Mathematical model for the study of the diffusion of Zika. Computational experimentation in Paramaribo and Santa Ana

Zika virus spreads to people primarily through the bite of an infected Aedes aegypti species mosquito. But it Zika can also be passed through sex from an infected to his or her sex partners and it can be spread from a pregnant woman to her fetus. Zika continues to spreading geographically to areas where competent vectors are present. Although a decline in cases of Zika virus infection has been reported in some countries, or in some parts of countries, vigilance needs to remain high. In this work, we propose a mathematical model that uses diffusion-advection equations to study the impact of the Zika epidemic. We present a numerical scheme linking finite elements (FEM) with finite differences to solve the model. The computer simulations are performed for Paramaribo and Santa Ana that have different demographic characteristics and allow us to extend the study to other regions.El virus del Zika se propaga a las personas principalmente a través de la picadura de un mosquito de la especie Aedes Aegypti infectado. El Zika también puede transmitirse a través del sexo de una persona infectada a sus parejas sexuales y se puede transmitir de una mujer embarazada a su feto. El Zika continúa expandiéndose geográficamente a áreas donde están presentes vectores competentes. Si bien se ha informado una disminución en los casos de infección por el virus del Zika en algunos países o en algunas partes de los países, la vigilancia debe mantenerse alta. En este trabajo proponemos un modelo matemático que utiliza ecuaciones de difusión-advección para estudiar el impacto de la epidemia de Zika. Presentamos un esquema numérico vinculando elementos finitos (FEM) con diferencias finitas para resolver el modelo. Las simulaciones computacionales se realizan para Paramaribo y Santa Ana, que tienen diferentes características demográficas y nos permiten ampliar el estudio a otras regiones

Assessing the spatio-temporal spread of COVID-19 via compartmental models with diffusion in Italy, USA, and Brazil

The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation (ODE) models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of an epidemic in space. Such research on PDE models within a Susceptible, Infected, Exposed, Recovered, and Deceased (SEIRD) framework has led to promising results in reproducing COVID-19 contagion dynamics. In this paper, we assess the robustness of this modeling framework by considering different geometries over more extended periods than in other similar studies. We first validate our code by reproducing previously shown results for Lombardy, Italy. We then focus on the U.S. state of Georgia and on the Brazilian state of Rio de Janeiro, one of the most impacted areas in the world. Our results show good agreement with real-world epidemiological data in both time and space for all regions across major areas and across three different continents, suggesting that the modeling approach is both valid and robust.Comment: 23 pages, 19 figure

Novel codynamics of the HIV-1/HTLV-Ⅰ model involving humoral immune response and cellular outbreak: A new approach to probability density functions and fractional operators

Both human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type Ⅰ (HTLV-Ⅰ) are retroviruses that afflict CD4 T cells. In this article, the codynamics of within-host HIV-1 and HTLV-Ⅰ are presented via piecewise fractional differential equations by employing a stochastic system with an influential strategy for biological research. It is demonstrated that the scheme is mathematically and biologically feasible by illustrating that the framework has positive and bounded global findings. The necessary requirements are deduced, ensuring the virus's extinction. In addition, the structure is evaluated for the occurrence of an ergodic stationary distribution and sufficient requirements are developed. A deterministic-stochastic mechanism for simulation studies is constructed and executed in MATLAB to reveal the model's long-term behavior. Utilizing rigorous analysis, we predict that the aforesaid model is an improvement of the existing virus-to-cell and cell-to-cell interactions by investigating an assortment of behaviour patterns that include cross-over to unpredictability processes. Besides that, the piecewise differential formulations, which can be consolidated with integer-order, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic processes, have been declared to be exciting opportunities for researchers in a spectrum of disciplines by enabling them to incorporate distinctive features in various temporal intervals. As a result, by applying these formulations to difficult problems, researchers can achieve improved consequences in reporting realities with white noise. White noise in fractional HIV-1/HTLV-Ⅰ codynamics plays an extremely important function in preventing the proliferation of an outbreak when the proposed flow is constant and disease extermination is directly proportional to the magnitude of the white noise

