### Session S01 - Modeling and Computation for Control and Optimization of Biological and Physical Systems

## Talks

Wednesday, July 14, 16:00 ~ 16:25 UTC-3

## Existence, comparison, monotonicity and convergence results for a class of elliptic hemivariational inequalities

### Domingo Alberto Tarzia

#### Universidad Austral and CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison and monotonicity of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.

Joint work with Claudia M. Gariboldi (Universidad Nacional de Río Cuarto, Argentina), Stanislaw Migórski (Jagiellonian University, Poland) and Anna Ochal (Jagiellonian University, Poland).

Wednesday, July 14, 16:30 ~ 16:55 UTC-3

## Variational Problems with Distributional and Weak Gradient Constraints

### Carlos Rautenberg

#### George Mason University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider variational problems with mixed boundary conditions and (distributional or weak) gradient constraints arising in the growth of non-homogeneous sand piles. In this setting, the upper bound in the constraint is either a function or a Borel measure; this allows the pile growth to observe discontinuities. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a non-standard space of Borel measures with square integrable divergences. We conclude the talk by introducing a mixed finite-element method and several numerical tests .

Joint work with Harbir Antil, Rafael Arndt, and Deepanshu Verma (George Mason University, USA).

Wednesday, July 14, 17:00 ~ 17:25 UTC-3

## Mathematical and asymptotic analysis of thermoelastic shells in normal damped response contact

### Sabrina Roscani

#### CONICET - Universidad Austral , Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a dynamic contact problem in thermoelasticity, where the contact is governed by a normal damped response function and the constitutive thermoelastic law is given by the Duhamel-Neumann relation. Under suitable hypotheses on data we show the existence and uniqueness of solution for this problem. Then, we study the particular case when the deformable body is a shell and use asymptotic analysis to study the convergence to a 2D limit problem when the thickness tends to zero.

Joint work with M. T. Cao-Rial (Universidade da Coruña, España) , G. Castiñeira (Universidade de Vigo, España) y A. Rodríguez-Arós (Universidade da Coruña, España).

Wednesday, July 14, 17:50 ~ 18:15 UTC-3

## Real Time Estimation of Advection Diffusion Equations

### John Burns

#### Virginia Tech, US - This email address is being protected from spambots. You need JavaScript enabled to view it.

Consider the convection diffusion equation \begin{equation} \label{eq:CDeq} \frac{\partial z(t,x)}{\partial t} = \nabla \cdot\left[K \nabla z(t,x))\right] - \nabla \cdot \left[K z(t,x)\right] +g(x) \eta(t) \end{equation} on a domain $\Omega \subset \mathbb{R}^{3}$. We assume the sensed output is given by \begin{equation} y_{i}(t)=\iiint\limits_{B_{\delta}(x_{i}(t))\cap\Omega}h_{i}(x)z(t,x)dx +E_{i}v(t), \end{equation} where $B_{\delta}(x_{i}(t))$ is a $\delta-$neighborhood of the trajectory $x_{i}(t)$ of a moving sensor platform and and $h_{i}(x)$ is a kernel function.

Let $\boldsymbol{y}(t)= [y_{1}(t) \ y_{2}(t) \ ... y_{p}(t)]^{T}$ and \[ C(t) = \iiint\limits_{B_{\delta}(\boldsymbol{\boldsymbol{{x}}}_{i}(t))\cap\Omega }h_{i}(\boldsymbol{\boldsymbol{{x}}})z(t,\boldsymbol{\boldsymbol{{x}}})d\boldsymbol{\boldsymbol{{x}}} \]

Given the system above, a stable (full) state estimator (Luenberger observer) will have the form% \begin{equation} \frac{\partial\hat{z}(t,x)}{\partial t} = \nabla \cdot\left[K \nabla z(t,x))\right] - \nabla \cdot \left[K z(t,x)\right] +g(x) \eta(t) +\mathcal{F}(t,x)[\boldsymbol{y}(t)-C(t)\hat{z}(t,x)],\label{eq:abstractobserver} \end{equation} where $\mathcal{F}(t,\cdot):\mathbb{R}^{p}\rightarrow Z=L_{2}(\Omega)$ is a bounded linear operator called the observer gain operator. The goal is to find an observer gain operator so that the error $e(t)=z(t)-\hat{z}(t)$ goes to zero as $ t$ approaches $ + \infty $ and the data driven dynamical estimator can be realized in real time. In this talk, we discuss this problem and show that real time implementation is possible.

