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Papers by Cruz Vargas-De-León

Applied Mathematics and Computation, Sep 1, 2014

In this paper, we introduce and analyze two structured models for the transmission of a vector-bo... more In this paper, we introduce and analyze two structured models for the transmission of a vector-borne infectious disease. The first of these models assumes that the level of contagiousness and the rate of removal (recovery) of infected individuals depends on the infection age. In the second model the hosts population is structured with respect to the physical age of the hosts, and the susceptibility of the hosts is assumed to be age-dependent. For these models, the threshold parameter for the existence of a positive (endemic) equilibrium state is determined, and the global asymptotic stability of the equilibrium states are established by the Lyapunov's direct method.

Acta Applicandae Mathematicae, May 13, 2014

We consider a mathematical model for malaria transmission which takes into account of the increas... more We consider a mathematical model for malaria transmission which takes into account of the increase of host's attractiveness to mosquitoes when the host harbours the parasite's gametocytes. We investigate how this behavioral manipulation by malaria parasite may impact the optimal interventions targeted to infectious humans like treatment and screening activities. In particular, our analysis suggests that it may produce an increase of total costs associated to the disease and its control. Keywords Malaria • Mathematical model • Optimal control 1 Introduction Malaria is a life-threatening disease which is vector-borne, in the sense that it is transmitted to susceptible humans through the bites of infected female mosquitoes of the genus Anopheles (the 'vectors'). The latest estimates indicate a worldwide substantial reduction of reported malaria cases and deaths, but, in absolute terms, data also show that malaria is still a global emergency. Malaria mortality rates have fallen by more than 25 % globally from 2000 to 2010 [21, 31]. However, in 2010 there were about 219 million cases and malaria was the underlying cause of death for 660 000 individuals [31] (1,24 million, according to [21]). Nowadays many aspects of the biology of malaria are known. In particular, much is known about the parasites that cause malaria in humans, the protozoan Plasmodium, the disease transmission conditions and the life cycle in both humans and vectors [28]. On the

Applied Mathematics and Computation, May 1, 2015

In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and... more In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and a discrete intracellular delay: the delay corresponds to the time between infection of a infected target hepatocytes and production of new HCV particles. We establish the global stability of the infection-free equilibrium and existence, uniqueness, local and global stabilities of the infected equilibrium, also we establish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. The unique infected equilibrium is globally-asymptotically stable for a special case, where the hepatotropic virus is non-cytopathic. We present a sensitivity analysis for the basic reproductive number. Numerical simulations are carried out to illustrate the analytical results.

International Journal of Contemporary Mathematical Sciences, 2013

In this paper we study the existence of Dulac functions for planar differential systems with pert... more In this paper we study the existence of Dulac functions for planar differential systems with perturbations or under algebraic operations (addition and multiplication) on vector fields, and also, the transformation of vector fields under affine transformations. We give some applications and examples in order to illustrate the applicability of the results.

PLOS ONE, Jul 19, 2017

Computational modeling has been applied to simulate the heterogeneity of cancer behavior. The dev... more Computational modeling has been applied to simulate the heterogeneity of cancer behavior. The development of Cervical Cancer (CC) is a process in which the cell acquires dynamic behavior from non-deleterious and deleterious mutations, exhibiting chromosomal alterations as a manifestation of this dynamic. To further determine the progression of chromosomal alterations in precursor lesions and CC, we introduce a computational model to study the dynamics of deleterious and non-deleterious mutations as an outcome of tumor progression. The analysis of chromosomal alterations mediated by our model reveals that multiple deleterious mutations are more frequent in precursor lesions than in CC. Cells with lethal deleterious mutations would be eliminated, which would mitigate cancer progression; on the other hand, cells with non-deleterious mutations would become dominant, which could predispose them to cancer progression. The study of somatic alterations through computer simulations of cancer progression provides a feasible pathway for insights into the transformation of cell mechanisms in humans. During cancer progression, tumors may acquire new phenotype traits, such as the ability to invade and metastasize or to become clinically important when they develop drug resistance. Non-deleterious chromosomal alterations contribute to this progression.

Mathematical Methods in The Applied Sciences, Mar 3, 2014

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellula... more In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra-type functions, composite quadratic functions and Volterra-type functionals, we provide the global stability for this model. If R 0 , the basic reproductive number, satisfies R 0 Ä 1, then the infection-free equilibrium state is globally asymptotically stable. Our system is persistent if R 0 > 1. On the other hand, if R 0 > 1, then infection-free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R 0 > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations.

