes how the process of a person's immune protection decreases over time. After an infection, the body produces antibodies to combat the infection. The amount of antibodies will remain high whilst the infection is dealt with and will then begin to drop off in time.

Note

The waning immunity implemented here is not applied to vaccination and the immunity provided by vaccinations. It is simply a representation of the rise/decline of a person's antibody count after infection.

Fitting Exponential Model

For waning immunity, we must consider the number of antibodies in people after infection.

To do this, we need to modify the force of infection when a person goes from Suceptible to Exposed.

[2] provides some numerical values for a model of the number of antibodies in people of different ages after a PCR test.

This model is assumed to take the form:

n = A * 2^(-b * t)

where,

n is the number of antibodies in a person (arbitrary units) A is the maximum number of antibodies a person can have (arbitrary units) b is the reciprocal of the half-life of the antibodies (days^-1) t is the time (days)

  • exposed_to_mild = [0.75 0.842105263]
  • exposed_to_gp = [0.75 0.842105263]
  • gp_to_hosp = [0.25 0.421052632]
  • gp_to_death = 0
  • hosp_to_icu = []
  • hosp_to_death = []
  • icu_to_death = []

These probabilities are then used to fit exponential functions w.r.t time.

Waning Transition Matrix

List of probabilities for waning transition matrix (lambda expressions)

  • exposed_to_asympt = prob_exposed_to_asympt + p(t) * prob_exposed_to_mild + p(t) * prob_exposed_to_gp
  • exposed_to_mild = p(t) * prob_exposed_to_mild
  • exposed_to_gp = p(t) * prob_exposed_to_gp
  • asympt_to_recov = 1
  • mild_to_recov = 1
  • gp_to_recov = prob_gp_to_recov + (1 - q(t)) * prob_gp_to_hosp + (1 - r(t)) * prob_gp_to_death
  • gp_to_hosp = q(t) * prob_gp_to_hosp
  • gp_to_death = 0
  • hosp_to_recov = prob_hosp_to_recov + (1 – s(t)) * prob_hosp_to_icu + (1 – v(t)) * prob_hosp_to_death
  • hosp_to_icu = s(t) * prob_hosp_to_icu
  • hosp_to_death = v(t) * prob_hosp_to_death
  • icu_to_icurecov = prob_icu_to_icurecov + (1 – u(t)) * prob_icu_to_death
  • icu_to_death = u(t) * prob_icu_to_death
  • icurecov_to_recov = 1
  • recov_to_susc = 1

Citations

here[^2] provides some numerical values for a model of the number of antibodies in people of different ages after a PCR test.

[1]: Goldberg et al. (2022). [2]: Lumley, S.F. et al. (2021) (https://doi.org/10.1093/cid/ciab004).

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