A dynamical model for COVID-19 spread relating to non-pharmaceutical interventions and vaccination is mathematically generated by adding a gradual vaccination compartment for the susceptible population and considering only a symptomatic infectious stage. In our model, there are seven compartments dividing a given population into susceptible (S), vaccinated (V ), exposed (E), infected (I), quarantined (Q), recovered (R) and death (D) groups, respectively. Then, theoretically analysis is given by investigating the COVID-19 free and endemic equilibrium points, and computing the vaccination reproduction number of this model denoted as R vac using the next generation matrix. If R vac > 1, then the COVID-19 transmission increases exponentially and depends on vaccine efficacy. On the other hand, if R vac < 1, then there occurs the COVID-19 disease eradication. The risk from infection can be importantly reduced whenever the intake of COVID-19 vaccines exceeds one dose. The numerical results reveal that the nonpharmaceutical ways and the administered COVID-19 vaccines can be effective against the current variants of COVID-19, and the additional efforts such as a third vaccine booster shot should be considered and implemented to greatly mitigate the risks of emerging variants of the COVID-19 virus. Moreover, combining different types of COVID-19 vaccines can be appeared as a possible way to give better protection against COVID-19 as well.

<abstract><p>The use of vaccines has always been controversial. Individuals in society may have different opinions about the benefits of vaccines. As a result, some people decide to get vaccinated, while others decide otherwise. The conflicting opinions about vaccinations have a significant impact on the spread of a disease and the dynamics of an epidemic. This study proposes a mathematical model of COVID-19 to understand the interactions of two populations: the low risk population and the high risk population, with two preventive measures. Unvaccinated individuals with chronic diseases are classified as high risk population while the rest are a low risk population. Preventive measures used by low risk group include vaccination (pharmaceutical way), while for the high risk population they include wearing masks, social distancing and regular hand washing (non-pharmaceutical ways). The susceptible and infected sub-populations in both the low risk and the high risk groups were studied in detail through calculations of the effective reproduction number, model analysis, and numerical simulations. Our results show that the introduction of vaccination in the low risk population will significantly reduce infections in both subgroups.</p></abstract>

In this article, a COVID-19 transmission mathematical model incorporating vaccination and non-pharmaceutical interventions was formulated and theoretically analysed. Here, the COVID-19 free and endemic equilibrium points, vaccine reproduction number were computed. The derived vaccination reproduction number largely depends on vaccine efficacy for disease eradication to occur. Infection risk is significantly reduced whenever the vaccine intake is greater than one dosage. The simulation results indicate that the administered COVID-19 vaccines and non-pharmaceutical interventions have been effective for the current variants, additional efforts such as a third vaccine booster shot should be considered and implemented to greatly mitigate the risk of the emerging variants of the COVID-19 pandemic.

The insight in structures of the blood vessels is a basis for study of blood flows to help understanding the abnormalities of blood vessels that can cause vascular diseases. Basic concept used for constructing structures of blood vessels in organs is arterial branching, which is usually characterized by fractal similarity in the bifurcation pattern. In this work, the concept of Lindenmayer system (L-system) is modified for three-dimensional (3D) tree-like structures to model structures of blood vessels in organs, and then, applied to construct and visualize structural blood vessels via our software created based on openGL and Lazarus program. The structure of blood vessels is constructed based on the physiological law of arterial branching proposed Murray (Murray’s law) under additional assumptions and constraints such as the spreading of blood vessels to cover all directions, the angle condition and the non-overlapping vessels condition. The concept is applied to simulate structures of blood vessels in 3 study cases, including symmetric arterial branching, non-symmetric arterial branching and structure of blood vessel on different domains. The results of structures of blood vessels generated from all cases are measured based on the number of segments, the total blood volume and the fractal dimension. The results of modeling and simulation in this work are illustrated by comparing with other results appeared literature. Moreover, the constructed structures of the blood vessels based on this 3D L-system could be useful for future research such as blood flow, pressure and other properties involving in structures of blood vessels in different organs of human and animals. HIGHLIGHTS A new 3D L-system is developed based on directional vectors for construction of 3D tree-like structures such as structures of blood vessels The model of structures of blood vessels is constructed based on the physiological laws of arterial branching (Murray’s law) with additional assumptions on the spreading of blood vessels, the angle condition, and the non-overlapping of blood vessels Algorithm and software are developed based on L-system to simulate and visualize 3D structures of blood vessels GRAPHICAL ABSTRACT