Discrete Models in Epidemiology: New Contagion Probability Functions Based on Real Data Behavior.

Mathematical modeling is a tool used for understanding diseases dynamics. The discrete-time model is an especial case in modeling that satisfactorily describes the epidemiological dynamics because of the discrete nature of the real data. However, discrete models reduce their descriptive and fitting potential because of assuming a homogeneous population. Thus, in this paper, we proposed contagion probability functions according to two infection paradigms that consider factors associated with transmission dynamics. For example, we introduced probabilities of establishing an infectious interaction, the number of contacts with infectious and the level of connectivity or social distance within populations. Through the probabilities design, we overcame the homogeneity assumption. Also, we evaluated the proposed probabilities through their introduction into discrete-time models for two diseases and different study zones with real data, COVID-19 for Germany and South Korea, and dengue for Colombia. Also, we described the oscillatory dynamics for the last one using the contagion probabilities alongside parameters with a biological sense. Finally, we highlight the implementation of the proposed probabilities would improve the simulation of the public policy effect of control strategies over an infectious disease outbreak.

A discrete model for the evaluation of public policies: The case of Colombia during the COVID-19 pandemic.

In mathematical epidemiology, it is usual to implement compartmental models to study the transmission of diseases, allowing comprehension of the outbreak dynamics. Thus, it is necessary to identify the natural history of the disease and to establish promissory relations between the structure of a mathematical model, as well as its parameters, with control-related strategies (real interventions) and relevant socio-cultural behaviors. However, we identified gaps between the model creation and its implementation for the use of decision-makers for policy design. We aim to cover these gaps by proposing a discrete mathematical model with parameters having intuitive meaning to be implemented to help decision-makers in control policy design. The model considers novel contagion probabilities, quarantine, and diffusion processes to represent the recovery and mortality dynamics. We applied mathematical model for COVID-19 to Colombia and some of its localities; moreover, the model structure could be adapted for other diseases. Subsequently, we implemented it on a web platform (MathCOVID) for the usage of decision-makers to simulate the effect of policies such as lock-downs, social distancing, identification in the contagion network, and connectivity among populations. Furthermore, it was possible to assess the effects of migration and vaccination strategies as time-dependent inputs. Finally, the platform was capable of simulating the effects of applying one or more policies simultaneously.

Mathematical Modeling for the Assessment of Public Policies in the Cancer Health-Care System Implemented for the Colombian Case.

The incidence of cancer has been constantly growing worldwide, placing pressure on health systems and increasing the costs associated with the treatment of cancer. In particular, low- and middle-income countries are expected to face serious challenges related to caring for the majority of the world's new cancer cases in the next 10 years. In this study, we propose a mathematical model that allows for the simulation of different strategies focused on public policies by combining spending and epidemiological indicators. In this way, strategies aimed at efficient spending management with better epidemiological indicators can be determined. For validation and calibration of the model, we use data from Colombia-which, according to the World Bank, is an upper-middle-income country. The results of the simulations using the proposed model, calibrated and validated for Colombia, indicate that the most effective strategy for reducing mortality and financial burden consists of a combination of early detection and greater efficiency of treatment in the early stages of cancer. This approach is found to present a 38% reduction in mortality rate and a 20% reduction in costs (% GDP) when compared to the baseline scenario. Hence, Colombia should prioritize comprehensive care models that focus on patient-centered care, prevention, and early detection.

The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model.

Some deterministic models deal with environmental conditions and use parameter estimations to obtain experimental parameters, but they do not consider anthropogenic or environmental disturbances, e.g., chemical control or climatic conditions. Even more, they usually use theoretical or measured in-lab parameters without worrying about uncertainties in initial conditions, parameters, or changes in control inputs. Thus, in this study, we estimate parameters (including chemical control parameters) and confidence contours under uncertainty conditions using data from the municipality of Bello (Colombia) during 2010-2014, which includes two epidemic outbreaks. Our study shows that introducing non-periodic pulse inputs into the mathematical model allows us to: (i) perform parameter estimation by fitting real data of consecutive dengue outbreaks, (ii) highlight the importance of chemical control as a method of vector control, and (iii) reproduce the endemic behavior of dengue. We described a methodology for parameter and sub-contour box estimation under uncertainties and performed reliable simulations showing the behavior of dengue spread in different scenarios.

Sensitivity, uncertainty and identifiability analyses to define a dengue transmission model with real data of an endemic municipality of Colombia.

