# DE Seminar: Caleb Phillips, Mac Luu, and Matthew Lastner

### Undergraduate Students

Location

Mathematics/Psychology : 401

Date & Time

April 29, 2024, 11:00 am – 12:00 pm

Description

**Title**: Parameter Sensitivity and Estimation for an

*in vivo*HIV model

**Speaker**: Caleb Phillips

**Abstract**: Human immunodeficiency Virus (HIV) is an incurable, infectious virus that affects over 30 million people globally. We investigate the relationship between state variables that represent the CD4+ T-cell counts and viral load and the parameters of a system of first-order coupled ordinary differential equations. We first investigate the sensitivity of the state variables to changes in the parameters using a Sobol sequence based parameter sensitivity algorithm. Identifying the sensitive parameters allows us to target the parameter estimation to those which significantly affect the CD4+ T-cell counts and viral load. We then use a data assimilation method, also known as nudging, to estimate the values of the more sensitive parameters. The model of CD4+ T-cells and virions contains parameters which are difficult to estimate using experimental data. Data assimilation methods use time series data on CD4+ T-cell counts and viral load and the model to estimate the parameters by minimizing the difference between the model output and the data. Being able to accurately model HIV gives us a better understanding of the disease, and can identify which treatment and prevention interventions lead to the largest decrease in disease prevalence.

*This work was funded, in part, by the UMBC START grant.*

**Title**: Parameter Estimation and Sensitivity Applied to a Novel Malaria Outbreak in Dire Dawa, Ethiopia

**Speaker**: Mac Luu

**Abstract**: Malaria, a devastating vector-borne disease, poses a significant global public health threat. The emergence of the exotic Anopheles stephensi presents a new urban vector of malaria and raises concerns about the dynamic spread of the disease in the traditionally non-malarial Dire Dawa region of Ethiopia. We consider a mechanistic differential equation Susceptible-Exposed-Infected-Recovered (SEIR) model for human malarial transmission where the primary parameters of interest are the rate at which individuals are exposed to malarial mosquitoes $\beta$ and the rate at which exposed individuals become infected $\alpha _1$. Our goal is to use a novel dataset that includes the number of individuals exposed to and infected with malaria in the Dire Dawa region to improve model predictions. In particular, we use Sobol sequence sensitivity techniques to identify the sensitive parameters of the model and subsequently use Bayesian inference and data assimilation techniques to estimate parameters from the novel dataset in the SEIR model. Given the inherent uncertainties and variability in real-world epidemiological data, Bayesian inference and nudging arise as a powerful tool for parameter estimation. Using Bayesian inference, we update prior knowledge on the transmission parameters based on the observed data, thereby refining the estimates of $\beta$ and $\alpha_1$ with derived posterior distributions. Concurrently, nudging was employed to iteratively adjust the SEIR model's state towards the observed state, enhancing the model's fidelity to real-world data. These methods ensure that the model remains closely aligned with the observed outbreak dynamics, improving the reliability of predictions and providing a framework for parameter estimation models where parameters can not be observed.

**Title**: Analyzing the Relationship Between Observable Outputs and Parameters in a Structured Treatment Interruptions Model for HIV Patients

**Speaker**: Matthew Lastner

**Abstract**: Highly active antiretroviral therapy (HAART) has increased the life expectancy of individuals infected with human immunodeficiency virus (HIV); however, consistent, constant treatment may not be readily available and can result in adverse reactions and drug resistance. Mathematical models of disease dynamics play a crucial role in understanding the viability of alternative intervention methods. Structured treatment interruptions (STI) has been proposed as an alternative treatment where thresholds in CD4+ T-cell count determine the duration of intermittent treatment. Previous literature explored the sensitivity of treatment duration based on the upper and lower thresholds, leading to a high failure rate depending on the choice of thresholds. We investigate the relationship between observable outputs, such as total CD4+ T-cell count and viral load, and the parameters in an STI model with first-order coupled ordinary differential equations. Applying Sobol sensitivity analysis and partial rank correlation coefficients, we measure the outputs’ sensitivity to small changes in the numerical values of the parameters. We also determine whether each parameter’s numerical value can be uniquely identified from sample data. Applying differential algebra, we first determine which subsets of parameters are structurally identifiable with respect to sample data without noise. Once structurally identifiable subsets of parameters are determined, Monte Carlo simulations are applied to determine if each of these parameter subsets can be practically identified through stochastic data. We then compute the numerical values of these practically identifiable parameter subsets through estimation techniques such as simulated annealing and nudging. By analyzing these output-parameter relationships, the viability of STI can be further characterized, potentially providing an alternative treatment method to tackle the ongoing HIV epidemic.

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