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Waning Immunity

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Abbie Evans edited this page Feb 26, 2024 · 30 revisions

Waning Immunity

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

General Information

Natural Waning Immunity (Goldberg et al. (2022)) 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.

This is achieved by allowing Removed people to return to the Susceptible compartment.

Furthermore, the probabilities of moving between compartments and the force of infection are modified depending on the person's modelled antibody levels.

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; we do this with an exponential model

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

Lumley, S.F. et al. (2021) 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 after the maximum number of antibodies (days)

The probabilities of people travelling between compartments are recorded for people who are 'partially waned' and 'fully waned'.

These probabilities are given below and are used to fit the exponential model:

  • exposed_to_mild
  • exposed_to_gp
  • gp_to_hosp
  • gp_to_death
  • hosp_to_icu
  • hosp_to_death
  • icu_to_death

Waning Transition Matrix

The probabilities for the waning transition matrix are calculated using the formulas (lambda expressions) below:

  • exposed_to_asympt = prob_exposed_to_asympt + (1 - p(t)) * prob_exposed_to_mild + (1 - 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
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