La dinámica de la transmisión de la enfermedad según la teoría de la complejidad / Disease transmission dynamics according to complexity theory

RESUMEN Objetivo Se ilustra el proceso de transmisión de una enfermedad, entendido como un sistema complejo a la luz de la teoría de la complejidad. Métodos Se simula el comportamiento de un modelo matemático SEIR que refleja el proceso de transmisión de una enfermedad a partir de la conexión de los estados de susceptibilidad, infección, enfermedad y recuperación y no linealidad en la interacción de susceptibles e infectados. Se asume una tasa de infección con oscilaciones en el tiempo, descrito por un mapeo logístico. Resultados La transmisión transcurre en el tiempo con la reducción de los susceptibles en la medida que estos se infectan y enferman y el aumento de la recuperación tras el diagnóstico y tratamiento. Con pequeños aumentos en el valor de la tasa de infección, se observan oscilaciones en el número de susceptibles y expuestos y aleatoriedad en la relación entre los susceptibles e infectados, hasta confluir a un patrón regular. Conclusión El modelo refleja la conexión entre los estados, la no linealidad y el comportamiento caótico tras pequeños aumentos del valor de la tasa de infección. Una perspectiva histórica y transdisciplinaria ayudaría a comprender la complejidad de la transmisión y a concertar opciones de control.

ABSTRACT Objective Illustrating disease transmission as a complex system according to complexity theory. Methods A SIR mathematical model (S=number susceptible, I=number infectious, and R=number recovered or immune) reflecting disease transmission from the connection between states of susceptibility, infection, disease, recovery and nonlinearity in the interaction between susceptible and infected was simulated. Infection rate temporal fluctuations were described by logistic mapping. Results Transmission occurs with the reduction of susceptible states as people become infected and sick, followed by an increase in individuals' recovery following diagnosis and treatment. Small increases in infection rate value led to fluctuations in the number of susceptible and exposed people and randomness in the relationship between being susceptible and infected, until converging towards a regular pattern. Conclusion The model reflected the connection between states of susceptibility, nonlinearity and chaotic behavior following small increases in infection rate. A historical and trans-disciplinary perspective could help in understanding transmission complexity and coordinating control options.

[Disease transmission dynamics according to complexity theory]. / La dinámica de la transmisión de la enfermedad según la teoría de la complejidad.

OBJECTIVE: Illustrating disease transmission as a complex system according to complexity theory. METHODS: A SIR mathematical model (S=number susceptible, I=number infectious, and R=number recovered or immune) reflecting disease transmission from the connection between states of susceptibility, infection, disease, recovery and nonlinearity in the interaction between susceptible and infected was simulated. Infection rate temporal fluctuations were described by logistic mapping. RESULTS: Transmission occurs with the reduction of susceptible states as people become infected and sick, followed by an increase in individuals' recovery following diagnosis and treatment. Small increases in infection rate value led to fluctuations in the number of susceptible and exposed people and randomness in the relationship between being susceptible and infected, until converging towards a regular pattern. CONCLUSION: The model reflected the connection between states of susceptibility, nonlinearity and chaotic behavior following small increases in infection rate. A historical and trans-disciplinary perspective could help in understanding transmission complexity and coordinating control options.

[Herd immunity against rubella according to a survey of the population in Medellin, Colombia]. / Inmunidad colectiva contra la rubéola según una encuesta poblacional en Medellín, Colombia.

OBJECTIVE: Calculate the critical proportion (Pc) for achieving herd immunity based on a 2009 population study conducted in Medellin, Colombia, by age, globally and disaggregated by sex, location, and socioeconomic stratum. METHODS: A survey of seroprevalence in the population was conducted by means of a random sample of 2 124 individuals aged 6 to 64 that was representative of age, sex, and location. The basic reproduction number was estimated using a quadratic regression of the average IgG titers for rubella by age in unvaccinated individuals with titers greater than or equal to 15 IU/ml. The effective reproduction number (Re) was calculated with the data on the weighted proportion of protection by age, sex, location, and socioeconomic stratum. RESULTS: Overall, the Pc was 90.0% (95% CI, 88.6-95.2%) and the Re was 0.95 (95% CI, 0.8-1.8), for a weighted proportion of protection of 89.4% (95% CI, 86.8- 91.6%). Protection was lower than the expected Pc in both sexes, in high and low socioeconomic strata, and in the rural area. In the urban area, protection was greater than the Pc (89.4%, with a 95% CI, 86.6-91.7%, compared to 87.4% and a 95% CI, 85.2-87.8%). CONCLUSIONS: The urban area has made progress toward herd immunity, but the overall proportion of protection in women, the rural area, and the high socioeconomic strata must be increased. The effective number may be greater than one, indicating the potential for the spread of the disease.

