Foundational Compartmental Models (Deterministic, ODE)

🏥 The Colonization–Infection Hospital Model: Dynamics of Nosocomial Spread 📈

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🏗️ Conceptual Overview Transmission dynamics within hospitals differ fundamentally from those in the community. High patient turnover, intensive contact with healthcare workers, and the presence of asymptomatic carriers create conditions in which pathogens can persist and spread silently. The Colonization–Infection Hospital Model is a specialized compartmental framework designed to capture these features by explicitly distinguishing … Read more

📉 The Closed Population SIR Model: Dynamics of a Single Epidemic Wave 📉

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🏗️ Conceptual Overview Within mathematical epidemiology, the Closed Population SIR Model is the canonical framework for analyzing acute, short-term epidemic outbreaks. The defining assumption is that the epidemic unfolds on a time scale that is short relative to host demographic processes. As a result, births, natural deaths, and migration are neglected, and the total population … Read more

🩸 The Catalytic Model: Decoding the Force of Infection from Serology 📈

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🏗️ Conceptual Overview In mathematical epidemiology, direct observation of transmission intensity over long time horizons is often impossible. The Catalytic Model provides a principled method to infer the force of infection (λ)—the per capita rate at which susceptible individuals acquire infection—using cross-sectional serological data stratified by age. Rather than tracking infections forward in calendar time, … Read more

📉 The Capasso–Serio Model: Modeling Saturated Incidence and Behavioral Adaptation 📈

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🏢 Conceptual Overview Classical epidemic models commonly assume bilinear (mass-action) transmission, where the incidence rate grows proportionally with the product of susceptible and infectious individuals. The Capasso–Serio model relaxes this assumption by introducing a saturated (nonlinear) incidence function, capturing situations in which transmission does not increase indefinitely as the number of infectious individuals grows. Such … Read more

📉 The Bilinear Incidence SIR Model: The Foundations of Mass-Action Kinetics 📈

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🏗️ Conceptual Overview In mathematical epidemiology, the Bilinear Incidence SIR Model, commonly known as the Mass-Action SIR model, represents the foundational deterministic framework for modeling infectious disease transmission. It assumes a homogeneous, well-mixed population in which the rate of new infections is proportional to the product of the number of susceptible individuals and the number … Read more

🐛 The Anderson–May Macroparasite Model: Dynamics of Helminth Infection

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📈 Conceptual Overview In contrast to microparasite models, which track whether hosts are infected or not, the Anderson–May macroparasite model focuses on infections where disease severity and transmission depend on the parasite burden within individual hosts. This framework is essential for helminth infections such as hookworm, schistosomiasis, and lymphatic filariasis, where morbidity increases nonlinearly with … Read more

🔄 The SIRS Compartmental Epidemic Model: Temporary Immunity and Endemic Cycles

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The Susceptible–Infectious–Recovered–Susceptible (SIRS) model extends the classic SIR framework by incorporating temporary immunity. After recovering, individuals transition to the Recovered (R) class, where they are immune for a period before returning to the Susceptible (S) pool. This reinfection cycle is ideal for modeling diseases like seasonal influenza, respiratory syncytial virus (RSV), or the common cold, … Read more

🔄 The SIS Compartmental Epidemic Model: Reinfection Without Immunity

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The Susceptible–Infectious–Susceptible (SIS) model is a classical framework used to describe the spread of infectious diseases that do not confer lasting immunity. Individuals transition between being susceptible (S) and infectious (I), with recovered individuals immediately returning to the susceptible class. This structure is appropriate for non-immunizing infections, such as the common cold, gonorrhea, or certain … Read more

🧪 The SEIR Compartmental Epidemic Model: Structure, Dynamics, and Extensions

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The Susceptible–Exposed–Infectious–Removed (SEIR) model is a classical and widely used framework in mathematical epidemiology. It extends the foundational SIR model by incorporating a latent (incubation) period during which individuals are infected but not yet infectious. This makes it more biologically realistic for diseases like measles, COVID-19, and influenza. The model has been extensively analyzed in … Read more

🦠 The Susceptible–Infectious–Removed (SIR) Model

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The Susceptible–Infectious–Removed (SIR) model is the foundational framework in mathematical epidemiology. Originally proposed by W. O. Kermack and A. G. McKendrick in 1927, this model provides a deterministic mechanism to analyze how infectious diseases evolve within a population over time. 1. Core Compartmental Structure The model partitions the total population (N) into three mutually exclusive … Read more