Joint work with James Cheung (Virginia Tech), Michael Demetriou (WPI), Nikolaos Gatsonis (WPI), Weiwei Hu (University of Georgia) and Xin Tian (WPI).

Wednesday, July 14, 18:20 ~ 18:45 UTC-3

## Semiclassic Ginzburg-Landau model for a nematic superconducting medium in the presence of an electromagnetic field

### Juan Pablo Borgna

#### ICIFI (CONICET) - CEDEMA (Universidad de San Martín), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we shall present the derivation of a system of equations related to the optical response of a superconducting nematic liquid crystal medium, in the presence of an electromagnetic field. We shall consider the semi-classical Ginzburg-Landau superconductor formulation and the related Helmholtz free energy. By a minimizing process on the suitable physical observable variables, we obtain a set of equations. In a first step, we consider a sample of superfluid nematic liquid crystals, which allows us to ignore the magnetic field and consider the electric one only. We shall discuss the existence of solutions of the system obtained by focusing on geometries of interest, such as a slab and an annular cylinder. We describe the contribution of nematicity in the behavior of the Fréederickz threshold in these different configurations. In the second step, from the Helmholtz free energy expression, we derive the system of equations for the case of a superconducting medium. In this case, it is intrinsic to the problem to consider an electromagnetic field and this must be taken into account throughout the formulation. We shall comment on advances of the partial results obtained.

Joint work with Diego García Ovalle (Aix-Marseille Université, Francia). and Mariano F. De Leo (INMBB, Conicet - Universidad Nacional del Sur, Argentina).

Wednesday, July 14, 18:50 ~ 19:15 UTC-3

## Optimal control of differential quasivariationalinequalities with applications in contact mechanics

### Julieta Bollati

#### CONICET-Depto de Matemática, FCE-Universidad Austral, Paraguay 1950, 2000 Rosario, Argentina, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a differential quasivariational inequality for which we state and prove the continuous dependence of the solution with respect to the data. This convergence result allows us to prove the existence of at least one optimal pair for an associated control problem. Finally, we illustrate our abstract results in the study of a free boundary problem which describes the equilibrium of a viscoelastic body in frictionless contact with a foundation made of a rigid body covered by a rigid-elastic layer.

Joint work with D.A. Tarzia (CONICET-Depto de Matemática, FCE-Universidad Austral, Paraguay 1950, 2000 Rosario, Argentina) and M. Sofonea (Département de Mathématiques, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France).

Wednesday, July 14, 19:20 ~ 19:45 UTC-3

## Stabilization for a class of damped Schroedinger equations.

### Mariano Fernando De Leo

#### INMABB (CONICET)-Depto. Matemática (Universidad Nacional del Sur), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we shall present the exponential decay of both the total charge and the free energy, given by the norm in some suitable Sobolev space, for the solutions of a family of non linear, non local Schroedinger equations with a localized damping in the whole line. We shall also discuss qualitative aspects of the dynamics and show that, for finite times, the stabilization rate becomes smaller as the free--damping region is chosen around the origin.

Thursday, July 15, 12:00 ~ 12:25 UTC-3

## Compressive Spectral Imaging Using Regularization Gradients on Calibrated Optimized Systems

### Henry Arguello

#### Senior Member, IEEE. Universidad Industrial de Santander , Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The design of compressive spectral imaging systems is a growing field to improve the reconstruction quality of spectral images. However, the performance achieved in the simulation is lost when it is carried out to real optical setups. Therefore, this work proposed a recovery method for compressive spectral imaging for the calibration problem in optimized coded systems. The proposed method modeled the calibration problem as the sum of the designed matrix and an additional matrix that stands for the calibrated error. Under this decomposition, an additional term is identified and minimized in the loss function to reduce the calibration gap. Simulation shows that the proposed method converges faster to the solution; specifically, the traditional approach requires up to twice the number of iterations to obtain the same performance as the proposed approach.