Dengue disease is a major problem for public health surveillance entities in tropical and subtropical regions having a significant impact not only epidemiological but social and economical. There are many factors involved in the dengue transmission process. We can evaluate the importance of these factors through the formulation of mathematical models. However, the majority of the models presented in the literature tend to be overparameterized, with considerable uncertainty levels and excessively complex formulations. We aim to evaluate the structure, complexity, trustworthiness, and suitability of three models, for the transmission of dengue disease, through different strategies. To achieve this goal, we perform structural and practical identifiability, sensitivity and uncertainty analyses to these models. The results showed that the simplest model was the most appropriate and reliable when the only available information to fit them is the cumulative number of reported dengue cases in an endemic municipality of Colombia.

An alternative model to explain the vectorial capacity using as example Aedes aegypti case in dengue transmission.

Vectorial capacity (VC), as a concept that describes the potential of a vector to transmit a pathogen, has had historical problems related to lacks in dimensional significance and high error propagation from parameters that take part in the model to output. Hence, values estimated with those equations are not sufficiently reliable to consider in control strategies or vector population study. In this paper, we propose a new VC model consistent at dimensional level, i.e., the definition and the equation of VC have same and consistent units, with a parameter estimation method and mathematical structure that reduces the uncertainty in model output, using as a case of study an Aedes aegypti population of the municipality of Bello, Colombia. After a literature review, we selected one VC equation following biological, measurability and dimensional criteria, then we rendered a local and global sensitivity analysis, identifying the mortality rate of mosquitoes as a target component of the equation. Thus, we studied the Weibull and Exponential distributions as probabilistic models that represent the expectation of mosquitoes infective life, intending to include the best distribution in a selected VC structure. The proposed mortality rate estimation method includes a new parameter that represents an increase or decrease in vector mortality, as it may apply. We noticed that its estimation reduces the uncertainty associated with the expectation of mosquitoes' infective life expression, which also reduces the output range and variance in almost a half.

Discrete COVID-19 Transmission Models and Preliminary Publications in Science: A Systematic review / Modelos discretos de transmisión de COVID-19 y publicaciones preeliminares en la ciencia: una búsqueda sistematizada

In the Chinese city of Wuhan at the end of 2019, a new respiratory disease known as COVID-19 emerged, caused by the SARS-CoV-2 virus. This disease spreads rapidly worldwide and presents numerous infections and deaths; therefore, the World Health Organization upgraded its category from epidemic to pandemic because of alarming levels of spread, severity, and inaction. Given this situation, different areas of science have approached the study of this disease, among them is mathematical epidemiology through the modeling of the phenomenon; therefore, in this document, we performed a systematic review related to transmission models of COVID-19, specifically discrete models because of the daily report of infection cases around the world. We identified different important disease features implemented in the models, e.g., metapopulations, migration, quarantine, inclusion of latency, and incubation periods, among others. Also, we identified its basic structure, and we found that many papers directly used SIR and SEIR models with no modification, being an excessive simplification of the COVID-19 transmission phenomenon. Likewise, some authors highlighted an important problem during the application of mathematical models: the quality or absence of the daily case data in some affected countries. Finally, the mathematical models should be constantly updated together with the publication of research related to virology and epidemiology of the disease.

A finales del año 2019, en la ciudad china de Wuhan, emergió una nueva enfermedad respiratoria conocida como COVID-19 que es producida por el virus SARS-CoV-2, similar al virus causante del síndrome respiratorio agudo grave (SARS-CoV). Actualmente, esta enfermedad se esparció rápidamente a nivel mundial y ha presentado una gran cantidad de afectados en diferentes regiones del mundo; por lo tanto, la Organización Mundial de la Salud elevó su categoría de epidemia a pandemia debido a los niveles alarmantes de propagación, gravedad e inacción. Dada esta situación, diferentes áreas de la ciencia han abordado su estudio, entre ellas esta la epidemiología matemática a través del modelado del fenómeno; por lo tanto en el presente documento se realizó una revisión sistematizada de literatura relacionada a modelos de transmisión del COVID-19, específicamente modelos discretos debido a la naturaleza de reporte diaria de casos de la enfermedad en diferentes localidades del mundo. Se lograron identificar diferentes características importantes de la enfermedad que son implementadas en los modelos matemáticos: división por grupos etarios, metapoblaciones, migración, cuarentena, inclusión de periodos de latencia e incubación, entre otros. Aun así, se encontró una gran cantidad de artículos que utilizaban directamente modelos SIR y SEIR sin ningún tipo de modificación, haciendo una simplificación desmedida del fenómeno de transmisión del COVID-19. Asimismo, se identificaron algunas problemáticas a la hora de implementar los modelos matemáticos: la presencia y calidad de los datos de casos diarios en algunos países afectados. Finalmente, se sugiere que los modelos matemáticos estén en constante actualización junto a la publicación de las investigaciones relacionadas con virología y epidemiología de la enfermedad.