Dynamics of an SEIR Epidemic Model with Vaccination, Treatment and Saturated Incidence: Threshold Analysis and Numerical Study

Abstract

Vaccination and treatment are the two main tools used to bring an outbreak under control, yet they act at different stages of the infection and are rarely compared inside a single, simple mathematical framework. In this paper we study a deterministic SEIR model in which susceptible people are vaccinated at a constant rate, infectious people are treated at a constant per-capita rate, and the force of infection saturates at high prevalence to reflect behavioural change and limited contact. We derive the control reproduction number Rc with the next-generation method and show that the disease-free equilibrium is globally asymptotically stable when Rc < 1, while a unique endemic equilibrium exists, with an explicit closed form, whenever Rc > 1. The model therefore has no backward bifurcation. A normalised sensitivity analysis ranks the transmission rate, vaccination rate and recovery-related rates as the most influential parameters. Numerical simulations with illustrative parameters show that the combined strategy lowers the epidemic peak by about 85% compared with no intervention, and that treatment is the faster-acting measure while vaccination is the more effective long-run one. The results give a compact, analytically tractable basis for comparing intervention mixes.

Introduction

The text presents a mathematical model for studying the spread and control of infectious diseases using a modified SEIR model. The study combines vaccination, treatment, and a saturated infection rate to understand how these factors influence disease transmission, epidemic persistence, and elimination.

1. Introduction

Mathematical models are important tools for understanding how infectious diseases spread through populations. The SEIR model divides the population into four compartments: susceptible, exposed, infectious, and recovered or immune. Unlike the basic SIR model, the SEIR model accounts for the latent period during which exposed individuals are infected but not yet infectious.

The study extends the traditional SEIR framework by incorporating vaccination, treatment, and saturated incidence. Saturated incidence represents the possibility that infection rates do not increase indefinitely with the number of infectious individuals because of behavioral changes, limited contact, or preventive measures.

2. Model Formulation

The population is divided into four groups:

• Susceptible (S): Individuals who can become infected.

• Exposed (E): Individuals who have been infected but are not yet infectious.

• Infectious (I): Individuals who can transmit the disease.

• Recovered (R): Individuals who have recovered or gained immunity.

The model considers population recruitment, natural deaths, vaccination, disease transmission, progression from exposure to infection, natural recovery, disease-related deaths, and treatment.

The infection rate is represented by the saturated incidence function:

βSI1+αI1+αIβSI?

where ββ is the transmission rate and αα represents the saturation effect. Vaccination reduces the susceptible population, while treatment helps infectious individuals recover more quickly.

3. Positivity and Boundedness

The study establishes that the model has biologically meaningful solutions. Starting with non-negative initial population values, all four population groups remain non-negative and bounded over time.

This means that the model avoids unrealistic negative population sizes and that the total population remains within a mathematically defined limit.

4. Equilibrium and Reproduction Number

The study identifies two possible equilibrium states:

• Disease-free equilibrium: The infection disappears from the population.

• Endemic equilibrium: The infection persists at a stable, positive level.

The control reproduction number is given by:

Rc=βσΛ(μ+ν)(μ+σ)(μ+d+γ+τ)Rc?=(μ+ν)(μ+σ)(μ+d+γ+τ)βσΛ?

The reproduction number indicates whether an infection can spread under the model's conditions.

• If Rc<1Rc?<1, the disease eventually disappears.

• If Rc>1Rc?>1, a unique endemic equilibrium exists.

• If Rc=1Rc?=1, the system is at the threshold between these regimes.

Vaccination and treatment reduce RcRc?, while a higher transmission rate increases it. The saturation parameter αα does not affect the reproduction number because infection levels are initially very low when the disease is introduced.

5. Stability Analysis

The study proves that the disease-free equilibrium is locally and globally asymptotically stable when Rc<1Rc?<1. In practical terms, if vaccination and treatment reduce the reproduction number below one, the disease eventually disappears regardless of the initial number of exposed or infectious individuals.

When Rc>1Rc?>1, the disease-free equilibrium becomes unstable, and numerical simulations suggest that the population approaches the endemic equilibrium. However, the study does not provide a rigorous proof of global stability for that endemic state.

The analysis also shows that the model exhibits a forward bifurcation at Rc=1Rc?=1, meaning that the endemic equilibrium emerges when the reproduction number crosses one. The model does not exhibit the backward bifurcation found in some more complicated vaccination and treatment models.

6. Sensitivity Analysis

Sensitivity analysis examines how changes in model parameters affect the reproduction number.

The reported results indicate that:

• Transmission rate (ββ): A 1% increase produces a 1% increase in RcRc?.

• Vaccination rate (νν): Reduces the susceptible population and lowers RcRc?.

• Treatment rate (ττ): Reduces the infectious period and lowers RcRc?.

• Natural recovery rate (γγ): Also reduces RcRc? by shortening the infectious period.

• Latent progression rate (σσ): Has a relatively small effect under the selected parameter values.

• Disease-related mortality (dd): Has a small negative sensitivity index in the illustrative scenario.

At the selected parameter values, vaccination has a stronger negative sensitivity index than treatment. The findings also emphasize that reducing transmission through preventive measures can contribute significantly to disease control.

7. Numerical Simulations

The study uses Python and an adaptive Runge–Kutta numerical integration method to simulate the model over time. The parameter values are illustrative rather than fitted to a particular disease dataset. The initial population contains 980 susceptible, 10 exposed, 10 infectious, and no recovered individuals.

These simulations are intended to illustrate the mathematical results and examine how vaccination, treatment, and saturation influence epidemic dynamics.

Conclusion

We have analysed an SEIR model with constant-rate vaccination, constant per-capita treatment and a saturated incidence rate. The model is well posed on a biologically feasible region. Its control reproduction number Rc is given in closed form, the disease-free equilibrium is globally asymptotically stable for Rc < 1, and a unique endemic equilibrium with an explicit expression exists for Rc > 1, which excludes backward bifurcation. Sensitivity analysis highlights the transmission rate, the vaccination rate and the recovery-related rates as the most influential parameters. In the numerical experiments, combining vaccination with treatment reduced the epidemic peak by about 85% and drove the infection to extinction, whereas either measure on its own left the disease endemic. The analysis supports the use of mixed strategies and provides explicit threshold formulas that can be used to size them.

References

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Copyright

Copyright © 2026 Dinesh Kumar, Dr. Vandana Malviya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.