This work was supported by the VIE-UIS under Projects 2699

Joint work with Hans Garcia (Student Member, IEEE), Jorge Bacca, (Member, IEEE) and Brendt Wohlberg (Senior Member IEEE).

Thursday, July 15, 12:30 ~ 12:55 UTC-3

## Coupling epidemiological models with social dynamics

### Nicolas Saintier

#### Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work we study a Susceptible-Infected-Susceptible model coupled with a continuous opinion dynamics model. We assume that each individual can take measures to reduce the probability of contagion, and the level of effort each agent applies can change due to social interactions. We propose simple rules to model the propagation of behaviors that modify the level of effort, and analyze their impact on the dynamics of the disease.

We derive a set of two ordinary differential equations describing the dynamic of the proportion of the number of infected individuals and the mean value of the effort parameter, and analyze the equilibria of the system. The results we obtain are in complete agreement with the simulations of the agent-based model, and show that the social dynamic strongly mitigates the impact of the disease. This is quanified by a modified $R_0$ number we call the behaviorally reduced reproduction number.

Joint work with Carlo Giambiagi Ferrari (Conicet y Univ. Buenos Aires) and Juan Pablo Pinasco (Conicet y Univ. Buenos Aires).

Thursday, July 15, 13:00 ~ 13:25 UTC-3

## $(Bio)-$fueling protein-protein interaction using data mining

### Gabriel Soto

#### Universidad Nacional de la Patagonia San Juan Bosco, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Biofuels are currently on high demand because of rising oil prices and limitations on oil reserves. Finding new sources for biofuels such as triacylglycerols (TAG), is highly important from a biotechnological point of view. Oleaginous organisms such as Rhodococcus jostii (RHA1) have the capacity, under stress conditions to accumulate TAGs at more than 20\% of their dry weight. For this lipid accumulation to occur, it requires an integral configuration of metabolism and regulatory processes rather than the sole existence of an efficient lipid biosynthesis pathway. Even thought there have been important advances in our basic understanding of bacterial TAG biosynthesis, many questions remain unanswered. One is the complete characterization of the protein-protein interaction map (PPI) of Rhodococcus jostii. In this presentation, we present results obtained by applying the clustering method \textit{affinity propagation} (AP) on proteomic data from RHA1 that allow us to uncover potentially new protein-protein interactions to be tested experimentally.

Joint work with Nelson Villagra (Universidad Nacional de la Patagonia San Juan Bosco, Argentina), Pablo Rodriguez (Universidade Federal de Pernambuco, Brazil), Héctor Álvarez (INBIOP-CONICET, Universidad Nacional de la Patagonia San Juan Bosco, Argentina) and Roxana Silva (INBIOP-CONICET, Universidad Nacional de la Patagonia San Juan Bosco, Argentina).

Thursday, July 15, 14:30 ~ 14:55 UTC-3

## A size-structured coagulation-fragmentation model on the space of measures: well-posedness and numerical approximation

### Azmy Ackleh

#### University of Louisiana at Lafayette, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We formulate a size-structured coagulation-fragmentation model on the space of Radon measures equipped with the bounded Lipschitz norm. Such formulation unifies the study of discrete and continuous coagulation-fragmentation models. We establish the well-posedness of this model and show that it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We then develop a numerical approximation scheme which conserves mass and prove its convergence to the unique weak solution.

Joint work with Rainey Lyons (University of Louisiana at Lafayette) and Nicolas Saintier (University of Buenos Aires).