Public Health, Aug 1, 2020

Objectives: Even when new cases of syphilis are notifiable since 1944, the Mexican National Epide... more Objectives: Even when new cases of syphilis are notifiable since 1944, the Mexican National Epidemiological Surveillance System lacks information on the changes of the rate of case reports considering the geographic and demographic variables. Therefore, it is necessary to have evidence, with particular attention to the study of the epidemiological behavior by the identification of risk factors and groups. The objective of this study was to analyze the epidemiology, geographical distribution, and forecast of syphilis in Mexico. Study design: The design of the study was a secondary research of epidemiological databases. Methods: A retrospective analysis of the national surveillance data (2007e2017) of acquired and congenital syphilis (CS) issued by the General Directorate of Epidemiology was performed. Results: Of all cases, 34,998 and 1030 cases were reported for acquired syphilis (AS) and CS , respectively, reflecting an increasing trend in the whole country for both diseases. Cases and incidence of AS per year showed that, male gender presented an increase in reproductive age. Distribution of the rate of case reports is mostly commanded by the states in the extreme north (Gulf of California and northern Gulf of Mexico) and south (Gulf of southern Mexico and the Caribbean Sea). Likewise, the Seasonal Autoregressive Integrated Moving Average model was selected as the best-fit model for the forecast analysis. This model was used to forecast AS cases during 2018e2019. AS may have a slight fluctuation (on the rise) during the following 24 months. Conclusions: These findings underscore the importance of intensifying, as well as expanding screening and treatment in adult population, including men, who are not routinely benefiting from maternal and reproductive service-based syphilis screening and treatment.

Mathematical Methods in The Applied Sciences, Sep 10, 2015

The purpose of this paper is to study the global stability properties of equilibria for age-depen... more The purpose of this paper is to study the global stability properties of equilibria for age-dependent epidemiological models in presence of recurrence phenomenon. In these systems, the recurrence rate depends on asymptomatic-infectionage. The models are appropriate for human herpes virus (HSV-1 and HSV-2) and varicella-zoster virus. We derived explicit formulas for the basic reproductive number, which completely characterizes the global behaviour of solutions to the models: if the basic reproductive number is less than or equal to unity, the disease will die out; if the basic reproductive number is greater than unity, the disease will be persistent. Volterra-type Lyapunov functions are constructed to establish the global asymptotic stability of the infection-free and endemic steady states.

Mathematical and Computer Modelling, Aug 1, 2009

This paper presents a deterministic model for monitoring the impact of drug resistance on the tra... more This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a disease-free equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is ...

Chaos Solitons & Fractals, Dec 1, 2011

In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic... more In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic models with constant recruitment, disease-induced death and standard incidence rate. We will make ingenious linear combination of known functions, common quadratic and Volterra-type, and of a new class of functions, we call composite-Volterra function, for obtain a suitable Lyapunov functions. In particular, for SIRS model we prove the global stability of the endemic equilibrium under a condition of parameters.

Computational and Mathematical Methods in Medicine, Aug 23, 2022

The aim of this paper is to model the dynamics of the human papillomavirus (HPV) in cervical epit... more The aim of this paper is to model the dynamics of the human papillomavirus (HPV) in cervical epithelial cells. We developed a mathematical model of the epithelial cellular dynamics of the stratified epithelium of three (basale, intermedium, and corneum) stratums that is based on three ordinary differential equations. We determine the biological condition for the existence of the epithelial cell homeostasis equilibrium, and we obtain the necessary and sufficient conditions for its global stability using the method of Lyapunov functions and a theorem on limiting systems. We have also developed a mathematical model based on seven ordinary differential equations that describes the dynamics of HPV infection. We calculated the basic reproductive number (R 0) of the infection using the next-generation operator method. We determine the existence and the local stability of the equilibrium point of the cellular homeostasis of the epithelium. We then give a sufficient condition for the global asymptotic stability of the epithelial cell homeostasis equilibrium using the Lyapunov function method. We proved that this equilibrium point is nonhyperbolic when R 0 = 1 and that in this case, the system presents a forward bifurcation, which shows the existence of an infected equilibrium point when R 0 > 1. We also study the solutions numerically (i.e., viral kinetic in silico) when R 0 > 1. Finally, local sensitivity index was calculated to assess the influence of different parameters on basic reproductive number. Our model reproduces the transient, acute, latent, and chronic infections that have been reported in studies of the natural history of HPV.

