Time-Varying Epidemic Transmission in Heterogeneous Networks and Applications to Measles

Abstract

In this paper, we analyze some epidemic models by considering a time-varying transmission rate in complex heterogeneous networks. The transmission rate is assumed to change in time, due to a switching signal, and since the spreading of the disease also depends on connections between individuals, the population is modeled as a heterogeneous network. We establish some stability results related to the behavior of the time-weighted average Basic Reproduction Number (BRN). Later, a Susceptible–Exposed–Infectious–Recovered (SEIR) model which describes the measles disease is proposed and we show that its dynamics is determined by a threshold value, below which the disease dies out. Moreover, compared with models without the Exposed compartment, we can find weaker stability results. A control strategy with an imperfect vaccine is applied, to determine how it could effect the size of the peak. Due to the periodic behavior of the switching rule, we compare the results with the BRN of the model. Some simulations are given, using a scale-free network, to illustrate the threshold conditions found.

A survey on Lyapunov functions for epidemic compartmental models

“…However, in order to increase realism, it is possible to use non-autonomous systems to describe the spread of an infectious disease. This is the case of systems, in which some parameters change in time [ 61 , 62 ], to describe seasonal changes, or in which the state variables depend on the previous state, i.e. the model includes a time delay [ 63 , 64 ].…”

A survey on Lyapunov functions for epidemic compartmental models

“…However, in order to increase realism, it is possible to use non-autonomous systems to describe the spread of an infectious disease. This is the case of systems in which some parameters change in time [42,56], to describe seasonal changes, or in which the state variables depend on the previous state, i.e. the model includes a time delay [4,67].…”