Thursday, July 15, 15:00 ~ 15:25 UTC-3

## Modelling the effectiveness of parasitoid biological control on an invasive fruit fly in Senegal , West Africa

### John E. Banks

#### California State University, Monterey Bay , United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Tephritid fruit flies are global pests of economically important crops, and their life history ecology and population dynamics have been well-studied as part of efforts to develop effective management strategies. The oriental fruit fly Bactrocera dorsalis (Hendel) is a widely distributed pest throughout the tropical regions of the Pacific Islands, Asia, and Africa, and has been the focus of many successful control campaigns that integrate augmentative, conservation, and classical biological control with IPM schemes. The recent establishment of B. dorsalis in West Africa, where it is a devastating pest on mangos, has created an urgent need to develop effective control strategies tailored to that region. We describe here a modelling effort describing the results of a biological control program aimed at reducing populations of B. dorsalis in Senegal, West Africa, based on the successful deployment of the braconid parasitoid Fopius arisanus (Sonan) to control B. dorsalis in other tropical regions. We employ a series of ordinary differential equations to model the population dynamics of B. dorsalis and its parasitoid F. arisanus, based on data from trap collections in three distinct regions of Senegal. The model tracks populations of egg and adult flies using temperature-dependent growth rates, linking biotic and abiotic factors parameterized with environmental data from each region. The rate of parasitism is modeled using a host-dependent functional response equation, and the elasticity, or model sensitivity, to the model parameters is analyzed using both local and global sensitivity methods in each of the three regions. We discuss differences in the effects of F. arisanus releases on B. dorsalis outbreaks in different regions across Senegal, and the implications for future biological control efforts.

Joint work with H.T. Banks (North Carolina State University, Raleigh, NC), Natalie G. Cody (North Carolina State University, Raleigh, NC), Annabel E. Meade (North Carolina State University, Raleigh, NC), Hang Nguyen (North Carolina State University, Raleigh, NC), Elhadji Omar Dieng (Crop Protection Directorate, Dakar, Senegal), Nicholas C. Manoukis (USDA, Agricultural Research Service, Hilo, HI), Stephanie Gayle (USDA, Agricultural Research Service, Hilo, HI) and Roger Vargas (USDA, Agricultural Research Service, Hilo, HI).

Thursday, July 15, 15:30 ~ 15:55 UTC-3

## A two-strain model of tumor growth

### Fabio Milner

#### Arizona State University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Cancer in most organs develops in the form of solid tumors that begin a spheroids consisting entirely of cancerous (parenchyma) cells. When the tumor reaches the size of approximately 1 mm, it needs blood vessels (vascularization) to develop in it in order to be able to develop further. A model of vascularized tumor growth with competing parenchyma cells of two different strains will be presented. A related model for a single strain of parenchyma was described by Chen and Friedman in [1]. The model consists of a free-boundary value problem for a system of nonlinear parabolic PDEs describing the dynamics of the parenchyma cells and vascular endothelial cells (VECs), coupled with an ODE describing the dynamics of vascular density and an algebraic equation describing a variable used as proxy for resource availability locally. The tumor is assumed to have radial symmetry and the mathematical domain is then taken to be a sphere with moving boundary, with the boundary moving at the unique physically determined velocity that changes local tumor volume exactly to maintain density.

Some theoretical analytical properties of the model will be presented, and then the focus will shift to simulations to study the possible development of hypertumors [2]. Results from several simulations will be shown and biological interpretations provided, including how selection for increased proliferation or for increased angiogenesis may lead to tumors that are structurally like integrated tissues or like segregated ecological niches, as well as to hypertumors.

References

[1] D. Chen, and A. Friedman, ``A two-phase free boundary problem with discontinuous velocity: Application to tumor model", $J. Math. Anal. Applic.$ 399 (1), 378-393 (2013).

[2] J. Nagy, ``Competition and Natural Selection in a Mathematical Model of Cancer", $Bull. Math. Biol.$ 66, 663-687 (2004).

Joint work with R. L. Alvarez (Podium Education-Data Science and Curriculum-Austin, TX , USA) and J.D. Nagy (Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ, USA).

## Posters

## Mathematical Model for Feline Sporotrichosis

### Aurélio de Aquino Araújo

#### Fundação Oswaldo Cruz, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work, a mathematical model with an age structure is proposed that describes the transmission dynamics of subcutaneous mycosis caused by dimorphic fungi of the genus \textit {Sporothrix}, known as sporotrichosis. This model is presented through a system of ordinary differential equations that involves three age groups (puppies, youngsters and adults) for the population of male and female semi-domiciled cats that live in the city of Rio de Janeiro - Brazil, and that comprise three states immunological (susceptible, infected and under treatment).

Joint work with Dayvison F S Freitas (Fundação Oswaldo Cruz), Priscila M Macedo (Fundação Oswaldo Cruz), Sandro A Pereira (Fundação Oswaldo Cruz), Cláudia T Codeço (Fundação Oswaldo Cruz) and Flávio C Coelho (Fundação Getulio Vargas).