Study of Virus Dynamics by Mathematical Models

This thesis studies virus dynamics within host by mathematical models, and topics discussed include viral release strategies, viral spreading mechanism, and interaction of virus with the immune system. Firstly, we propose a delay differential equation model with distributed delay to investigate the evolutionary competition between budding and lytic viral release strategies. We find that when antibody is not established, the dynamics of competition depends on the respective basic reproduction numbers of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproduction numbers of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, lytic virus can outcompete provided that its reproductive ratio is very high. An explicit threshold is derived. Secondly, we consider model containing two modes for viral infection and spread, one is the diffusion-limited free virus transmission and the other is the direct cell-to-cell transfer of viral particles. By incorporating infection age, a rigorous analysis of the model shows that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number, which is identified explicitly. The formula for the basic reproduction number of our model reveals the effects of various model parameters including the transmission rates of the two modes, and the impact of the infection age. We show that basic reproduction number is underestimated in the existing models that only consider the cell-free virus transmission, or the cell-to-cell infection, ignoring the other. Assuming logistic growth for target cells, we find that if the basic reproduction number is greater than one, the infection can persist and Hopf bifurcation can occur from the positive equilibrium within certain parameter ranges. Thirdly, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed, and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed shows that the repulsion of superinfecting vision promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, some recent experimental results. Finally, the effect of chemotactic movement of CD8+ cytotoxic T lymphocytes (CTLs) on HIV-1 infection dynamics is studied by a reaction diffusion model with chemotaxis. Choosing a typical chemosensitive function, we find that chemoattractive movement of CTLs due to HIV infection does not change stability of the positive steady state of the model. However, chemorepulsion movement of CTLs destabilizes the positive steady state as the strength of the chemotactic sensitivity increases. In this case, Turing instability occurs, which can be Hopf bifurcation or steady state bifurcation, and spatial heterogeneous patterns may form

Analysis of HIV-host interaction on different scales

The human immunodeficiency virus depends on molecular pathways of the host for efficient replication and spread. The intricate network of host-virus interactions shapes the virus\u27; evolution by driving the pathogen to evade immune recognition and constraining it to maintain its capacity to replicate. Study of the HIV-host interactions provides important insights into viral evolution, pathogenicity and potential treatment strategies. This thesis presents an analysis of HIV-host interactions on several scales, ranging from individual protein interactions to whole genomes. On the scale of individual interaction we analyze structural and physical determinants of the interaction between host TRIM5alpha and virus capsid — an interaction of potential therapeutic interest due to the capacity of TRIM5alpha to block retroviral infections. On the scale of viral population we present two studies of a highly variable region of the virus genome involved in the interaction with host cell coreceptors upon virus cell entry. The studies provide insights into the virus evolution and the physicochemical and structural properties related to its interaction with cellular coreceptors. On the scale of the single cell we develop models of HIV cell entry involving virus, host and environmental factors. The models represent a comprehensive picture of the virus phenotype and allow one to view the variability of virus phenotypes on 2D phenotype maps. On the genomic scale we perform a large-scale analysis of all HIV-host interactions. This study reveals insights into general patterns of the host-pathogen evolution and suggests candidate host proteins involved in interactions potentially important for the infection and interesting for further study on other scales. Interactions and processes crucial for the HIV infection reemerge across the scales pointing to the importance of integrative, multi-scale studies of host-pathogen biology.Das Humane Immundefizienz-Virus hängt von molekularen Mechanismen des Wirts für seine effiziente Replikation und Ausbreitung ab. Das komplizierte Netzwerk von Wirt-Virus Interaktionen formt die Evolution des Virus, indem es den Erreger dazu bringt, sich der Erkennung durch das Immunsystem zu entziehen und seine Replikationskapazität aufrecht zu erhalten. Das Studium der HIV-Wirt Interaktionen erlaubt wichtige Einblicke in die viralen Evolution, die Pathogenität des Virus, sowie mögliche Behandlungsstrategien. Diese Arbeit stellt eine Analyse der HIV-Wirt-Interaktionen in mehreren Größenordnungen vor, von einzelnen Protein-Interaktionen bis hin zur Analyse ganzer Genome. In Hinblick auf einzelne Interaktionen untersuchen wir strukturelle und physikalische Determinanten der Interaktion zwischen dem Wirtfaktor TRIM5alpha; und dem viralen Kapsid - eine Interaktion, die von therapeutischem Interesse ist wegen der Fähigkeit von TRIM5alpha, retrovirale Infektionen zu blockieren. In Hinblick auf virale Populationen präsentieren wir zwei Studien einer hochvariablen Region des viralen Genoms, die in der Interaktion des Virus mit zellulären Rezeptoren des Wirts beim viralen Zelleintritt involviert sind. Diese Studien geben Einblick in die virale Evolution und die physikalisch-chemischen und strukturellen Eigenschaften des Virus bezüglich dessen Interaktion mit zellulären Ko-Rezeptoren. Auf der Skala der einzelnen Zelle entwickeln wir Modelle des HIV Zelleintritts welche das Virus, den Wirt und Umgebungsfaktoren berücksichtigen. Diese Modelle bieten ein umfassendes Bild des viralen Phänotyps und erlauben es, die Variabilität des Virus auf 2D-Phänotyp-Karten zu visualisieren. Im genomweiten Maßstab führen wir eine groß angelegte Analyse aller HIV-Wirt-Interaktionen durch. Diese Studie erlaubt Einblicke in allgemeine Muster der Wirt-Pathogen-Evolution und identifiziert Kandidaten für Wirtsproteine, deren Interaktionen potenziell wichtig für die virale Infektion sind und deren weitere Untersuchung in anderen Größenordnungen von Interesse ist. Interaktionen und Prozesse, die von entscheidender Bedeutung für die HIV-Infektion sind zeigen sich wiederholt in allen untersuchten Maßstäben und unterstreichen die Bedeutung einer integrativen und multi-skalaren Untersuchung der Wirt-Pathogen-Biologie