[Simulating measles and rubella elimination levels according to social stratification and interaction]. / Simulación del nivel de eliminación de sarampión y rubéola según la estratificación e interacción social.

OBJECTIVE: The study was aimed at comparing measles and rubella disease elimination levels in a homogeneous and heterogeneous population according to socioeconomic status with interactions amongst low- and high-income individuals and diversity in the average number of contacts amongst them. METHODS: Effective reproductive rate simulations were deduced from a susceptibleinfected- recovered (SIR) mathematical model according to different immunisation rates using measles (1980 and 2005) and rubella (1998 and 2005) incidence data from Latin-America and the Caribbean. Low- and high-income individuals' social interaction and their average number of contacts were analysed by bipartite random network analysis. MAPLE 12 (Maplesoft Inc, Ontario Canada) software was used for making the simulations. RESULTS: The progress made in eliminating both diseases between both periods of time was reproduced in the socially-homogeneous population. Measles (2005) would be eliminated in high- and low-income groups; however, it would only be achieved in rubella (2005) if there were a high immunity rate amongst the low-income group. If the average number of contacts were varied, then rubella would not be eliminated, even with a 95 % immunity rate. CONCLUSION: Monitoring the elimination level in diseases like measles and rubella requires that socio-economic status be considered as well as the population's interaction pattern. Special attention should be paid to communities having diversity in their average number of contacts occurring in confined spaces such as displaced communities, prisons, educational establishments, or hospitals.

Simulación del nivel de eliminación de sarampión y rubéola según la estratificación e interacción social / Simulating measles and rubella elimination levels according to social stratification and interaction

Objetivo Comparar el nivel de eliminación de enfermedades como sarampión y rubéola en población homogénea y heterogénea según la existencia de estratos sociales con interacción entre individuos de estrato social alto y bajo y diversidad en el número promedio de contactos entre ellos. Métodos Simulaciones del ritmo reproductivo efectivo, derivado de un modelo matemático tipo SIR (Susceptibles Infectados Recuperados), según diferentes ritmos de inmunidad. Se utilizaron datos de incidencia de sarampión (1980 y 2005) y rubéola (1998 y 2005) de América Latina y el Caribe. Se analizó la interacción entre individuos del estrato social alto y bajo con diferente número promedio de contactos mediante análisis de red aleatoria bipartita. Las simulaciones se ejecutaron en MAPLE 12 (Maplesoft Inc, Ontario Canada). Resultados En la población socialmente homogénea se reprodujo el avance en la eliminación de ambas enfermedades entre los dos períodos de tiempo. En el estrato alto y bajo, se lograría la eliminación en sarampión (2005) pero en rubéola (2005) sólo se lograría si hay alto ritmo de inmunidad en el estrato bajo. Si varía el número promedio de contactos habituales, no se lograría la eliminación de rubéola ni con un ritmo de inmunidad de 95 por ciento. Conclusión El seguimiento del nivel de eliminación de enfermedades como sarampión y rubéola demanda la consideración de la situación socioeconómica y del patrón de interacción de la población. Especial atención se debe prestar a comunidades con diversidad en el número promedio de contactos en espacios confinados como comunidades desplazadas, carcelarias, educativas, hospitalarias, etc.

Objective The study was aimed at comparing measles and rubella disease elimination levels in a homogeneous and heterogeneous population according to socioeconomic status with interactions amongst low- and high-income individuals and diversity in the average number of contacts amongst them. Methods Effective reproductive rate simulations were deduced from a susceptibleinfected- recovered (SIR) mathematical model according to different immunisation rates using measles (1980 and 2005) and rubella (1998 and 2005) incidence data from Latin-America and the Caribbean. Low- and high-income individuals' social interaction and their average number of contacts were analysed by bipartite random network analysis. MAPLE 12 (Maplesoft Inc, Ontario Canada) software was used for making the simulations. Results The progress made in eliminating both diseases between both periods of time was reproduced in the socially-homogeneous population. Measles (2005) would be eliminated in high- and low-income groups; however, it would only be achieved in rubella (2005) if there were a high immunity rate amongst the low-income group. If the average number of contacts were varied, then rubella would not be eliminated, even with a 95 percent immunity rate. Conclusion Monitoring the elimination level in diseases like measles and rubella requires that socio-economic status be considered as well as the population's interaction pattern. Special attention should be paid to communities having diversity in their average number of contacts occurring in confined spaces such as displaced communities, prisons, educational establishments, or hospitals.