## A strategy based in formal concept analysis to support approved COVID-19 vaccines prioritization

### Javier Burgos-Salcedo

#### Corporación para la investigación-CIINAS, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The World Health Organization (WHO) proposed a set of criteria to be considered for the prioritization of COVID-19 candidate vaccines for further development of phase II/III clinical trials, thinking in a target audience that includes vaccine scientists, product developers, manufacturers, regulators, and funding agencies. In this paper, a mathematical knowledge-based strategy is employed to perform a prioritization matrix of approved COVID-19 vaccines: BBIBP-CorV, JANSSEN, CORONAVAC, SPUTNIK V, MODERNA, PFIZER, and VAXZEVRIA, based on those proposed criteria by WHO, related to safety, efficacy, stability, implementation, and availability. We found that JANSSEN vaccine is the one with the highest score in the present study, but our analysis suggests that the WHO criteria could be more useful if they are considered separately, taking into account the social, demographic and economic characteristics of each country.

## Discretization and Variational Formulation of a Continuous Model in PDE of Tumor-induced Angiogenesis

### Ana Kristhel Esteban López

#### Universidad Juárez Autónoma de Tabasco, México - This email address is being protected from spambots. You need JavaScript enabled to view it.

The term angiogenesis literally means blood vessel formation. The concept that "tumor growth is dependent on angiogenesis" has prompted continued progress in the development of angiogenesis inhibitors toward the goal of future tumor therapy. This hypothesis, which was first proposed in 1971, can be expressed in its simplest terms: once the tumor is "taken", each increase in the tumor cell population must be preceded by an increase in new capillaries that converge towards the tumor. Thus, this work aims to show the discretization and variational formulation of a system of nonlinear partial differential equations, which describes the dynamics of the density of endothelial cells that migrate through a tumor and form neovascular structures in response to a chemical signal specific known as tumor angiogenic factor (TAF). The complete system of equations describing the interactions of endothelial cells, TAF and fibronectin is \begin{eqnarray*} \begin{split} \frac{\partial n}{\partial t} &=D_{n} \nabla^{2}n-\nabla \cdot \left( \frac{\chi_{0}k_1}{k_1+c}n \nabla c\right) -\nabla \cdot \left(\varrho _{0}n \nabla f\right), \\ \frac{\partial f}{\partial t}&= w n-\mu nf,\\ \frac{\partial c}{\partial t}&= -\lambda nc, \end{split} \end{eqnarray*}

where, $n$ is the endothelial cell density at time $t$, $f$ is the fibronectin concentration at time $t$, $c$ is the TAF concentration at time $t$, $D_{n}$ is the cell random-motility coefficient, $\chi_{0}$ is the chemotactic coefficient, $\varrho _{0}$ is the haptotactic coefficient and $k_1$, $\lambda$, $w$ y $\mu$ are positive constants.

## Mathematical modeling as a tool to explain different proliferation patterns in the diagnosis and relapse of leukemias

### Miguel Angel Martínez Hernández

#### Central University Marta Abreu of Las Villas, Cuba - This email address is being protected from spambots. You need JavaScript enabled to view it.

Recent experimental evidence suggests that acute myeloid leukemias may arise from multiple clones of malignant cells. However, it is unknown how the observed clones may differ with respect to cellular properties, such as proliferation and self-renewal. There is little information on how the properties of cells change due to chemotherapy and relapse. We apply a mathematical model based on ordinary differential equations to investigate the impact of cell properties on the multi-clonal composition of leukemias. The simulations carried out imply that increased self-renewal is a key mechanism in the clonal selection process. Simulations suggest that rapid proliferation and high cell self-renewal dominate in the primary diagnosis, while relapse followed after therapy-induced remission is triggered mainly by high self-renewal, but slowly by proliferating cells.

Keywords: multi-compartment models, systems of differential equations, systems biology, clonal evolution, leukemias.

## Comparative study of the hematopoietic reconstitution of a group of patients of the Arnaldo Milian Clinical-Surgical Hospital of Santa Clara

### Miguel Angel Martínez Hernández

#### Central University Marta Abreu of Las Villas, Cuba - This email address is being protected from spambots. You need JavaScript enabled to view it.

