🔬Mathematical Biology Unit 5 – Epidemiological Models: SIR and SIS

Key Concepts and Definitions

• Epidemiological models mathematical frameworks used to study the spread of infectious diseases within a population
• Compartmental models divide the population into distinct groups (compartments) based on their disease status (Susceptible, Infected, Recovered)
• SIR model (Susceptible-Infected-Recovered) assumes individuals move from Susceptible to Infected to Recovered compartments
  • Susceptible individuals have not been infected but are vulnerable to the disease
  • Infected individuals have contracted the disease and can spread it to others
  • Recovered individuals have recovered from the disease and gained immunity
• SIS model (Susceptible-Infected-Susceptible) assumes individuals move from Susceptible to Infected and back to Susceptible compartments
  • Recovered individuals do not gain immunity and can become susceptible again
• Basic reproduction number ($R_0$) average number of secondary infections caused by one infected individual in a fully susceptible population
• Herd immunity occurs when a significant portion of the population becomes immune, reducing the likelihood of disease spread

Historical Context and Development

• Epidemiological models have roots in the early 20th century with the work of public health pioneers like William Hamer and Ronald Ross
• In 1927, William Kermack and Anderson McKendrick introduced the SIR model, laying the foundation for modern epidemiological modeling
• The SIR model was initially used to study the spread of measles and other childhood diseases
• The SIS model was developed to study diseases where individuals do not gain permanent immunity after recovery (sexually transmitted infections, common cold)
• Advancements in mathematical biology and computing power have enabled the development of more complex and realistic epidemiological models
• Recent outbreaks (SARS, H1N1, COVID-19) have highlighted the importance of epidemiological modeling in public health decision-making

SIR Model Basics

• The SIR model divides the population into three compartments: Susceptible (S), Infected (I), and Recovered (R)
• Individuals move from S to I at a rate proportional to the number of infected individuals and a transmission rate ($\beta$)
• Infected individuals recover and move to the R compartment at a rate proportional to the number of infected individuals and a recovery rate ($\gamma$)
• The total population size (N) remains constant, assuming no births, deaths, or migration ($S + I + R = N$)
• The model is described by a system of ordinary differential equations (ODEs) that govern the rates of change of S, I, and R over time
  • $\frac{dS}{dt} = -\beta SI$
  • $\frac{dI}{dt} = \beta SI - \gamma I$
  • $\frac{dR}{dt} = \gamma I$
• The basic reproduction number for the SIR model is given by $R_0 = \frac{\beta}{\gamma}$

SIS Model Fundamentals

• The SIS model assumes individuals move from Susceptible (S) to Infected (I) and back to Susceptible (S) compartments
• Infected individuals recover but do not gain immunity, becoming susceptible again
• The total population size (N) remains constant ($S + I = N$)
• The model is described by a system of two ODEs:
  • $\frac{dS}{dt} = -\beta SI + \gamma I$
  • $\frac{dI}{dt} = \beta SI - \gamma I$
• The basic reproduction number for the SIS model is also given by $R_0 = \frac{\beta}{\gamma}$
• If $R_0 > 1$, the disease persists in the population, and if $R_0 < 1$, the disease dies out
• The SIS model is suitable for studying diseases with no long-lasting immunity (gonorrhea, chlamydia)

Mathematical Formulation and Analysis

• Epidemiological models are formulated using ODEs that describe the rates of change of compartment sizes over time
• The SIR model ODEs are:
  • $\frac{dS}{dt} = -\beta SI$
  • $\frac{dI}{dt} = \beta SI - \gamma I$
  • $\frac{dR}{dt} = \gamma I$
• The SIS model ODEs are:
  • $\frac{dS}{dt} = -\beta SI + \gamma I$
  • $\frac{dI}{dt} = \beta SI - \gamma I$
• Equilibrium points (steady states) are found by setting the ODEs equal to zero and solving for S, I, and R
  • Disease-free equilibrium (DFE) occurs when I = 0
  • Endemic equilibrium (EE) occurs when I > 0
• Stability analysis determines the long-term behavior of the system around equilibrium points
  • If the DFE is stable, the disease will die out
  • If the EE is stable, the disease will persist in the population
• The basic reproduction number ($R_0$) is a threshold value that determines the stability of equilibrium points
  • If $R_0 < 1$, the DFE is stable, and if $R_0 > 1$, the EE is stable

Model Parameters and Assumptions

• Transmission rate ($\beta$) represents the probability of disease transmission per contact between a susceptible and an infected individual
  • Depends on factors such as population density, social behavior, and disease characteristics
• Recovery rate ($\gamma$) represents the rate at which infected individuals recover and move to the recovered (SIR) or susceptible (SIS) compartment
  • Depends on factors such as the disease's natural history and available treatments
• The SIR and SIS models assume homogeneous mixing, meaning all individuals have an equal chance of contacting others
• The models also assume that the population size remains constant, with no births, deaths, or migration
• More advanced models can incorporate additional compartments (Exposed, Quarantined) and heterogeneous mixing patterns
• Parameter estimation is crucial for accurate modeling and can be done using statistical methods and real-world data

Applications in Real-World Scenarios

• Epidemiological models have been used to study the spread of various diseases (influenza, measles, COVID-19)
• Models can help estimate the potential impact of public health interventions (vaccination, social distancing, quarantine)
  • Vaccination reduces the susceptible population, lowering the effective reproduction number
  • Social distancing reduces the contact rate between individuals, lowering the transmission rate
• Models can guide resource allocation and decision-making during outbreaks
  • Predicting hospital bed demand, ventilator requirements, and healthcare staffing needs
• Real-time data can be used to update model parameters and improve predictions
• Models have been used to develop optimal vaccination strategies and determine the level of herd immunity needed to control a disease

Limitations and Criticisms

• Epidemiological models are simplifications of complex real-world systems and may not capture all relevant factors
• Models are sensitive to parameter values, which can be difficult to estimate accurately
  • Inaccurate parameter estimates can lead to incorrect predictions and conclusions
• Homogeneous mixing assumption may not hold in real-world populations with complex social networks and heterogeneous contact patterns
• Models may not account for individual-level variations in susceptibility, infectiousness, and behavior
• Stochastic effects (random fluctuations) can be important in small populations or during the early stages of an outbreak
• Models should be used in conjunction with other sources of information (clinical data, expert opinion) to inform decision-making

Advanced Topics and Extensions

• Age-structured models incorporate age-specific contact patterns and disease outcomes
  • Relevant for diseases with age-dependent severity (COVID-19) or vaccination strategies
• Spatial models consider the geographic spread of diseases and the impact of human mobility
  • Metapopulation models divide the population into subpopulations with different characteristics
• Network models explicitly represent the contact structure between individuals using graphs
  • Can capture heterogeneous mixing patterns and the role of superspreaders
• Stochastic models incorporate random fluctuations in disease transmission and recovery processes
  • Relevant for small populations or rare events (disease extinction)
• Multi-strain models consider the interaction between different strains of a pathogen
  • Relevant for studying the evolution of drug resistance or the impact of cross-immunity
• Coupled disease-behavior models incorporate the feedback between disease dynamics and human behavior
  • Can capture the impact of risk perception, information spread, and adherence to public health measures