SAGE Open, 2022

The goal of the study was to evaluate and adjust the model that associates mathematical motivatio... more The goal of the study was to evaluate and adjust the model that associates mathematical motivation and learning strategies as quantitative instruments. The items related to the task value, cost, and self-efficacy were validated with Mexican students of rural areas in south-west region of Mexico between 12 and 16 years old, using 14 items that measure self-reported motivation levels. The construct validity of the mathematics motivation scale was checked using confirmatory factor analysis (CFA), by analyzing first- and second-order CFA models. The factor reliability was investigated by means of Dillon-Goldstein’s rho and Cronbach’s alpha. The model had adequate goodness of fit of in the confirmatory analysis. The instrument proved that it is convenient and reliable for application in mathematics motivation in Mexican adolescents. An advantage of the use of this instrument to be applied by teachers of mathematics is its simplicity, ease of application, and interpretation.

DOAJ (DOAJ: Directory of Open Access Journals), Jul 1, 2019

In this work, we considered a family of SIRS models for a fatal disease, with seasonal variation ... more In this work, we considered a family of SIRS models for a fatal disease, with seasonal variation in the contact rate and isolation control strategies. We establish the existence of periodic orbits of seasonal SIQRS disease, by using Leray-Schauder degree theory. Examples related to the seasonal variation in respiratory syncytial virus infection are included.

Mathematical Biosciences and Engineering, 2022

Age as a risk factor is common in vector-borne infectious diseases. This is partly because childr... more Age as a risk factor is common in vector-borne infectious diseases. This is partly because children depend on adults to take preventative measures, and adults are less susceptible to mosquito bites because they generally spend less time outdoors than children. We propose a dengue disease model that considers the human population as divided into two subpopulations: children and adults. This is in order to take into consideration that children are more likely than adults to be bitten by mosquitoes. We calculated the basic reproductive number of dengue, using the next-generation operator method. We determined the local and global stability of the disease-free equilibrium. We obtained sufficient conditions for the global asymptotic stability of the endemic equilibrium using the Lyapunov functional method. When the infected periods in children and adults are the same, we that the endemic equilibrium is globally asymptotically stable in the interior of the feasible region when the threshold quantity $ R_0 > 1 .Additionally,weperformedanumericalsimulationusingparametervaluesobtainedfromtheliterature.Finally,alocalsensitivityanalysiswasperformedtoidentifytheparametersthathavethegreatestinfluenceonchangesin. Additionally, we performed a numerical simulation using parameter values obtained from the literature. Finally, a local sensitivity analysis was performed to identify the parameters that have the greatest influence on changes in .Additionally,weperformedanumericalsimulationusingparametervaluesobtainedfromtheliterature.Finally,alocalsensitivityanalysiswasperformedtoidentifytheparametersthathavethegreatestinfluenceonchangesin (R_0) $, and thereby obtain a better biological interpretation of the results.

Applied Mathematics and Computation, Sep 1, 2012

A viral infection model of HBV infection of hepatocytes with ''cure'' of infected cells and intra... more A viral infection model of HBV infection of hepatocytes with ''cure'' of infected cells and intracellular delay is studied. The delay corresponds to the time necessary for a newly produced virion to become infectious particles. We prove that the stability is completely determined by the basic reproductive number R 0 ðsÞ. If R 0 ðsÞ 1, the infection-free steady state is globally asymptotically stable. If R 0 ðsÞ > 1 then infection-free steady state becomes unstable and a unique infected steady state exists and is locally asymptotically stable. On the other hand, we derive sufficient conditions for the global asymptotic stability of the infected steady state. Numerical simulations are presented to illustrate the results.

Mathematical Biosciences and Engineering, 2023

Mathematical Methods in The Applied Sciences, Oct 28, 2021

A diffusive multispecies Lotka–Volterra mutualism model is studied in this paper, where the mutua... more A diffusive multispecies Lotka–Volterra mutualism model is studied in this paper, where the mutualistic interaction coefficients and the diffusion coefficient are spatially heterogeneous. The architecture of these models is one of a hyper‐connected central species that interacts with n − 1 peripheral species, but the peripheral species do not interact with each other. By using Lyapunov functional method, the global asymptotic stability of the nonhomogeneous coexisting equilibrium state is established. We extend our results to the multispecies mutualism model in which interaction and diffusion coefficients are spatially homogeneous.

Nonlinear Analysis-Modelling and Control, Jul 20, 2017

In this paper, we construct Dulac functions for a family of planar differential equations. We pro... more In this paper, we construct Dulac functions for a family of planar differential equations. We provide some conditions on the components of a vector field, which ensure the existence of Dulac functions for such vector field. We also present some applications and examples in biomathematical models to illustrate our results.