A mathematical model, based on ordinary differential equations, is applied to the study of the hematopoietic reconstitution of a group of sixteen patients with Hodgkin and non-Hodgkin Lymphoma, after having undergone Autologous Transplantation of Hematopoietic Stem Cells in the Hematology Service of the University Hospital "Arnaldo Milián" . Starting from the model, the dynamics of the recovery process are computationally simulated and the hematopoietic recovery times of each patient collected in the clinical data are compared with the times predicted by the simulations. Despite the simplicity of the mathematical model, it is surprising how well the reconstitution curves match the collected clinical data, so that the model is able to predict with great accuracy the day the hematopoietic recovery criterion is met. The model's prognoses can be used to estimate the average duration of the pre-implantation period, in which severe aplasia and immunodeficiency condition the occurrence of hemorrhagic and infectious complications, often serious. In addition, it constitutes an important tool for planning the next patient to be transplanted by the team, by reducing the time spent in planning the logistics required between one transplant and another, which will allow increasing the number of patients benefiting from this procedure.

Key words: systems of differential equations, multi-compartment models, hematopoiesis, bone marrow transplantation.

## Mathematical model of a predator-prey food chain: plankton-anchovy

### Neisser Pino Romero

#### Universidad Peruana Cayetano Heredia, Peru - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the present work, a mathematical model is built that represents the dynamics that exist between phytoplankton (F), zooplankton (Z) and anchovy (A) from the model of Samares and Anal. This food chain occurs in the Peruvian maritime area where there is a three-link ecosystem, in addition to the fact that the food chain is a main axis in the ecological balance within the sea.

The behavior of the populations (maritime species) will be studied where the stability of the model (Routh-Hurwitz criterion) in the long term will be determined, where it will be complemented with the Dulac-Bendixon criterion for the state of coexistence of the three species.

Therefore, the dynamics of this food chain allows to have a fundamental axis in the economy and ecology for the country where it is sought to have a balance and stability of these two important activities. Without a doubt, the application of a model that helps us locate predation rates that do not disturb or affect ecological populations. This complication directly affects the ecosystem, especially because man is a predator that can lead to the extinction of the anchovy population and its upper chains. Consequently, this balance must be determined by the Peruvian Sea Institution (IMARPE) to maintain the biological balance.

Joint work with Christian Ulises Salazar Fernández (Universidad Nacional de Ingeniería, Perú).

## Time sampling optimization for a growth model of $Pomacea\:canaliculata$ (Prosobranchia:Ampullaridae).

### Andrés O. Porta

#### Universidad Nacional de San Martín, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Freshwater mollusks of the American genus $ Pomacea$ (Gastropoda:Prosobranchia:Ampullariidae), commonly called apple snails, have been object of intensive ecological studies because of the important damage over crops caused by invasive species of this genus, particularly in southern and eastern Asia. Most of the modeling of the growth of the species has been done adjusting a von Bertalanffy's growth curve or a polinomial curve to empirical data. Only recently (Sutton $et\:al.$, 2017), an population approach have been implemented using a set of two partial differential equation, one per sex, for modeling the growth (in terms of weight) of $P.\:maculata$. For the females population density is: $ u_t(x,t)-(g(x)u(x,t))_{x}=-\mu u(x,t) $ where $u(t,x)$ represents the density of female snails of size $x$ (represented by the weight) at time $t$, $g(x)$ is the per capita growth rate of female apple snails of weight $x$ and death rate $\mu$, which is assumed as constant over the median lifespan of a female apple. An analogous equation is used for modeling the male snails density. As a preliminary of a work that has as objective the developing of populations models for some species of ampullarids of Argentina, here we present an optimal design for the time sampling of the growth of the snail $ Pomacea\:canaliculata$ based in both first hand and literature data. The underlying model consist of only one equation that, taking into account empirical data on the species growth, incorporates sexual differential growth as a delay in a first growth phase in the female. The optimal schema derived will be used for prospective studies in the population growth dynamics of the species both in laboratory and natural conditions.

Joint work with Diana Rubio (Universidad de San Martín, Argentina).

