One of the most challenging tasks in constructing a mathematical model of cancer treatment is the calculation of biological parameters from empirical data. This task becomes increasingly difficult if a model involves several cell populations and treatment modalities. A sophisticated model constructed by de Pillis et al., Mixed immunotherapy and chemotherapy of tumours: Modelling, applications and biological interpretations, J. Theor. Biol. 238 (2006), pp. 841-862; involves tumour cells, specific and non-specific immune cells (natural killer (NK) cells, CD8 þ T cells and other lymphocytes) and employs chemotherapy and two types of immunotherapy (IL-2 supplementation and CD8 þ T-cell infusion) as treatment modalities. Despite the overall success of the aforementioned model, the problem of illustrating the effects of IL-2 on a growing tumour remains open. In this paper, we update the model of de Pillis et al. and then carefully identify appropriate values for the parameters of the new model according to recent empirical data. We determine new NK and tumour antigen-activated CD8 þ T-cell count equilibrium values; we complete IL-2 dynamics; and we modify the model in de Pillis et al. to allow for endogenous IL-2 production, IL-2-stimulated NK cell proliferation and IL-2-dependent CD8 þ T-cell self-regulations. Finally, we show that the potential patient-specific efficacy of immunotherapy may be dependent on experimentally determinable parameters.

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While cholera has been a recognized disease for two centuries, there is no strategy for its effective control. We formulate a mathematical model to include essential components such as a hyperinfectious, short-lived bacterial state, a separate class for mild human infections, and waning disease immunity. A new result quantifies contributions to the basic reproductive number from multiple infectious classes. Using optimal control theory, parameter sensitivity analysis, and numerical simulations, a cost-effective balance of multiple intervention methods is compared for two endemic populations. Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods.

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We investigate optimal control of a first order partial differential equation (PDE) system representing a competitive population model with age structure. The controls are the proportions of the populations to be harvested, and the objective functional represents the profit from harvesting. The existence and unique characterization of the optimal control pair are established.