see also:

• pandemics - influenza, coronavirus, Ebola
• epidemiology
• COVID-19 coronavirus (2019-nCoV / SARS-CoV-2)
• Covid-19 modeling for Victoria
• 2009 Infectious Disease Modelling: Estimating epidemic exponential growth rate and basic reproduction number
• Modeling COVID-19 epidemics in an Excel spreadsheet: Democratizing the access to first-hand accurate predictions of epidemic outbreaks

• pandemics spread by contagions generally follow relatively similar quantitative “rules” based upon various features of the contagion, the immunity of the host population, the ease of transmission within the host population, the duration of infectivity, nature of spread (eg. droplets, aerosols, fecal-oral, vectors) etc

• time from exposure to symptoms

• period where an infected person can transmit the infection

• a measure of contagiousness
• the number of secondary infections each infected individual produces
• this will vary with:
  • The rate of contact between individuals in the host population
    • eg. social distancing will reduce R0 if it is droplet spread
  • The probability of the infection being transmitted during contact;
    • ease of transmission
    • level of PPE
    • viral factors
  • The duration of infectiousness

• R = R0 x proportion of population susceptible
• the disease spreads to all the community if it has an R0>1
• R0 needs to be less than 1 to contain an epidemic and modelling suggests the community need to be at least 80% compliant with strict social distancing rules ¹⁾
• the lower the R0 is, the flatter the growth curve of infection and the lower the impact on the healthcare system and overall mortality and morbidity, although it will tend to prolong the epidemic
• see Reproduction Ratio (RO)

• the proportion of the population who are NOT susceptible (ie. vaccinated or immune following infection) which results in effective R value falling below 1
• herd immunity threshold proportion of population = 1 - 1/R0
• also equates to the critical immunisation threshold or the proportion of the population needed to be vaccinated to eradicate the disease

• this is the number of deaths divided by the number of CONFIRMED cases
• it is thus NOT the chance of dying if you catch the disease as this would be much lower if the number of undetected or unconfirmed cases is much high higher than confirmed cases
• CFR is NOT constant, it is determined by many factors including:
  • susceptibility characteristics of a population (eg. age profile of a population)
  • access to medical care (CFR can go up substantially if health care systems become overwhelmed)
  • the time the contagion takes to kill someone which can cause an artificially low rate early if it takes some weeks
  • degree of testing - the more non-fatal cases including asymptomatic or mild illness cases detected by higher levels of testing, the lower the CFR will be

• the probability that any individual in the population will die from the disease;
• calculated by dividing the number of deaths from the disease by the total population.

• this is the chance of dying if you catch the disease and is calculated by number of deaths / number of people infected
• unfortunately, number of people infected has to be estimated in most pandemics as there are usually many untested or unconfirmed cases

• time to hospitalization from onset of symptoms (or from exposure)
• hospitalization rate
  • this will vary depending upon:
    • hospital admission practices (China admitted ALL confirmed cases of Covid-19 to ensure quarantine)
    • rate of severe illness

• length of hospital stay
• recovery time for milder cases

• patients will be in one of the following stages at any point and may progress through the stages if they become exposed
  • Susceptible
  • Exposed
  • Infected
  • Removed

• dS/dt = −(S(t)/N)*(I(t)*R(t)/Tinf​)
• dN/dt = daily new positive cases = (E(t)/Tinc)
• dE/dt ​= (S(t)/N)*I(t)*R(t)/Tinf​​​ − (E(t)/Tinc) = - dS/dt - dN/dt
• dI/dt = E(t)/Tinc - (I(t)/Tinf) = dN/dt - dR/dt - dD/dt
• dR/dt = I(t)/Tinf
• dD/dt = IFR * I(t)
• dS/dt + dE/dt + dI/dt + dR/dt = 0
• S(t) + E(t) + I(t) + R(t) = N
• where, R(t) = R value based on current transmission rates which may change with interventions, Tinf = number days infective, Tinc = incubation period in days, S(t) is number susceptible at time t, I(t) = number infected at time t, E(t) = number exposed at time t, D(t) = deaths from disease, IFR is infection fatality rate, N is initial population number
• see also:
  • https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology
  • http://gabgoh.github.io/COVID/index.html
  • https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)30260-9/fulltext
  • https://towardsdatascience.com/infectious-disease-modelling-beyond-the-basic-sir-model-216369c584c4
  • https://www.pnas.org/content/117/29/16732
  • https://www.nber.org/papers/w26901.pdf
  • Teslya et al. Compartment model for vaccination vs social distancing compliance with equations

• dS/dt = −(S(t)/N)*Infectious*β(t)
  • where,
    • β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject
    • β(t) is the transmission rate over time = R(t)*γ = R(t)/Tinf
    • γ = the rate constant for the transition from infectious to recovered (hence this rate is γ * number infectious) and is equal to 1/(duration person is infectious)
    • Force of infection(t) = β(t) * I(t)

• assuming those who have been infected develop some long lasting immunity, epidemic infections are likely to transition to endemic infections which primarily infect children²⁾
  • the IFR of such endemic states thus generally reflects the IFR of children with that infection in the initial epidemic
• immunity (whether natural or vaccine induced) has several separate, time-dependent features which impact how long it takes to transition to endemicity in addition to the R0 of the pathogen:
  • sterilizing immunity - this can prevent a pathogen from replicating, thereby rendering the host refractory to reinfection, with coronaviruses, this may wane over the year following infection
  • transmissibility of reinfections
  • attenuation of pathology due to reinfection (ie. reinfections may be milder with lower IFR)
• for an R0 of 2, the transition to a mainly paediatric endemicity may take 10yrs, while it may occur in only 3yrs if R0 is 4 or if the sterilizing immunity wanes rapidly