Sigmoidal growth models constitute an important class of phenomenological forecasting tools in mathematical epidemiology, designed to project the propagation and temporal trajectory of infectious populations during epidemic outbreaks. Unlike classical compartmental models, these approaches are primarily statistical rather than mechanistic. They capture the characteristic S-shaped epidemic curve, describing how case counts evolve from an initial exponential phase toward eventual saturation. Such models have proven especially valuable for rapid assessment and short-term forecasting during large-scale viral outbreaks.
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📊 Structure and Flow (Non-Compartmental)
Sigmoidal growth models operate without explicit compartmentalization of the population. They do not distinguish between susceptible, exposed, infectious, or recovered individuals.
Instead, these models focus on fitting observed epidemic data—typically cumulative case counts or total infected individuals over time—to a smooth sigmoidal curve. The objective is to represent the overall epidemic trajectory and its transition toward saturation, rather than to model biological transmission mechanisms.
Two broad modeling approaches are commonly employed:
General Sigmoid Models
These models are used to capture monotonic epidemic growth and saturation patterns. Typical formulations include generalized fractional-power models, generalized inverse tangent models, and sigmoidal Boltzmann-type models.
Multiple-Terms Sigmoidal Models
These extensions, often based on logistic growth principles, are designed to incorporate multiple transitions in epidemic dynamics. They allow the growth rate to change over time, reflecting shifts due to interventions, behavioral changes, or policy measures.
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⚛️ Mathematical Formulation
Sigmoidal growth models describe the temporal evolution of the infectious population, denoted by N_inf(t), using nonlinear growth equations that impose an upper bound on epidemic expansion.
The general modeling objective is to represent the rate of change of the infected population as a function of its current size, time, and a set of descriptive parameters:
dN_inf/dt = F(N_inf, parameters, t)
Here, the functional form F is chosen to generate a sigmoidal trajectory. Commonly used formulations include Gompertz-type growth, von Bertalanffy-type growth, cubic polynomial representations, and Boltzmann sigmoidal equations.
The specific differential equations vary by model type and application. However, all formulations share the core objective of reproducing the observed slowing of epidemic growth as saturation is approached.
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🔢 Table 1. Parameter Definitions
| Parameter | Definition |
|---|---|
| N_inf(t) | Total number of infected or cumulative cases at time t |
| Carrying capacity | Maximum epidemic size or saturation level of total cases |
| Growth rate | Describes the intrinsic speed of epidemic expansion |
| Inflection point | Time at which the growth rate transitions from acceleration to deceleration |
| Shape parameter(s) | Controls curvature and symmetry of the sigmoidal trajectory |
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📈 Table 2. Typical Parameter Ranges
| Parameter | Typical Range | Interpretation |
|---|---|---|
| Carrying capacity | Data-dependent | Determined by epidemic scale, population size, and reporting practices |
| Growth rate | Data-dependent | Reflects epidemic velocity during the expansion phase |
| Inflection time | Data-dependent | Indicates the timing of epidemic slowdown |
| Shape parameters | Model-specific | Control flexibility and smoothness of curve fitting |
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🎯 Applicability and Limitations
Primary Application
Sigmoidal growth models are particularly effective for short-term forecasting and for estimating the timing and magnitude of epidemic peaks. They are well suited for rapid situational awareness during emerging outbreaks.
Utility of Extensions
Multiple-terms sigmoidal models are useful for capturing changes in epidemic growth associated with interventions, behavioral adaptations, or policy shifts that alter transmission intensity over time.
Limitations
These models lack explicit biological or epidemiological structure and therefore do not provide mechanistic insight into disease transmission processes. They cannot be used to evaluate intervention mechanisms or population-level susceptibility and are restricted to descriptive analysis of observed epidemic curves.
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📚 Selected References
Konstantinov, M., Konstantinov, K., and Konstantinov, S. Sigmoid models for pandemic growth analysis.
Essadok-Jemai, A., and Al-Rabiah, A. Predictive approaches using multiple-terms sigmoidal transition models.
Ahmadi, A., Fadaei, Y., Shirani, M., and Rahmani, F. Modeling and forecasting epidemic trends using phenomenological growth models.
El Aferni, A., Guettari, M., and Tajouri, T. Mathematical modeling of epidemic waves using Boltzmann-type sigmoidal equations.