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In the current study, we have formulated an S I epidemic model incorporating the influence of Allee effect on the dynamics of the model system in space and time. During the spread of the disease, inhibition among the susceptible and infected individuals is seen, which is measured using the Crowley–Martin type incidence function. Extensive analysis explores the system's stability and different bifurcation scenarios, including Hopf, saddle-node, transcritical, and Bogdanov–Takens. We also examine how these bifurcations react to parameter variations within the proposed model. The temporal model has further been extended to a spatial one to investigate the impact of the Allee effect on the formation of patterns for different values of the cross-diffusion coefficient. The well-posedness of the stationary solution and the global stability of the spatial system are derived. Also, the Turing instability, Hopf, and Turing bifurcation are calculated considering the transmission rate as critical parameter. In a heterogeneous environment, the spatial distribution of the susceptible population shows a complex structure with flat tabular surface and a few almost circular holes in surface plots for both strong and weak Allee effects. To further explore the role of Allee effect on pattern development for various cross-diffusion coefficient values, the temporal model has been spatially extended. Additionally, the transmission rate is taken into account while calculating the Turing instability and Hopf and Turing bifurcations. We ran numerical simulations to validate our analytical results for both spatial and nonspatial models. Our current model system can explain the recent occurrence of respiratory distress syndrome in endangered penguin species. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to explore a new delayed epidemiological model assuming that the disease is transmitted among the susceptible population and possessing nonlinear incidence function along with a saturated treatment rate. Due attention is paid to the positivity and boundedness of the solutions and the bifurcation of the dynamical system as well. Basic reproduction number is being calculated, and considering the latent period as a bifurcation parameter, it has been examined that a Hopf-bifurcation occurs near the endemic equilibrium point while the parameter attains critical values. We have also discussed the stability and direction of Hopf-bifurcation near the endemic equilibrium point, the global stability analysis and the optimal control theory. We conclude that the system reveals chaotic dynamics through a specific time-delay value. Numerical simulations are being performed in order to explain the accuracy and effectiveness of the acquired theoretical results. [ABSTRACT FROM AUTHOR]