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$x''(t)=-f(x(t), x(t-\tau)).$

When $f$ possesses a symmetric property and grows asymptotically linear both at zero and at infinity, some new results for the existence and multiplicity of periodic solutions are obtained by using the critical point theory and $S^1$ geometrical index theory.

Oncolytic virotherapy is an experimental treatment of cancer patients. This method is based on the administration of replication-competent viruses that selectively destroy tumor cells but remain healthy tissue unaffected. In order to obtain optimal dosage for complete tumor eradication, we derive and analyze a new oncolytic virotherapy model with a fixed time period $τ $ and non-local infection which is caused by the diffusion of the target cells in a continuous bounded domain, where $τ $ is assumed to be the duration that oncolytic viruses spend to destroy the target cells and to release new viruses since they enter into the target cells. This model is a delayed reaction diffusion system with nonlocal reaction term. By analyzing the global stability of tumor cell eradication equilibrium, we give different treatment strategies for cancer therapy according to the different genes mutations (oncogene and antioncogene).

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