A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world

This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS–CoV–2 Severe Acute Respiratory Syndrome, corona virus n.2. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives.publishedVersionFil: Bellomo, Nicola. Universidad de Granada. Departamento de Matemática Aplicada; España.Fil: Bingham, Richard. University of York. Departments of Mathematics and Biology. Cross-disciplinary Centre for Systems Analysis; United Kingdom.Fil: Chaplain, Mark A. J. University of St Andrews. School of Mathematics and Statistics; Scotland.Fil: Dosi, Giovanni. Scuola Superiore Sant’Anna. Institute of Economics and EMbeDS; Italia.Fil: Forni, Guido. Accademia Nazionale dei Lincei; Italia.Fil: Knopoff, Damian A. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.Fil: Knopoff, Damian A. Consejo Nacional de Investigaciones Científicas y Técnicas de Argentina. Centro de Investigacion y Estudios de Matematica; Argentina.Fil: Lowengrub, John. University California Irvine. Department of Mathematics; United States.Fil: Twarock, Reidun. University of York. Departments of Mathematics and Biology. Cross-disciplinary Centre for Systems Analysis; United Kingdom.Fil: Virgillito, Maria Enrica.Scuola Superiore Sant’Anna. Institute of Economics and EMbeDS; Italia

Nonlocal Models in Biology and Life Sciences: Sources, Developments, and Applications

Nonlocality is important in realistic mathematical models of physical and biological systems at small-length scales. It characterizes the properties of two individuals located in different locations. This review illustrates different nonlocal mathematical models applied to biology and life sciences. The major focus has been given to sources, developments, and applications of such models. Among other things, a systematic discussion has been provided for the conditions of pattern formations in biological systems of population dynamics. Special attention has also been given to nonlocal interactions on networks, network coupling and integration, including models for brain dynamics that provide us with an important tool to better understand neurodegenerative diseases. In addition, we have discussed nonlocal modelling approaches for cancer stem cells and tumor cells that are widely applied in the cell migration processes, growth, and avascular tumors in any organ. Furthermore, the discussed nonlocal continuum models can go sufficiently smaller scales applied to nanotechnology to build biosensors to sense biomaterial and its concentration. Piezoelectric and other smart materials are among them, and these devices are becoming increasingly important in the digital and physical world that is intrinsically interconnected with biological systems. Additionally, we have reviewed a nonlocal theory of peridynamics, which deals with continuous and discrete media and applies to model the relationship between fracture and healing in cortical bone, tissue growth and shrinkage, and other areas increasingly important in biomedical and bioengineering applications. Finally, we provided a comprehensive summary of emerging trends and highlighted future directions in this rapidly expanding field.Comment: 71 page

Spatio-temporal dynamics of some reaction-diffusion population models in heterogeneous environments

Spatial and temporal evolutions are very important topics in epidemiology and ecology. This thesis is devoted to the study of global dynamics of some reaction-diffusion models incorporating environmental heterogeneities. As biological invasions significantly impact ecology and human society, how invasive species’ growth and spatial spread interact with the environment becomes a significant challenging problem. We start with an impulsive time-space periodic model to describe a single species with a birth pulse in the reproductive stage in Chapter 2. In-host viral infections commonly involve hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To explore the effects of the spread heterogeneity on the spread of within-host virus, we propose a time-delayed nonlocal reaction-diffusion model and obtain the threshold-type results in terms of the basic reproduction ratio in Chapter 3. In Chapter 4, we then explore the existence and nonexistence of traveling wave solutions for such a non-monotone system on an unbounded domain, and show that there is a minimum wave speed for traveling waves connecting the infection-free equilibrium and the endemic equilibrium. Mosquito-borne diseases are transmitted by the bite of infected mosquitoes, including Zika, West Nile, Chikungunya, dengue, and malaria. To investigate the effects of spatial and temporal heterogeneity on the spread of the Chikungunya virus, we develop a nonlocal periodic reaction-diffusion model of Chikungunya disease with periodic time delays in Chapter 5. We further establish two threshold-type results regarding the global dynamics of mosquito growth and disease transmission, respectively. At the end of this thesis, a brief summary and some future works are presented