**A generalization of the Kermack-McKendrick deterministic epidemic model**

In this paper the Kermack-McKendrick settled model is generalized, introducing associate degree interaction term during which the dependence upon the quantity of infectives happens via a nonlinear delimited operate which can take into consideration saturation phenomena for giant numbers of infectives. associate degree extension of the well-known threshold theorem is obtained, once a stability analysis of the equilibrium points of the system. A numerical example is dispensed well. **[1]**

**Pulse vaccination strategy in the SIR epidemic model**

Theoretical results show that the morbilli ‘pulse’ vaccination strategy are often distinguished from the standard ways in resulting in disease demolition at comparatively low values of vaccination. mistreatment the SIR epidemic model we have a tendency to showed that beneath a planned pulse vaccination regime the system converges to a stable answer with the quantity of infectious people adequate zero. we have a tendency to showed that pulse vaccination results in epidemics demolition if sure conditions concerning the magnitude of vaccination proportion and on the amount of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the fundamental SIR model results in periodic and chaotic dynamics of epidemics. **[2]**

**Dynamical behavior of an epidemic model with a nonlinear incidence rate**

In this paper, we tend to study the world dynamics of a deadly disease model with important dynamics and nonlinear incidence rate of saturated mass-action principle. By polishing off international qualitative and bifurcation analyses, it’s shown that either the quantity of infective people tends to zero as time evolves or there’s a locality specified the illness are going to be persistent if the initial position lies within the region and also the illness can disappear if the initial position lies outside this region. once such a locality exists, it’s shown that the model undergoes a Bogdanov–Takens bifurcation, i.e., it exhibits a saddle–node bifurcation, Hopf bifurcations, and a homoclinic bifurcation. Existence of none, one or 2 limit cycles is additionally mentioned. **[3]**

**A regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion**

In this paper, we tend to gift a regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion. we tend to utilize the Markov semigroup theory to get the existence of a novel stable stationary distribution. we tend to prove that the densities of the distributions of the solutions will converge in L1 to an invariant density underneath sure condition. Moreover, the spare conditions for the extinction of the sickness, which suggests the sickness can die out with chance one, are given in 2 cases. Meanwhile, we tend to acquire a threshold parameter which might be utilised in distinctive the random extinction and persistence of the sickness.** [4]**

**A Multi-Regions SIRS Discrete Epidemic Model With a Travel-Blocking Vicinity Optimal Control Approach on Cells**

In Susceptible-Infected-Removed-Susceptible (SIRS) compartmented models, we will suppose that a removed population has lost its immunity once being well from Associate in Nursing infection, and then, it moves to the inclined compartment. during this paper, we have a tendency to devise a multi-regions SIRS separate epidemic model that describes infection dynamics in regions that area unit connected with their neighbors by any reasonably social science movement. we have a tendency to introduce managements variables into our model to indicate the effectiveness of movements restrictions of the infected people returning from the section of {a region|a neighborhood|an area|a district|a locality|a section|a part|a section} we have a tendency to target by an effect strategy we have a tendency to decision here by the travel-blocking vicinity best control strategy. A gridded surface of coloured cells is conferred let’s say the total domain littered with the epidemic whereas every cell represents a sub-domain or region.** [5]**

**Reference**

**[1]** Capasso, V. and Serio, G., 1978. A generalization of the Kermack-McKendrick deterministic epidemic model. Mathematical Biosciences, 42(1-2), (Web Link)

**[2]** Shulgin, B., Stone, L. and Agur, Z., 1998. Pulse vaccination strategy in the SIR epidemic model. Bulletin of mathematical biology, 60(6), (Web Link)

**[3]** Ruan, S. and Wang, W., 2003. Dynamical behavior of an epidemic model with a nonlinear incidence rate. Journal of Differential Equations, 188(1), (Web Link)

**[4]** A regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion

Zhongwei Cao, Xu Liu, Xiangdan Wen, Liya Liu & Li Zu

Scientific Reportsvolume 9, Article number: 10696 (2019) (Web Link)

**[5]** Abouelkheir, I., Kihal, F., Rachik, M., Zakary, O. and Elmouki, I. (2017) “A Multi-Regions SIRS Discrete Epidemic Model With a Travel-Blocking Vicinity Optimal Control Approach on Cells”, Journal of Advances in Mathematics and Computer Science, 20(4), (Web Link)