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We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a semilinear parabolic equation, including a chemotaxis term which is used to describe the movement of the immune response induced by gradients of the infected cell density. Under the assumption that the chemotactic sensitivity coefficient is small compared with the diffusion coefficient of the immune response, we prove the global existence and uniqueness of the solution of this free boundary problem. For large chemotactic coefficient, the global existence is still open.

This paper deals with a procedure for cancer radiovirotherapy which requires not only injection of replication-competent viruses but also administration of radioiodide. The viruses infect tumor cells, replicate inside them and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. Radioiodide is in a continuous state of flux between the tumor and the remaining part. Iodide undergoes beta particle decay and the emitted beta particles have a significant effect on tumor cells. The combination of virotherapy with radiotherapy has recently been shown to be significantly more effective than treatment with virotherapy alone. Cancer radiovirotherapy can be described by a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of a tumor. Global existence and uniqueness of solutions to this free boundary problem is proved, and a new explicit parameter condition corresponding to the success of therapy is also found. Furthermore, numerical simulations are given to show that there is an optimal timing for radio-iodine administration, and that there is an optimal dose for the radioactive iodide.

In this paper, we introduce a mathematical model for the virus medical imaging (VMI). In this method, first, by proposing a mathematical model, we show that there are two types of viruses that each of them produce one type of signal. Some of these signals can be received by males and others by females. Then, we will show that in the VMI technique, viruses can communicate with cells, interior to human’s body via two ways. (1) Viruses can form a wire that passes the skin and reaches to a special cell. (2) Viruses can communicate with viruses interior to body in the wireless form and send some signals for controlling evolutions of cells interior to human’s body.

Based on the complex network and non-equilibrium statistic theory, the birth and death equation (BDE) is proposed for the virus spreading on biological networks. This equation, in which the Susceptible Infected Susceptible model is extended, can predict the distribution rule of infection density by generating function. It is a better approach to a real infection process.

A new non-autonomous predator-prey system with the effect of viruses on the prey is investigated. By using the method of coincidence degree, some sufficient conditions are obtained for the existence of a positive periodic solution. Moreover, with the help of an appropriately chosen Lyapunov function, the global attractivity of the positive periodic solution is discussed. In the end, a numerical simulation is used to illustrate the feasibility of our results.

Crystallization is recognized among structural biologists as a necessary process before three-dimensional structure can be solved at an atomic level. Crystallization has a dose of mysticism among protein chemist. Some treats it as an “art” and others as “black magic”. These concepts aroused from a limited knowledge in the physical chemistry of proteins in solution. Crystallization appears only in a metastable state. To define crystallization conditions the experiments are guided either by a chance search or by dedicated factorial design. Here we will briefly describe a factorial design method to rationally approach the metastable state. In summary, there is nothing mysterious in crystallization of biological macromolecules, and the success can often be achieved within a limited number of experiments.