[Application of Whittle's stochastic threshold theorem to a chickenpox outbreak]. / Aplicación del teorema del umbral estocástico de Whittle a un brote de varicela.

OBJECTIVE: To estimate the basic reproductive rate of a chickenpox outbreak, to apply the stochastic threshold theorem to estimate the probability of an outbreak occurrence and to identify preventive measures. METHODS: The study was carried out in a daycare center comprising 16 children, 13 susceptible, one infected and two children with acquired immunity by previous disease. A stochastic susceptible - infected - removed model was applied. The basic reproductive rate (R0) was estimated using the maximum likelihood method based on probability distribution for the total size of the epidemic and making an approach of almost complete epidemic. Based on R0, the theorem was applied to establish some preventive measures for preventing a chickenpox outbreak. RESULTS: Each initially infected subject produced three new cases of infection requiring minimum vaccination coverage of 62% to prevent the outbreak or to reduce in 62% the contact among members of the group or to increase in 170% removal of infected subjects. CONCLUSIONS: The stochastic threshold theorem allows to identifying measures that could be implemented to prevent and control chickenpox outbreaks. Although the distribution of the epidemic size showed similar probability of occurrence of large and small outbreaks in a typical bimodal pattern, it illustrates the uncertainty of epidemic process in small groups, requiring close detection of outbreaks in such groups.

Aplicación del teorema del umbral estocástico de Whittle a un brote de varicela / Application of Whittle's stochastic threshold theorem to a chickenpox outbreak

OBJETIVO: Estimar el ritmo reproductivo básico en un brote de varicela, aplicar el teorema umbral estocástico para estimar la probabilidad de la ocurrencia del brote e identificar medidas preventivas. MÉTODOS: El estudio fue realizado en una guardería de 16 niños, con 13 susceptibles, un infectado inicial y dos niños inmunes por antecedente de enfermedad. Se partió de un modelo estocástico: susceptible - infectado - removido. Se estimó el ritmo de reproducción básico de la enfermedad R0, usando un método de máxima verosimilitud basado en el conocimiento de la distribución de probabilidades para el tamaño total de la epidemia y haciendo una aproximación de epidemia casi-completa. Con el R0 obtenido se aplicó el teorema de umbral estocástico para obtener algunas medidas preventivas que podrían impedir la irrupción del brote de varicela. RESULTADOS: Cada infectado inicial produjo tres casos nuevos de infección, requiriendo para impedir el brote, una cobertura mínima de vacunación del 62 por ciento, o disminuir en 62 por ciento el contacto entre miembros del grupo o aumentar en 170 por ciento la remoción de infectados. CONCLUSIONES: El teorema del umbral estocástico permite identificar medidas que se podrían implementar para prevenir y controlar brotes de varicela. Aunque la distribución del tamaño de la epidemia en forma bimodal con similar probabilidad de ocurrencia de brotes grandes y pequeños, señala la incertidumbre del proceso epidémico en grupos pequeños, requiriéndose un estrecho seguimiento de los brotes en tales grupos.

OBJECTIVE: To estimate the basic reproductive rate of a chickenpox outbreak, to apply the stochastic threshold theorem to estimate the probability of an outbreak occurrence and to identify preventive measures. METHODS: The study was carried out in a daycare center comprising 16 children, 13 susceptible, one infected and two children with acquired immunity by previous disease. A stochastic susceptible - infected - removed model was applied. The basic reproductive rate (R0) was estimated using the maximum likelihood method based on probability distribution for the total size of the epidemic and making an approach of almost complete epidemic. Based on R0, the theorem was applied to establish some preventive measures for preventing a chickenpox outbreak. RESULTS: Each initially infected subject produced three new cases of infection requiring minimum vaccination coverage of 62 percent to prevent the outbreak or to reduce in 62 percent the contact among members of the group or to increase in 170 percent removal of infected subjects. CONCLUSIONS: The stochastic threshold theorem allows to identifying measures that could be implemented to prevent and control chickenpox outbreaks. Although the distribution of the epidemic size showed similar probability of occurrence of large and small outbreaks in a typical bimodal pattern, it illustrates the uncertainty of epidemic process in small groups, requiring close detection of outbreaks in such groups.