Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls

Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone incidence rate

“…Due to the existence of time delay, we have used the method in [ 35 ] to obtain an explicit expression for the effective reproduction number ( ) which gives the actual number of secondary infections per infectious person at any time [ 16 , 36 ]. Then we have obtained the threshold dynamics of the system with respect to : (i) we have shown that the disease-free equilibrium is globally stable when ; and (ii) we have discussed the system persistence and the coexistence of endemic equilibrium when .…”

“…Denote the infection force function by . Following [ 16 , 31 ], we assume that the infection force function decreases when the number of infectious individuals increases since the individuals tend to reduce the number of contacts among them per unit time when they are under intervention policies. Consequently, we consider the following assumptions: …”

“…In system ( 2 ), when , the disease-free equilibrium is always exists where The equations for the diseased classes E and I in the linearized system of ( 2 ) about can be rewritten as where Let be the number of classes E ( t ) and I ( t ) at , then from ( 6 ) the distribution of the remaining population of classes E ( t ) and I ( t ) at time is The total number of newly infected individuals is due to the nonsingularity of the matrix . Then it follows that, the next infection operator is In the literature (see e.g., [ 35 ]), the reproduction number for system ( 2 ) is the spectral radius of the matrix , which is Since we introduce a vaccination program in system ( 2 ), is called the effective reproduction number which gives the actual number of secondary infections per infectious person at any time [ 16 , 36 ]. Biologically, is the time spent as an infectious individual and is the survival rate of infected individual in latent period.…”

A time-delayed SVEIR model for imperfect vaccine with a generalized nonmonotone incidence and application to measles

“…They played a major role in the eradication of many infectious diseases completely, such as smallpox, or partially, like measles [2] . Many authors in the literature have studied the dynamics of epidemics models with different types of vaccination schedules [3] , [4] , [5] , [6] , [7] . For instance, in [3] , the dynamics of cholera epidemics with impulsive vaccination have been studied.…”

The dynamics of a delayed generalized fractional-order biological networks with predation behavior and material cycle

“…In the past few decades, fractional calculus theory has been improved significantly and has been successfully applied to various research fields [7,16,23,26,31,33,34,36]. In fact, most population systems have long-term memory.…”