Canadian Journal of Infectious Diseases & Medical Microbiology, Apr 25, 2022

Background. Evidence from across the world suggests that the pediatric population shows di erent ... more Background. Evidence from across the world suggests that the pediatric population shows di erent clinical manifestations and has a lower risk of severe presentation of SARS-CoV-2 infection compared to adults. However, Mexico has one of the highest mortality rates in the pediatric population due to SARS-CoV-2 infection. erefore, our objective was to explore the epidemiological and clinical characteristics associated with a positive con rmatory test in the Mexican pediatric population admitted to a tertiary care hospital in Mexico City. Methods. Clinical, imaging and laboratory data were retrospectively collected from 121 children hospitalized during the period from March 4 th , 2020, to August 8 th , 2021. e patients were identi ed as suspicious cases according to the guidelines of the General Directorate of Epidemiology of Mexico. Real-time polymerase chain reaction (RT-PCR) tests were used to con rm SARS-CoV-2 infection. Categorical variables were compared using the Chi-square test, and propensity score matching was performed to determine univariate and multivariate odds ratios of the population regarding a positive vs. negative SARS-CoV-2 result. Results. Of the 121 children, 36 had laboratory-con rmed SARS-CoV-2 infection. e main risk for SARS-CoV-2-associated pediatric hospitalization was contact with a family member with SARS-CoV-2. It was also found that fever and fatigue were statistically signi cantly associated with a positive SARS-CoV-2 test in multivariate models. Clinical and laboratory data in this Mexican hospitalized pediatric cohort di er from other reports worldwide; the mortality rate (1.6%) of the population studied was higher than that seen in reports from other countries. Conclusion. Our study found that fever and fatigue at hospital presentation as well as an antecedent exposure to a family member with SARS-CoV-2 infection were important risk factors for SARS-CoV-2 positivity in children at hospital admission.

Chaos Solitons & Fractals, Mar 1, 2023

The aim of this paper is to provide a mathematical study of the amount of drug administered as a ... more The aim of this paper is to provide a mathematical study of the amount of drug administered as a continuous intravenous infusion or oral dose. For this purpose, we consider fractional-order mammillary-type models describing the anomalous dynamics of exchange of concentrations between compartments at, constant input rates, power-law type, and in the form of oral doses at given (discrete) times. We have developed a general analysis strategy for these models, in which we have found closed-form analytical solutions written in terms of the multivariate Mittag-Leffler function. Numerical simulations have been performed using our formulas, with parameters from the literature.

Applied Mathematics and Computation, Sep 1, 2014

In this paper, we introduce and analyze two structured models for the transmission of a vector-bo... more In this paper, we introduce and analyze two structured models for the transmission of a vector-borne infectious disease. The first of these models assumes that the level of contagiousness and the rate of removal (recovery) of infected individuals depends on the infection age. In the second model the hosts population is structured with respect to the physical age of the hosts, and the susceptibility of the hosts is assumed to be age-dependent. For these models, the threshold parameter for the existence of a positive (endemic) equilibrium state is determined, and the global asymptotic stability of the equilibrium states are established by the Lyapunov's direct method.

Acta Applicandae Mathematicae, May 13, 2014

We consider a mathematical model for malaria transmission which takes into account of the increas... more We consider a mathematical model for malaria transmission which takes into account of the increase of host's attractiveness to mosquitoes when the host harbours the parasite's gametocytes. We investigate how this behavioral manipulation by malaria parasite may impact the optimal interventions targeted to infectious humans like treatment and screening activities. In particular, our analysis suggests that it may produce an increase of total costs associated to the disease and its control. Keywords Malaria • Mathematical model • Optimal control 1 Introduction Malaria is a life-threatening disease which is vector-borne, in the sense that it is transmitted to susceptible humans through the bites of infected female mosquitoes of the genus Anopheles (the 'vectors'). The latest estimates indicate a worldwide substantial reduction of reported malaria cases and deaths, but, in absolute terms, data also show that malaria is still a global emergency. Malaria mortality rates have fallen by more than 25 % globally from 2000 to 2010 [21, 31]. However, in 2010 there were about 219 million cases and malaria was the underlying cause of death for 660 000 individuals [31] (1,24 million, according to [21]). Nowadays many aspects of the biology of malaria are known. In particular, much is known about the parasites that cause malaria in humans, the protozoan Plasmodium, the disease transmission conditions and the life cycle in both humans and vectors [28]. On the

Applied Mathematics and Computation, May 1, 2015

In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and... more In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and a discrete intracellular delay: the delay corresponds to the time between infection of a infected target hepatocytes and production of new HCV particles. We establish the global stability of the infection-free equilibrium and existence, uniqueness, local and global stabilities of the infected equilibrium, also we establish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. The unique infected equilibrium is globally-asymptotically stable for a special case, where the hepatotropic virus is non-cytopathic. We present a sensitivity analysis for the basic reproductive number. Numerical simulations are carried out to illustrate the analytical results.

