Nowadays, there are a variety of descriptive studies of available clinical data for coronavirus disease (COVID-19). Mathematical modelling and computational simulations are effective tools that help global efforts to estimate key transmission parameters. The model equations often require computational tools and dynamical analysis that play an important role in controlling the disease. This work reviews some models for coronavirus first, that can address important questions about the global health care and suggest important notes. Then, we model the disease as a system of differential equations. We develop previous models for the coronavirus, some key computational simulations and sensitivity analysis are added. Accordingly, the local sensitivities for each model state with respect to the model parameters are computed using three different techniques: non-normalizations, half normalizations and full normalizations. Results based on sensitivity analysis show that almost all model parameters may have role on spreading this virus among susceptible, exposed and quarantined susceptible people. More specifically, communicate rate person–to–person, quarantined exposed rate and transition rate of exposed individuals have an effective role in spreading this disease. One possible solution suggests that healthcare programs should pay more attention to intervention strategies, and people need to self-quarantine that can effectively reduce the disease.

Calculating analytical approximate solutions for non-linear infectious disease models is a difficult task. Such models often require computational tools to analyse analytical approximate methods which appear in some theoretical and practical applications in systems biology. They represent key critical elements and give some approximate solutions for such systems. The SIR epidemic disease model is given as the non-linear system of ODE's. Then, we use a proper scaling to reduce the number of parameters. We suggest Elzaki transform method to find analytical approximate solutions for the simplified model. The technique plays an important role in calculating the analytical approximate solutions. The local and global dynamics of the model are also studied. An investigation of the behaviour at infinity was conducted via finding the lines and singular points at infinity. Model dynamic results are computed in numerical simulations using Pplane8 and SimBiology Toolbox for Mathlab. Results provide a good step forward for describing the model dynamics. More interestingly, the simplified model could be accurate, robust, and used by biologists for different purposes such as identifying critical model elements.

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The purpose of this paper is to propose some coefficient conditions, characterizing the stability of periodic solutions bifurcated from zero-Hopf bifurcations of the general quadratic jerk system, and apply these theoretical results to a special jerk system in order to predict chaos. First, we characterize the zero-Hopf bifurcations of the general quadratic jerk system in $\mathbb{R}ˆ3$.The coefficient conditions on stability of periodic solutions are obtained via the averaging theory of first order. Next, we apply the theoretical results to a two-parameter jerk system. Finally special attention is paid to a jerk system with one non-negative parameter $\epsilon$ and one non-linearity. By studying the continuation of periodic solution initiating at the zero-Hopf bifurcation, we numerically find a sequence of period doubling bifurcations which leads to the creation of chaotic attractor.

ABASTRACTWe prove that near the bifurcation point unstable limit cycle arises from the Lorenz system. In the analysis, we use the method of local bifurcation theory, especially the center manifold and the normal form theorem. A computer algebra system using Maple to derive all the formulas and verify the results presented in this paper. Keywords: Lorenz system, limit cycle, center manifold and normal form, Hopf bifurcation. ‫ال‬ ‫د‬ ‫ا‬ ‫ال‬ ‫ات‬ ‫ر‬ ‫ل‬ ‫غائية‬ ‫نظام‬ Lorenz ‫بتفرع‬ Hopf ‫اد‬ ‫آز‬ ‫أم‬ ‫ي‬ ‫ن‬ ‫رزكار‬ ‫صالح‬ ‫بية‬ ‫التر‬ ‫كلية‬ ‫األساسية‬ ‫العلوم‬ ‫كلية‬ ‫الدين‬ ‫صالح‬ ‫جامعة‬ ‫السليمانية‬ ‫جامعة‬ ‫االستالم:‬ ‫تاريخ‬ 09 / 07 / 2006 ‫ت‬ ‫اريخ‬ ‫القبول:‬ 05 / 03 / 2007 ‫الملخص‬