Highlights A novel mathematical model of COVID-19 explicitly includes social-distancing. Short delay of distancing mandates has no appreciable effects on flattening the curve. Effect of periodic relaxation of distancing is highly sensitive to timing. Rate of gradual relaxation determines presence of a second wave, and its severity.

Motivated by the current COVID-19 epidemic, this work introduces an epidemiological model in which separate compartments are used for susceptible and asymptomatic "socially distant" populations. Distancing directives are represented by rates of flow into these compartments, as well as by a reduction in contacts that lessens disease transmission. The dynamical behavior of this system is analyzed, under various different rate control strategies, and the sensitivity of the basic reproduction number to various parameters is studied. One of the striking features of this model is the existence of a critical implementation delay in issuing separation mandates: while a delay of about four weeks does not have an appreciable effect, issuing mandates after this critical time results in a far greater incidence of infection. In other words, there is a nontrivial but tight "window of opportunity" for commencing social distancing. Different relaxation strategies are also simulated, with surprising results. Periodic relaxation policies suggest a schedule which may significantly inhibit peak infective load, but that this schedule is very sensitive to parameter values and the schedule's frequency. Further, we considered the impact of steadily reducing social distancing measures over time. We find that a too-sudden reopening of society may negate the progress achieved under initial distancing guidelines, if not carefully designed. 0 These authors (listed alphabetically) contributed equally.

Elaborate regulatory feedback processes are thought to make biological development robust, that is, resistant to changes induced by genetic or environmental perturbations. How this might be done is still not completely understood. Previous numerical simulations on reaction‐diffusion models of Dpp gradients in Drosophila wing imaginal disc have showed that feedback (of the Hill function type) on (signaling) receptors and/or non‐(signaling) receptors are of limited effectiveness in promoting robustness. Spatial nonuniformity of the feedback processes has also been shown theoretically to lead to serious shape distortion and a principal cause for ineffectiveness. Through mathematical modeling and analysis, the present article shows that spatially uniform nonlocal feedback mechanisms typically modify gradient shape through a shape parameter (that does not change with location). This in turn enables us to uncover new multi‐feedback instrument for effective promotion of robust signaling gradients.

The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy [11,10]. In this work, we expound on the details relating to an optimal control problem outlined in [10]. The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebra techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.

No abstract

One of the most important factors limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy (Greene et al., 2018a(Greene et al., , 2019. In this work, we expound on the details relating to an optimal control problem outlined in Greene et al. (2018a). The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebraic techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.

Genetic instability promotes cancer progression (by increasing the probability of cancerous mutations) as well as hinders it (by imposing a higher cell death rate for cells susceptible to cancerous mutation). With the loss of tumor suppressor gene function known to be responsible for a high percentage of breast and colorectal cancer (and a good fraction of lung and other types as well), it is important to understand how genetic instability can be orchestrated toward carcinogenesis. In this context, this paper gives a complete characterization of the optimal (time-varying) cell mutation rate for the fastest time to a target cancerous cell population through the loss of both copies of a tumor suppressor gene (TSG). Similar to the (1-step) oncogene activation model previously analyzed, the optimal mutation rate of the present 2-step model changes qualitatively with the convexity of the (mutation rate dependent) cell death rate. However, the structure of the Hamiltonian for the new model differs significantly and intrinsically from that of the 1-step model and a completely new approach is needed for the solution of the present 2-step problem. Considerable insight on the biology of optimal switching (between corner controls) is extracted from numerical results for cases with nonconvex death rates.

Elaborate feedback regulatory processes are thought to make biological developments robust, i.e., resistant to changes induced by genetic or environmental perturbations. How this might be done is still not completely understood. Previous numerical simulations on reaction-diffusion models of Dpp gradients in Drosophila wing imaginal disc showed that feedback (of the Hill's function type) on (signaling) receptors and/or non-(signaling) receptors are of limited effectiveness in promoting robustness. Spatial nonuniformity of the feedback processes is thought to lead to serious shape distortion and a principal cause for ineffectiveness. Through mathematical modeling of a spatially uniform nonlocal feedback mechanism, the present paper provides a theoretical support of these observations. More significantly, the new approach also enables us to uncover in this paper a new, theory-based multi-feedback instrument for broadly effective promotion of robust signaling gradients. _____________