International Journal of Contemporary Mathematical Sciences, 2013

In this paper we study the existence of Dulac functions for planar differential systems with pert... more In this paper we study the existence of Dulac functions for planar differential systems with perturbations or under algebraic operations (addition and multiplication) on vector fields, and also, the transformation of vector fields under affine transformations. We give some applications and examples in order to illustrate the applicability of the results.

PLOS ONE, Jul 19, 2017

Computational modeling has been applied to simulate the heterogeneity of cancer behavior. The dev... more Computational modeling has been applied to simulate the heterogeneity of cancer behavior. The development of Cervical Cancer (CC) is a process in which the cell acquires dynamic behavior from non-deleterious and deleterious mutations, exhibiting chromosomal alterations as a manifestation of this dynamic. To further determine the progression of chromosomal alterations in precursor lesions and CC, we introduce a computational model to study the dynamics of deleterious and non-deleterious mutations as an outcome of tumor progression. The analysis of chromosomal alterations mediated by our model reveals that multiple deleterious mutations are more frequent in precursor lesions than in CC. Cells with lethal deleterious mutations would be eliminated, which would mitigate cancer progression; on the other hand, cells with non-deleterious mutations would become dominant, which could predispose them to cancer progression. The study of somatic alterations through computer simulations of cancer progression provides a feasible pathway for insights into the transformation of cell mechanisms in humans. During cancer progression, tumors may acquire new phenotype traits, such as the ability to invade and metastasize or to become clinically important when they develop drug resistance. Non-deleterious chromosomal alterations contribute to this progression.

Mathematical Methods in The Applied Sciences, Mar 3, 2014

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellula... more In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra-type functions, composite quadratic functions and Volterra-type functionals, we provide the global stability for this model. If R 0 , the basic reproductive number, satisfies R 0 Ä 1, then the infection-free equilibrium state is globally asymptotically stable. Our system is persistent if R 0 > 1. On the other hand, if R 0 > 1, then infection-free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R 0 > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations.

Public Health, Aug 1, 2020

Objectives: Even when new cases of syphilis are notifiable since 1944, the Mexican National Epide... more Objectives: Even when new cases of syphilis are notifiable since 1944, the Mexican National Epidemiological Surveillance System lacks information on the changes of the rate of case reports considering the geographic and demographic variables. Therefore, it is necessary to have evidence, with particular attention to the study of the epidemiological behavior by the identification of risk factors and groups. The objective of this study was to analyze the epidemiology, geographical distribution, and forecast of syphilis in Mexico. Study design: The design of the study was a secondary research of epidemiological databases. Methods: A retrospective analysis of the national surveillance data (2007e2017) of acquired and congenital syphilis (CS) issued by the General Directorate of Epidemiology was performed. Results: Of all cases, 34,998 and 1030 cases were reported for acquired syphilis (AS) and CS , respectively, reflecting an increasing trend in the whole country for both diseases. Cases and incidence of AS per year showed that, male gender presented an increase in reproductive age. Distribution of the rate of case reports is mostly commanded by the states in the extreme north (Gulf of California and northern Gulf of Mexico) and south (Gulf of southern Mexico and the Caribbean Sea). Likewise, the Seasonal Autoregressive Integrated Moving Average model was selected as the best-fit model for the forecast analysis. This model was used to forecast AS cases during 2018e2019. AS may have a slight fluctuation (on the rise) during the following 24 months. Conclusions: These findings underscore the importance of intensifying, as well as expanding screening and treatment in adult population, including men, who are not routinely benefiting from maternal and reproductive service-based syphilis screening and treatment.

Mathematical Methods in The Applied Sciences, Sep 10, 2015

The purpose of this paper is to study the global stability properties of equilibria for age-depen... more The purpose of this paper is to study the global stability properties of equilibria for age-dependent epidemiological models in presence of recurrence phenomenon. In these systems, the recurrence rate depends on asymptomatic-infectionage. The models are appropriate for human herpes virus (HSV-1 and HSV-2) and varicella-zoster virus. We derived explicit formulas for the basic reproductive number, which completely characterizes the global behaviour of solutions to the models: if the basic reproductive number is less than or equal to unity, the disease will die out; if the basic reproductive number is greater than unity, the disease will be persistent. Volterra-type Lyapunov functions are constructed to establish the global asymptotic stability of the infection-free and endemic steady states.

Mathematical and Computer Modelling, Aug 1, 2009

This paper presents a deterministic model for monitoring the impact of drug resistance on the tra... more This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a disease-free equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is ...

Chaos Solitons & Fractals, Dec 1, 2011

In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic... more In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic models with constant recruitment, disease-induced death and standard incidence rate. We will make ingenious linear combination of known functions, common quadratic and Volterra-type, and of a new class of functions, we call composite-Volterra function, for obtain a suitable Lyapunov functions. In particular, for SIRS model we prove the global stability of the endemic equilibrium under a condition of parameters.

