Infectious Disease Modeling

Infectious Disease Models

• Infectious disease models simplify a system for analysis
• They capture essential behaviors and incorporate properties of interest
• These models clarify thinking and allow others to examine them
• Mathematical models allow precise, rigorous analysis and are used for quantitative prediction
• All models are simplifications, and may be considered wrong
• Simplifications useful to facilitate calculations

Infectious Disease Modeling Basics

• Models show population-level effects of processes at the individual level
• Risk of infection for an uninfected individual depends on prevalence of infectious individuals, contact rate, and individual infectiousness
• Transmission is a dynamic process, so individual risk can change over time
• More transmission correlates to prevalence of infection and risk
• More transmissions correlate to prevalence of infection increase
• Outbreaks can become extinct
• Infection can be acquired from external sources
• Herd immunity prevents transmission of infection
• No transmission occurs as susceptibles are not infected

Designing Compartmental Models

• Natural infection history considerations include latency, infectious period, and immunity
• Transmission of infection considerations include the directness, indirectness
• Population structure and demography includes stratification by age, sex, and geographic location
• Interventions target parts of the disease transmission process
• No single "correct" model exists for a particular infection; rather, there are levels of complexity

Natural History Modeling

• Divide a population into compartments or categories based on infection stage
• Each compartment contains individuals in different states
• Individuals in the same compartment are considered to have same properties
• These properties represent the average characteristics of individuals in the real world

Building Blocks of Compartmental Models

• Models are constructed from compartments and rates of change
• Compartments contain individuals in each infection state
• Values are stored in state variables describing the system
• Rates of change of numbers in compartments affect infection and recovery
• Rates typically depend on values of state variables, leading to feedback
• Changes in the state of the system as population grows leads to changes in transmission rate

Relationships Between Building Blocks

• Relationships between state variables and rates of change are expressed by functions
• Each compartment contains a tracking variable for individuals in that compartment
• Each involves a differential equation describing the rate of change of its state variable

Flow Diagram Notations

• S stands for Susceptible
• I stands for Infected
• R stands for Recovered
• Each term is a function of state variables so the term value changes as the state variables change
• Death rate of S and R is a background rate
• Death rate of Infected individuals is "background rate" plus disease-induced death rate

Adding Complexity to Models

• Complexity can be added to models
• Add maternal antibodies in the young
• Add incubation and latent periods
• Add asymptomatic infection
• Add infectious period with multiple stages
• Add resolution of infection
• Resolution of infection considerations include death, immunity, or return to susceptibility
• Add immunity considerations
• Immunity considerations include sterilizing, waning or permanent characteristics
• Vector-borne transmission can be added

Simple vs Complex Models

• Simple models may lack disease dynamic detail
• Simple models are suitable for understanding basic disease dynamics
• It is helpful to start with simple model and add complexity as needed

Flows Between Compartments

• Each flow rate is the number of individuals entering or leaving a compartment per unit of time and depends on two things:
• the per-capita rate
• the number of individuals subjected to that per-capita rate [exposed to the hazard]
• The population rate is the product of these (per-capita rate by the number of individuals.

Rate of Progression

• The per-capita rate of recovery is labeled σ
• Population recovery rate is σΙ with rate * number infected
• The per-capita rate of infection of susceptibles is the "force of infection" and is not fixed
• Depends on number Infectious at a point in time and the rate of contact with Susceptible individuals
• Also depends on transmission probability
• The population rate of infection depends on the number of Infecteds and Susceptibles

Death Rates

• The per-capita "background" death rate is labeled μ
• The population death rate of susceptibles is μS
• The population death rate of recovereds is μR
• Death rates of infecteds involve background rate μ + disease-induced death rate α, so their per-capita death rate is (μ + α),
• The population death rate of infected is (μ + α)Ι.

Birth Rates

• The per-capita birth rate is labeled be b
• The no. giving birth is S+I+R, which is the total population size, N = S+I+R, so
• the population birth rate is bN.
• set the per-capita birth rate equal to the per-capita background death rate to maintain a constant population size in the absence of disease
• using different parameters means that there is no requirement to make them equal

Transmission Rate Formula (direct)

• Contact rate is labeled C
• The rate of contacting infectious individuals is cI/N, where;
• I = number of infectious individuals, N = total population size
• I/N = proportion of population infectious
• The rate of transmission from infectious individuals: pcI/N, where:
• p = Probability of transmission when an infectious individual contacts a susceptible
• Force of infection is labeled λ
• Total transmission rate in population: pcSI/N, where;
• S = number of susceptible individuals
• Often β is written in place of pc.
• S, I, N are state variables which can change intrinsically, whilst p, c are constant parameters

Solving the Equations

• The models plot individuals in each compartment change over time.
• Derivatives are specified in differential equations relating to state of the system (i.e. the numbers in each compartment) at any point in time. – To get the lines themselves, differential equations are solved by integration.
• Most models use use computers for numerical integration to get results

SIR Model

• Susceptible individuals move to the infected state, then to the recovered state
• Recovered individuals are immune
• SIR stands for Susceptible Infected Recovered

SIR Quick Example Epidemic

• The rate of spread accelerates and transmits increase number of Infecteds, which;
• increases the force of infection
• increases further rate of spread
• Spreading slows as number of Susceptibles decreases
  • even though force of infection continues to increase
• If Infecteds recover to become immune, epidemic fades unless:
  • New susceptible individuals enter the population, or
  • Immunity wanes, returning individuals to susceptibility

Latent Period

• Simple models assume individual are infectious as soon as they are infected.
• A significant time period may occur between being infected and becoming infectious
• Latently-infected individuals are often called “Exposed”.

Incubation Period

• Incubation period is the time between infection and becoming symptomatic.
• Infections are treated once a person becomes symptomatic.
• Symptoms may occur before an individual becomes infectious or symptoms occur after the person is infectious
• Symptoms and infectiousness may occur together.

Branching Natural Histories

• Infections have different natural histories based on individual characteristics such as:
• age
• sex
• comorbidities
• For example, some people with gonorrhoea develop symptomatic infection, and other asymptomatic infection

Incorporating Behaviour into Models

• Categorize people by their behavioral response and physiological status
• People may need separate compartments for different behavioral categories if physiological characteristics are similar
• Examples of behavior to model is testing rate and treatments

Heterogeneity

• Models represent population in aggregate
• Entities have identical characteristics because they are indistinguishable
• Variation is represented by stratifying population into different groups
• Use individual or agent-based models to track individuals

Parameter Estimation

• Effect needs to be quantified for mathematical modeling
• Models have parameters with numerical values that are either: measured or varied
• Measurement is hard, requires high-quality data
• Variation done across plausible ranges via scenario analysis
• Parameter estimation is the hardest part of modelling