## Application of a fractional SIR model built with Mittag-Leffler distribution

### Sandro Rodrigues Mazorche

#### Universidade Federal de Juiz de Fora, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

Fractional Calculus has proven to be an especially useful tool in capturing the dynamics of the physical process of several scientific objects, being in general related to the “memory effect”. Generally, it consists of making a classic model more flexible by replacing an entire order derivative with an arbitrary one. Notably, compartmental models have been widely studied with arbitrary orders (e.g., [2], [4]).

Throughout the first author's master's research, we sought to investigate the use of arbitrary derivatives in SIR-type models, theoretically, analytically also numerically. We are interested in the questions of persistence of characteristics, starting from the classical modeling to discuss the difficulties in the construction of a non-artificial arbitrary model. What characteristics are maintained when exchanging orders? Are consistent models automatically established, regarding the definition of parameters, physical significance, conservation, and units? What about non-negativity, monotonicity (if any), among other issues? The analytical and numerical techniques provide an interesting field for research. However, from the modeling point of view, it is important to try to verify how, where, and why Fractional Calculus interfere in the model.

We believe that derivatives of an arbitrary order may arise through power-laws time-since-infection dependence in the infectiousness and removal functions. Thus, we present in [3] a physical derivation following in the footsteps of Angstmann, Henry & McGann [1], where they use the probabilistic language of Continuous Time Random Walks (CTRW) and Mittag-Leffler functions. The Riemann-Liouville derivative appears throughout the construction and the SIR model, with $ 1 \geq \beta \geq \alpha> 0 $, is given by \[ \dfrac{\omega(t)S(t)\theta(t,0)}{N\tau^\beta}D^{1-\beta}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\gamma(t)S(t),\] \[ \dfrac{dI(t)}{dt}=\dfrac{\omega(t)S(t)\theta(t,0)}{N\tau^\beta}D^{1-\beta}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\dfrac{\theta(t,0)}{\tau^\alpha}D^{1-\alpha}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\gamma(t)I(t), \] \[ \dfrac{dR(t)}{dt}=\dfrac{\theta(t,0)}{\tau^\alpha}D^{1-\alpha}\bigg(\dfrac{I(t)}{\theta(t,0)}\bigg)-\gamma(t)R(t),\] where $ \gamma (t) $ is the vital dynamic; $ \omega (t) $, extrinsic infectivity; $ N $, the total population; $ \tau $, a scale parameter and, $ \theta (t, t ') $, the probability that an infectious person since $ t' $ would not die of natural death until $ t $. If $ \beta = \alpha = 1 $, $ \gamma (t) \equiv \gamma $ and $ \omega (t) \equiv \omega $, we get the traditional SIR model. In [3], we revisited the authors' work, started the discussion of the reproduction number, and used optimization to apply the model to the data of the Brazilian and Italian pandemic of COVID-19. As explained in more detail in [3], the time of removal of the individual from the infectious compartment follows a Mittag-Leffler distribution related to $\alpha$, while the parameter $\beta$ relates to the law of the function of infectivity.

Here we pretend use this basis to explore current pandemic data of each state of Brazil, comparing similarities and differences about the laws of infectivity and recovery and equilibrium points. An interesting discussion would be a proposal to optimize the distribution of resources such as vaccines.

References

[1] Angstmann, C. N., Henry, B. I. and McGann, A. V. A fractional-order infectivity and recovery SIR model, Fractal and Fractional, 1(1), 11, 2017. DOI: 10.3390/fractalfract1010011.

[2] Cardoso, L. C., Santos, F. L. P. dos, and Camargo, R. F. Analysis of fractional-order models for hepatitis B, Computational and Applied Mathematics, 37(4), 4570-4586, 2018. DOI:10.1007/s40314-018-0588-4.

[3] Monteiro, N. Z. and Mazorche, S. R. Fractional Derivatives Applied to Epidemiology. TCAM,Trends in Computational and Applied Mathematics, 2021. (to appear)

[4] Santos, J. P. C. dos, Monteiro, E. and Vieira, G. B. Global stability of fractional SIR epidemic model, Proceeding Series of the Brazilian Society of Computational and Applied Mathematics,5(1), 2017. DOI: 10.5540/03.2017.005.01.0019.

Joint work with Noemi Zeraick Monteiro (Universidade Federal de Juiz de Fora, Brasil).