Computational and Mathematical Methods in Medicine, Aug 23, 2022

The aim of this paper is to model the dynamics of the human papillomavirus (HPV) in cervical epit... more The aim of this paper is to model the dynamics of the human papillomavirus (HPV) in cervical epithelial cells. We developed a mathematical model of the epithelial cellular dynamics of the stratified epithelium of three (basale, intermedium, and corneum) stratums that is based on three ordinary differential equations. We determine the biological condition for the existence of the epithelial cell homeostasis equilibrium, and we obtain the necessary and sufficient conditions for its global stability using the method of Lyapunov functions and a theorem on limiting systems. We have also developed a mathematical model based on seven ordinary differential equations that describes the dynamics of HPV infection. We calculated the basic reproductive number (R 0) of the infection using the next-generation operator method. We determine the existence and the local stability of the equilibrium point of the cellular homeostasis of the epithelium. We then give a sufficient condition for the global asymptotic stability of the epithelial cell homeostasis equilibrium using the Lyapunov function method. We proved that this equilibrium point is nonhyperbolic when R 0 = 1 and that in this case, the system presents a forward bifurcation, which shows the existence of an infected equilibrium point when R 0 > 1. We also study the solutions numerically (i.e., viral kinetic in silico) when R 0 > 1. Finally, local sensitivity index was calculated to assess the influence of different parameters on basic reproductive number. Our model reproduces the transient, acute, latent, and chronic infections that have been reported in studies of the natural history of HPV.

SAGE Open, 2022

The goal of the study was to evaluate and adjust the model that associates mathematical motivatio... more The goal of the study was to evaluate and adjust the model that associates mathematical motivation and learning strategies as quantitative instruments. The items related to the task value, cost, and self-efficacy were validated with Mexican students of rural areas in south-west region of Mexico between 12 and 16 years old, using 14 items that measure self-reported motivation levels. The construct validity of the mathematics motivation scale was checked using confirmatory factor analysis (CFA), by analyzing first- and second-order CFA models. The factor reliability was investigated by means of Dillon-Goldstein’s rho and Cronbach’s alpha. The model had adequate goodness of fit of in the confirmatory analysis. The instrument proved that it is convenient and reliable for application in mathematics motivation in Mexican adolescents. An advantage of the use of this instrument to be applied by teachers of mathematics is its simplicity, ease of application, and interpretation.

DOAJ (DOAJ: Directory of Open Access Journals), Jul 1, 2019

In this work, we considered a family of SIRS models for a fatal disease, with seasonal variation ... more In this work, we considered a family of SIRS models for a fatal disease, with seasonal variation in the contact rate and isolation control strategies. We establish the existence of periodic orbits of seasonal SIQRS disease, by using Leray-Schauder degree theory. Examples related to the seasonal variation in respiratory syncytial virus infection are included.

Mathematical Biosciences and Engineering, 2022

Age as a risk factor is common in vector-borne infectious diseases. This is partly because childr... more Age as a risk factor is common in vector-borne infectious diseases. This is partly because children depend on adults to take preventative measures, and adults are less susceptible to mosquito bites because they generally spend less time outdoors than children. We propose a dengue disease model that considers the human population as divided into two subpopulations: children and adults. This is in order to take into consideration that children are more likely than adults to be bitten by mosquitoes. We calculated the basic reproductive number of dengue, using the next-generation operator method. We determined the local and global stability of the disease-free equilibrium. We obtained sufficient conditions for the global asymptotic stability of the endemic equilibrium using the Lyapunov functional method. When the infected periods in children and adults are the same, we that the endemic equilibrium is globally asymptotically stable in the interior of the feasible region when the threshold quantity $ R_0 > 1 .Additionally,weperformedanumericalsimulationusingparametervaluesobtainedfromtheliterature.Finally,alocalsensitivityanalysiswasperformedtoidentifytheparametersthathavethegreatestinfluenceonchangesin. Additionally, we performed a numerical simulation using parameter values obtained from the literature. Finally, a local sensitivity analysis was performed to identify the parameters that have the greatest influence on changes in .Additionally,weperformedanumericalsimulationusingparametervaluesobtainedfromtheliterature.Finally,alocalsensitivityanalysiswasperformedtoidentifytheparametersthathavethegreatestinfluenceonchangesin (R_0) $, and thereby obtain a better biological interpretation of the results.

