In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current $I_{app}$
I
a
p
p
, the maximal conductance of the M-current $g_{M}$
g
M
and the conductance of the leak current $g_{L}$
g
L
. We give precise conditions for the model that ensure the existence of a Bogdanov–Takens (BT) point and show that such a point can occur by varying $I_{app}$
I
a
p
p
and $g_{M}$
g
M
. We discuss the case when the BT point becomes a Bogdanov–Takens–cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package . We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.
In this paper, the global dynamics of a periodic disease transmission model with two delays in incubation and asymptomatic carriage periods is investigated. We first derive the model system with a general nonlinear incidence rate function by stage-structure. Then, we identify the basic reproduction ratio [Formula: see text] for the model and present numerical algorithm to calculate it. We obtain the global attractivity of the disease-free state when [Formula: see text] and discuss the disease persistence when [Formula: see text]. We also explore the coexistence of endemic state in the nonautonomous system and prove the uniqueness with constants coefficients. Numerical simulations are provided to present a case study regarding the meningococcal meningitis disease transmission and discuss the influence of carriers on [Formula: see text].
In this paper, we study the dynamics of an infectious disease in the presence of a continuous-imperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number
. Then, we obtain the threshold dynamics of the system with respect to
by studying the global stability of the disease-free equilibrium when
and the system persistence when
. For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of
, construct approximate stability charts for the endemic equilibrium over different parameter spaces and explore the sensitivity of the proposed model solutions.
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