In this article we consider poro-elastic and poro-visco-elastic models inspired by problems in medicine and biology [12], and we perform sensitivity analysis on the solutions of these fluid-solid mixtures problems with respect to the imposed boundary data, which are the main drivers of the system. Moreover, we compare the results obtained in the elastic case vs. viscoelastic case, as it is known that structural viscosity of biological tissues decreases with age and disease. Sensitivity analysis is the first step towards optimization and control problems associated with these models, which is our ultimate goal.

We investigate the complex-step method as applied to compute sensitivities with respect to model "parameters" for several types of examples. We first consider time delayed differential equations (DDEs) whose sensitivities are known to have lack of smoothness or even discontinuities with respect to parameters such as the delays. The second type of "parameter sensitivity" we consider is that of solutions to partial differential equations (PDEs) with respect to boundary conditions which again may not possess smoothness. These sensitivities are fundamental in any type of boundary control formulation such as those we motivate in Section 4 below.Our main focus here is to evaluate the so-called "complex-step methods" for computing such sensitivities. This is of interest since the complex-step method was derived based on the Cauchy-Riemann equations for analytic complex functions. Our computational findings are compared to those using the standard chain rule-based sensitivity differential equations which can be rigorously developed even for derivatives possessing much less regularity than analyticity. Our findings suggest that the complex-step methods are in very good agreement with the usual sensitivity equation results up to some critical step size we call h crit . They can offer significant savings in computational costs for problems driven by complicated dynamical systems with reasonable parameter size.Given an analytic function f , the following is an outline of the general steps for implementing the complex-step method for computing the first derivative, df /dx.1. Define all functions and operators that are not defined for complex arguments such as for example max, min and abs.

We use dynamical systems modeling to help understand how selected intra-personal factors interact to form mechanisms of behavior change in problem drinkers. Our modeling effort illustrates the iterative process of modeling using an individual's clinical data. Due to the lack of previous work in modeling behavior change in individual patients, we build our preliminary model relying on our understandings of the psychological relationships among the variables. This model is refined and the psychological understanding is then enhanced through the iterative modeling process. Our results suggest that this is a promising direction in research in alcohol use disorders as well as other behavioral sciences.

Because of limited data, much remains uncertain about parameters related to transmission dynamics of Zika virus (ZIKV). Estimating a large number of parameters from the limited information in data may not provide useful knowledge about the ZIKV. Here, we developed a method that utilizes a mathematical model of ZIKV dynamics and the complex-step derivative approximation technique to identify parameters that can be estimated from the available data. Applying our method to epidemic data from the ZIKV outbreaks in French Polynesia and Yap Island, we identified the parameters that can be estimated from these island data. Our results suggest that the parameters that can be estimated from a given data set, as well as the estimated values of those parameters, vary from Island to Island. Our method allowed us to estimate some ZIKV-related parameters with reasonable confidence intervals. We also computed the basic reproduction number to be from 2.03 to 3.20 across islands. Furthermore, using our model, we evaluated potential prevention strategies and found that peak prevalence can be reduced to nearly 10% by reducing mosquito-to-human contact by at least 60% or increasing mosquito death by at least a factor of three of the base case. With these preventions, the final outbreak-size is predicted to be negligible, thereby successfully controlling ZIKV epidemics.

In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.