Applied Mathematics and Computation, Sep 1, 2012

A viral infection model of HBV infection of hepatocytes with ''cure'' of infected cells and intra... more A viral infection model of HBV infection of hepatocytes with ''cure'' of infected cells and intracellular delay is studied. The delay corresponds to the time necessary for a newly produced virion to become infectious particles. We prove that the stability is completely determined by the basic reproductive number R 0 ðsÞ. If R 0 ðsÞ 1, the infection-free steady state is globally asymptotically stable. If R 0 ðsÞ > 1 then infection-free steady state becomes unstable and a unique infected steady state exists and is locally asymptotically stable. On the other hand, we derive sufficient conditions for the global asymptotic stability of the infected steady state. Numerical simulations are presented to illustrate the results.

Mathematical Biosciences and Engineering, 2023

Mathematical Methods in The Applied Sciences, Oct 28, 2021

A diffusive multispecies Lotka–Volterra mutualism model is studied in this paper, where the mutua... more A diffusive multispecies Lotka–Volterra mutualism model is studied in this paper, where the mutualistic interaction coefficients and the diffusion coefficient are spatially heterogeneous. The architecture of these models is one of a hyper‐connected central species that interacts with n − 1 peripheral species, but the peripheral species do not interact with each other. By using Lyapunov functional method, the global asymptotic stability of the nonhomogeneous coexisting equilibrium state is established. We extend our results to the multispecies mutualism model in which interaction and diffusion coefficients are spatially homogeneous.

Nonlinear Analysis-Modelling and Control, Jul 20, 2017

In this paper, we construct Dulac functions for a family of planar differential equations. We pro... more In this paper, we construct Dulac functions for a family of planar differential equations. We provide some conditions on the components of a vector field, which ensure the existence of Dulac functions for such vector field. We also present some applications and examples in biomathematical models to illustrate our results.

Canadian Journal of Infectious Diseases & Medical Microbiology, Apr 25, 2022

Background. Evidence from across the world suggests that the pediatric population shows di erent ... more Background. Evidence from across the world suggests that the pediatric population shows di erent clinical manifestations and has a lower risk of severe presentation of SARS-CoV-2 infection compared to adults. However, Mexico has one of the highest mortality rates in the pediatric population due to SARS-CoV-2 infection. erefore, our objective was to explore the epidemiological and clinical characteristics associated with a positive con rmatory test in the Mexican pediatric population admitted to a tertiary care hospital in Mexico City. Methods. Clinical, imaging and laboratory data were retrospectively collected from 121 children hospitalized during the period from March 4 th , 2020, to August 8 th , 2021. e patients were identi ed as suspicious cases according to the guidelines of the General Directorate of Epidemiology of Mexico. Real-time polymerase chain reaction (RT-PCR) tests were used to con rm SARS-CoV-2 infection. Categorical variables were compared using the Chi-square test, and propensity score matching was performed to determine univariate and multivariate odds ratios of the population regarding a positive vs. negative SARS-CoV-2 result. Results. Of the 121 children, 36 had laboratory-con rmed SARS-CoV-2 infection. e main risk for SARS-CoV-2-associated pediatric hospitalization was contact with a family member with SARS-CoV-2. It was also found that fever and fatigue were statistically signi cantly associated with a positive SARS-CoV-2 test in multivariate models. Clinical and laboratory data in this Mexican hospitalized pediatric cohort di er from other reports worldwide; the mortality rate (1.6%) of the population studied was higher than that seen in reports from other countries. Conclusion. Our study found that fever and fatigue at hospital presentation as well as an antecedent exposure to a family member with SARS-CoV-2 infection were important risk factors for SARS-CoV-2 positivity in children at hospital admission.

Chaos Solitons & Fractals, Mar 1, 2023

The aim of this paper is to provide a mathematical study of the amount of drug administered as a ... more The aim of this paper is to provide a mathematical study of the amount of drug administered as a continuous intravenous infusion or oral dose. For this purpose, we consider fractional-order mammillary-type models describing the anomalous dynamics of exchange of concentrations between compartments at, constant input rates, power-law type, and in the form of oral doses at given (discrete) times. We have developed a general analysis strategy for these models, in which we have found closed-form analytical solutions written in terms of the multivariate Mittag-Leffler function. Numerical simulations have been performed using our formulas, with parameters from the literature.

The nomination for "Who'sWho in the World (2015)" was awarded for his work in the field of mathe... more The nomination for "Who'sWho in the World (2015)" was awarded for his work in the field of mathematical biology. Their paper has been in several different lists of Top 25 Hottest Articles (Elsevier). He has published ten paper in international journal during the years 2011 and 2015, between the journal are: Mathematical Biosciences; IMA Mathematical Medicine and Biology; Mathematical Biosciences and Engineering; Applied Mathematics and Computation; Chaos, Solitons and Fractals; Journal of Mathematical Analysis and Applications; Mathematical and Computer Modelling; Journal of Biological Systems; Computational and Mathematical Methods in Medicine.

