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Dynamics of immunotherapy antitumor models with impulsive control strategy

In this paper, using the methods of killing tumors and impulsive differential equations, two immunotherapy antitumor models for describing therapies of general tumors and advanced solid tumors are established. By using the theories of impulsive equations, small amplitude perturbation techniques, and the comparison technique, we obtain the conditions which guarantee the global asymptotical stability of the tumor‐eliminated periodic solution and system permanence, when immunotherapy alone is performed. The numerical results of the influences of the impulsive perturbation on the inherent oscillation show rich dynamics, such as period‐doubling bifurcation and chaos. Moreover, the effects of the combination of radiotherapy with immunotherapy on antitumor are obtained, including the threshold value of stability conditions of tumor‐eradication periodic solution when the mixed combination treatment of immunotherapy and radiotherapy is performed. Some numerical simulations for the effects of the timing of radiotherapy application and the timing of injection T cells on the threshold value are performed. Finally, we present some theoretical methods for suppressing the growth of tumors.