Tranmission 2: Models

Question 1

Modelling and uses

Answer 1

Modelling is a common approach to understand the spread of disease within a population.

Models can be created with different compartments can different parameters to represent the dynamics of a pathogen.

They are often used to simulate and predict the spread of infectious diseases, assess the impact of interventions such as vaccination or social distancing measures, and inform public health policy decisions

Question 2

SIR model

Answer 2

Predator- prey model with damped oscillations as the susceptible population is recovered due to births/ deaths or waning immunity. Eventually equilibirum is reached where compartments remain constant.

Susceptible -> Infected -> recovered

Parameters:
- Lambda or beta x I = force of infection
- >B= infectivity per infected person
- Sigma= rate of recovery (1/sigma= duration of infection)

Optional:
- w= rate of waning immunity
- d= death rate
- b= birth rate

Examples: Measles/ Mumps/ Rubella
- Measles R0 estimated to be between 12 and 18 making it very hard to control -> high HIT -> vaccination is possible

Vaccination is possible as they mimic natural immunity

Equations:
dS/dt = - betaSI - muS + (muS+muI+muR) + (wR)
dI/dt = betaSI - sigmaI - muI
dR/dt = sigmaI - muR - (wR)

Question 3

SI model

Answer 3

Coloniser model explaining chronic infections where individuals do not leave the infected compartment.

Susceptible-> Infected

parameters:
Gamma= rate of infection

Vaccines are unsuccesful as there is no natural immunity to be mimiced (pathogen evades immune response)

SI Equations:
dS/dt= -BSI
dI/dt= BSI

SIS equations:

dS/dt= -BSI + sigmaI
dI/dt= BSI - sigmaI

Question 4

More complex models with many compartments

Answer 4

Many more compartments can be added especially when there are multiple strains.

Example: Influenza
- High level of immune evasion leading to multiple strains circulating.
- intermediate levels of cross immunity leads to a chaotic strain system.
- Complex models are used to describe dynamics.

Example: Dengue
- Mosquito born virus leading to Dengue fever
- Currently no treatment/ vaccine but early detection and care reduces mortality.
- 4 serotypes -> DEN-1, DEN-2, DEN-3, and DEN-4 (short lived, high levels of cross immunity)
- Many strains can co-exist and infection by multiple strains is possible

Question 5

SEIRV

Answer 5

Can make models increasingly more complex

Example: Chicken pox
- Exposed: incubation period before infectious
- Vaccinated: individuals can be removed from susceptible into vaccinate population.

Question 6

herd immunity threshold

Answer 6

This is the proportion of immune individuals where R=1.

Below this threshold each infected individual will infect less than 1 other individual stopping the spread of the virus.

p(immune) at HIT= 1-1/R0

Increasing R0 increases the HIT, so an endemic is possible with a larger number of immune/ unsusceptible individuals.

Question 7

R0 and Rt

Answer 7

R0 is the average number of secondary cases generated by a primary case in a totally susceptible population.

R0= B x D
(infectiousness x duration of infection (1/sigma)

R0= Beta/Sigma
- Higher infectiousness and lower rate of recovery increases R0

Rt is the number of infections from an infected individual at time T during the epidemic.

Rt= R0 x P(S)
Rt= B(t) x D(t)

R0> 1 for an infection to take of
Infection occures if number of infected increased:
1> BSI- Sigma I
BSI/Sigma>I

I is 0 and sigma is 1 at the start of infection so:

0>Beta/Sigma (=R0)

Question 8

SUstained oscillations

Answer 8

Often endemics are sustained

1) demographic stochasticity

• In small populations, all the immune individuals may die leading to a fully susceptible population so the oscillaitions are not damped.
• Example: Study looked at Measles epidemics in cities of different sizes and found that in smaller cities, the epidemics are more scattered with alrge periods of no tranmission.

2) Seasonality

Example: Measles has biennial epidemics

Question 9

Seasonality

Answer 9

Seasonality leads to sustained oscillations due to changing R0 values through the seasons. Increasing R0 increases the herd immunity threshold levels -> HIT= (1-1/R0) so increasing R0 leads to a higher HIT.

Exmaple: Measles

Factors:

o Effect of seasons on parasite physiology/survival
- Temperatur, Humidity, UV exposure and viruses
- Example: Norovirus, a common cause of gastroenteritis, can persist on surfaces for extended periods under cooler temperatures

o Effects of seasons on host physiology
- Lower temperatures reduces immunity (increases duration of infection and R0)
- Allergy season may trigger immune response in individuals compromising host ability to fight infection.

o Effects of seasons on vector physiology
- Mosquito behvaior (e.g. disitribution and breeding in rainy season)

o Effect of seasons on host behaviour
- Inside when cold
- School terms
- Travelling at Christmas
- Summer picnics/ BBQ (food handling and warm temperatures)

Question 10

Summary

Answer 10

Models used to understand the dynamics of disease and provide key insights for disease control (e.g. the effect of vaccination)

Can make them progressively more complicated adding compartments, parameters and factors like cross immunity (e.g. Influenza, Dengue).

HIT

R0

Seasonality

Question 11

Multi strained models

Answer 11

When introducing multiple strains into models, cross immunity can be introduced using the parameter gamma

gamma: the degree of immunological cross protection conferred by exposure to any related antigenic type.

Different dynamics depending on the level of cross immunity.

HIgh: Discrete strain system (dominants strains, intermediate strains and extinct strains)
- Depends on how many alleles determining their epitopes they have in common (e.g. if there is 2, the dominant strain has none in common with the other strains therefor not competing).
- Example: potentially measles

Intermediate: Chaotic strain system (sequential replacement)
- Example: Streptococcus pneumoniae
- -> Gram Positive bacteria with 90 different capsule types leading to different strains with co-exist.
- -> 800,000+ annual deaths under 5
- Example: Influenza

Low: no strain structure
- Example: HIV

Question 12

COVID-19 and seasonality

Answer 12

Disease dynamics depends on when virus enters the system.

New York: COVID hits when HIT is high
-> High first peak with lower later peaks
->HIT is high so there is a large first peak leading to high levels of immunity.
->High HIT = high R0 (e.g. in the winter)

Arizona: COVID hits when HIT is low
-> Multiple peaks
-> HIT is low so it cannot cause such a large initial epidemic, but later epidemics are possible.