Transmissibility of epidemic diseases caused by delay with local proportional fractional derivative

Abstract

Abstract Epidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of change and constant solutions of each compartment are studied with incorporation of a special case of proportional fractional derivative (PFD). In addition, existence and uniqueness of the system are also comprehensively elaborated. Moreover, novel dynamic assessment of the system is carried out in context with the fractional order index. Succinctly, the manuscript accomplishes cyclic epidemiological behavior of the infectious disease due to the delay in treatment of the infected individuals.

Keywords

• Proportional fractional derivative
• Hopf bifurcation
• Stability
• Limit cycles