Impulsive vaccination of an SEIRS model with time delay and varying total population size

Overview

Immunological memory, a pivotal pillar of the adaptive immune system, is responsible for the body's ability to remember and respond more swiftly and effectively to previously encountered pathogens. This remarkable feature is what makes vaccines so effective in preventing diseases.
What is Immunological Memory?
Immunological memory is an integral function of the immune system that allows it to recognize and react more rapidly and effectively to pathogens previously encountered. This feature is...

In the ever-evolving field of public health, statistical analysis serves as a cornerstone for understanding and managing disease outbreaks. By leveraging various statistical tools, health professionals can predict potential outbreaks, analyze ongoing situations, and devise effective responses to mitigate impact. For that to happen, there are a few possible stages of the analysis:

Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is the relative...

Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...

Vaccines are among the most effective tools in preventive medicine, designed to prepare the immune system to recognize and combat infectious agents. By introducing antigens—substances that the immune system identifies as foreign—vaccines stimulate an adaptive immune response that leads to immunological memory. This immunological memory enables the body to mount a faster and more effective response upon future exposures to the actual pathogen.Vaccines can be categorized based on the type of...