We propose a simple discrete-time host–parasitoid model to investigate the impact of external input of parasitoids upon the host–parasitoid interactions. It is proved that the input of the external parasitoids can eventually eliminate the host population if it is above a threshold and it also decreases the host population level in the unique interior equilibrium. It can simplify the host–parasitoid dynamics when the host population practices contest competition. We then consider a corresponding optimal control problem over a finite time period. We also derive an optimal control model using a chemical as a control for the hosts. Applying the forward–backward sweep method, we solve the optimal control problems numerically and compare the optimal host populations with the host populations when no control is applied. Our study concludes that applying a chemical to eliminate the hosts directly may be a more effective control strategy than using the parasitoids to indirectly suppress the hosts.

Immunotherapy is one of the most recent approaches for controlling and curing malignant tumors. In this paper, we consider a mathematical model of periodically pulsed immunotherapy using CD4+ T cells and an antitumor cytokine. Mathematical analyses are performed to determine the threshold of a successful treatment. The interindividual variability is explored by one-, two-, and three-parameter bifurcation diagrams for a nontreatment case. Numerical simulation conducted in this paper shows that (i) the tumor can be regulated by administering CD4+ T cells alone in a patient with a strong immune system or who has been diagnosed at an early stage, (ii) immunotherapy with a large amount of an antitumor cytokine can boost the immune system to remit or even to suppress tumor cells completely, and (iii) through polytherapy the tumor can be kept at a smaller size with reduced dosages.

Oncolytic ability to direct target and lyse tumor cells makes oncolytic virus therapy (OVT) a promising approach to treating cancer. Despite its therapeutic potential to stimulate anti-tumor immune responses, it also has immunosuppressive effects. The efficacy of OVTs as monotherapies can be enhanced by appropriate adjuvant therapy such as anti-CTLA-4. In this paper, we propose a mathematical model to explore the interactions of combined therapy of oncolytic viruses and a checkpoint inhibitor, anti-CTLA-4. The model incorporates both the susceptible and infected tumor populations, natural killer cell population, virus population, tumor-specific immune populations, virus-specific immune populations, tumor suppressive cytokine IFN-g, and the effect of immune checkpoint inhibitor CTLA-4. In particular, we distinguish the tumor-specific immune abilities of CD8⁺ T, NK cells, and CD4⁺ T cells and describe the destructive ability of cytokine on tumor cells as well as the inhibitory capacity of CTLA-4 on various components. Our model is validated through the experimental results. We also investigate various dosing strategies to improve treatment outcomes. Our study reveals that tumor killing rate by cytokines, cytokine decay rate, and tumor growth rate play important roles on both the OVT monotherapy and the combination therapy. Moreover, parameters related to CD8⁺ T cell killing have a large impact on treatment outcomes with OVT alone, whereas parameters associated with IFN-g strongly influence treatment responses for the combined therapy. We also found that virus killing by NK cells may halt the desired spread of OVs and enhance the probability of tumor escape during the treatment. Our study reveals that it is the activation of host anti-tumor immune system responses rather than its direct destruction of the tumor cells plays a major biological function of the combined therapy.

We present a class of numerical methods for the reaction‐diffusion‐chemotaxis system which is significant for biological and chemistry pattern formation problems. To solve reaction‐diffusion‐chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. Along with the implementation of the method of lines, implicit or semi‐implicit schemes are typical time stepping solvers to reduce the effect on time step constrains due to the stability condition. However, these two schemes are usually difficult to employ. In this paper, we propose an adaptive optimal time stepping strategy for the explicit m‐stage Runge‐Kutta method to solve reaction‐diffusion‐chemotaxis systems. Instead of relying on empirical approaches to control the time step size, variable time step sizes are given explicitly. Yet, theorems about stability and convergence of the algorithm are provided in analyzing robustness and efficiency. Numerical experiment results on a testing problem and a real application problem are shown.