Introducción a la Biología Matemática La Biología Matemática es un campo de estudio interdisci... more Introducción a la Biología Matemática

La Biología Matemática es un campo de estudio interdisciplinario que se centra en el modelación de procesos biológicos usando técnicas de las matemáticas. Con este curso se pretende dar una introducción a los modelos básicos en ecología de poblaciones y epidemiología. En este curso el alumno aprenderá los principios básicos de modelación matemática en biología.

1. Dinámica de poblaciones de una especie
1.1 Crecimiento exponencial (Maltusianismo)
1.2 Modelo de crecimiento logístico
1.3 Ecuación logística en la propagación de rumores y epidemias
1.4. Ecuación de Gompertz y el crecimiento de tumores
1.5 Análisis cualitativo
1.6 Dinámica poblacional y cosechas en poblaciones

2. Modelos en la propagación de epidemias
2.1 Modelo Susceptible-Infeccioso-Susceptible
2.2 Modelo Susceptible-Infeccioso-Recuperado sin dinámica vital
2.3 Modelo Susceptible-Infeccioso-Recuperado con dinámica vital
2.4. Otras variantes de los modelos epidemiológicos

3. Modelos de dos especies interactuantes
3.1 Modelo Presa-Depredador (Lotka-Volterra)
3.2 Especies competidoras (Exclusión competitiva)
3.3 Especies mutualistas

Bibliografía

[1] Lourdes Esteva, Manuel Falconi Magaña. Biología matemática. Un enfoque desde los sistemas dinámicos. Colección Temas de Matemáticas,2002, 207 pp.
[2] Faustino Sánchez, Pedro Miramontes y José Luis Gutiérrez Sánchez (coordinadores). Clásicos de la biología matemática. Biblioteca Aprender a Aprender, Coedición CEIICH-UNAM/Siglo XXI Editores, 2002, 177 pp.
[3] J. D. Murray. Mathematical Biology, Volume 1. Springer-Verlag, 2002.
[4] C. Vargas-De-León. Lyapunov functions for two-species cooperative systems. Applied Mathematics and Computation Vol 219(5) 2493–2497 (2012).
[5] C. Vargas-De-León. On the global stability of SIS, SIR and SIRS epidemic models with standard incidence. Chaos, Solitons & Fractals, Vol 44(12), 1106-1110 (2011).

"Se presentan una revisión de los modelos matemáticos continuos desarrollados para analizar el ro... more "Se presentan una revisión de los modelos matemáticos continuos desarrollados para analizar el rol de la respuesta inmune (ya sea celular o bien por anticuerpos) en las infecciones virales.

Las infecciones producidas por el VIH, la Hepatitis B y C, entre otras, se han estudiado a través de sistemas de ecuaciones diferenciales ordinarias (EDO), ecuaciones diferenciales con retardo (EDR) y ecuaciones diferenciales parciales (EDP).

Los modelos propuestos en esta revisión van desde los que incluyen la dinámica básica de la infección hasta los que consideran la respuesta inmune celular o la por anticuerpos.

Los modelos escritos en el lenguaje de las EDO consideran que el contacto de las partículas virales y las células susceptibles producen "instantáneamente" células infectadas, mientras que biológicamente existe un periodo de latencia en las infecciones virales, esto se traduce en un tiempo de retardo, y por ello el uso de las EDR.

Por otro lado, en la literatura existen refinamientos que consideran que la tasa de producción de partículas virales y la tasa de mortalidad de las células infectadas varían y dependen de la edad de la célula infectada, tal situación se modela con EDP.

En esta conferencia discutiremos que en esta clase de modelos la estabilidad global de las soluciones esta completamente determinada por el número reproductivo básico R0. Se presentará la construcción de funciones (o funcionales) de Lyapunov para cada uno de los tipos de modelos usando la técnica desarrollada por Korobeinikov y McCluskey en la última década."

Host-vector disease models with discrete time delays or distributed time delays are studied. By m... more Host-vector disease models with discrete time delays or distributed time delays are studied. By means of Lyapunov’s direct method, we establish the global stability conditions of the equilibrium states. We proved that the global stability are completely determined by the threshold parameter, R_0 . If the threshold parameter is less than or equal to unity, the disease-free equilibrium state is globally asymptotically stable for any time delays in the feasible region. If the threshold parameters greater than one, a unique endemic equilibrium state exists and is globally asymptotically stable for any time delays in the interior of the feasibleregion. We extended our analysis to sexually-transmitted diseases models.

Keywords Host-vector model; distributed time-delay; discrete time-delay; global asymptotic stability; method of Lyapunov functionals.