Mediterr. J. Math.
Springer International Publishing AG 2017
Stochastic Epidemic SEIRS Models with
a Constant Latency Period
Xavier Bardina, Marco Ferrante and Carles Rovira
Abstract. In this paper, we consider the stability of a class of deter-
ministic and stochastic SEIRS epidemic models with delay. Indeed, we
assume that the transmission rate could be stochastic and the presence
of a latency period of r consecutive days, where r is a ﬁxed positive
integer, in the “exposed” individuals class E. Studying the eigenvalues
of the linearized system, we obtain conditions for the stability of the free
disease equilibrium, in both the cases of the deterministic model with
and without delay. In this latter case, we also get conditions for the sta-
bility of the coexistence equilibrium. In the stochastic case, we are able
to derive a concentration result for the random ﬂuctuations and then,
using the Lyapunov method, to check that under suitable assumptions
the free disease equilibrium is still stable.
Mathematics Subject Classiﬁcation. 92D30, 60J10, 60H10.
Keywords. SEIRS model, stochastic delay diﬀerential equations,
The mathematical models developed to describe the spread of a communica-
ble disease are both deterministic and stochastic and they may involve many
factors such as infectious agents, mode of transmission, incubation periods,
infectious periods, susceptibility, etc.
A well-known deterministic model in a closed population consisting of
susceptible (S), infective (I ) and recover (R) individuals was considered by
Kermack and McKendrick in . Since then various epidemic deterministic
models have been developed, such as SIR, SIS, SEIR and SEIRS models with
or without a time delay (see e.g., McCluskey [16,17], Huang et al. and
X. Bardina is partially supported by the Grant MTM2015-67802-P from
MINECO/FEDER. M. Ferrante is partially supported by the Grant 60A01-8451
from Universit`a di Padova. C. Rovira is partially supported by the Grant MTM2015-
65092-P from MINECO/FEDER, UE and by Visiting Professor Program 2015 of the
Universit`a di Padova.