Local Basic Reproduction Number Research Articles

Asymptotic profiles of basic reproduction number for a degenerate reaction–diffusion host–pathogen model

In this paper, the basic reproduction number (BRN) is explored in terms of a degenerate reaction–diffusion host–pathogen model. The expressions of BRN and local basic reproduction number (LBRN) for the model are calculated. Some basic properties of with respect to the general diffusion rates and the asymptotic profiles of with respect to large diffusion rates and small diffusion rates are established by introducing two compact operators and . Our results reveal that is the spectral radius of a sum of the product of additive operators and strongly positive compact linear operators with spectral radii one.

Dynamics analysis of a diffusive SIRI epidemic system under logistic source and general incidence rate

This study presents and examines a diffusive SIRI epidemic model incorporating logistic source and a general incidence rate. Differing from existing works, the system incorporates two factors: the general incidence rate and the logistic source. We first consider the well-posedness of the system. Then, utilizing the construction of four Lyapunov functions, we thoroughly examine the global asymptotic stability of equilibria in both specific and general scenarios, assuming all coefficients remain constant. In addition, we establish the basic reproduction number, denoted as R0, and subsequently derive the correlation between R0 and the local basic reproduction number. Furthermore, we provide a detailed discussion of the persistence and extinction of the infective population. In particular, in the case where R0 equals one and certain assumptions are met, we demonstrate the global asymptotic stability of the disease-free equilibrium. Lastly, we substantiate the validity of our theoretical findings through the presentation of five illustrative examples.

Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences

In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number R_{0} and the local basic reproduction number {overline{R}}_{0}(x) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R_{0} and {overline{R}}_{0}(x) as well as the asymptotic properties of R_{0} when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators L_{1} and L_{2}. Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when R_{0}<1, while the disease is uniformly persistent when R_{0}>1. Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle’s invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if R_{0}le 1, and the endemic equilibrium is globally asymptotically stable if R_{0}>1 and an additional condition is satisfied.

Spatio-temporal spread of COVID-19: Comparison of the inhomogeneous SEPIR model and data from South Carolina.

During the COVID-19 pandemic authorities have been striving to obtain reliable predictions for the spreading dynamics of the disease. We recently developed a multi-“sub-populations” (multi-compartments: susceptible, exposed, pre-symptomatic, infectious, recovered) model, that accounts for the spatial in-homogeneous spreading of the infection and shown, for a variety of examples, how the epidemic curves are highly sensitive to location of epicenters, non-uniform population density, and local restrictions. In the present work we test our model against real-life data from South Carolina during the period May 22 to July 22 (2020). During this period, minimal restrictions have been employed, which allowed us to assume that the local basic reproduction number is constant in time. We account for the non-uniform population density in South Carolina using data from NASA’s Socioeconomic Data and Applications Center (SEDAC), and predict the evolution of infection heat-maps during the studied period. Comparing the predicted heat-maps with those observed, we find high qualitative resemblance. Moreover, the Pearson’s correlation coefficient is relatively high thus validating our model against real-world data. We conclude that the model accounts for the major effects controlling spatial in-homogeneous spreading of the disease. Inclusion of additional sub-populations (compartments), in the spirit of several recently developed models for COVID-19, can be easily performed within our mathematical framework.

Effect of daily periodic human movement on dengue dynamics: The case of the 2010 outbreak in Hermosillo, Mexico

The human movement plays an important role in the spread of infectious diseases. On an urban scale, people move daily to workplaces, schools, among others. Here, we are interested in exploring the effect of the daily local stay on the variations of some characteristics of dengue dynamics such as the transmission rates and local basic reproductive numbers. For this, we use a two-patch mathematical model that explicitly considers that daily mobility of people and real data from the 2010 dengue outbreak in Hermosillo, Mexico. Based on a preliminary cluster analysis, we divide the city into two regions, the south and north sides, which determine each patch of the model. We use a Bayesian approach to estimate the transmission rates and local basic reproductive numbers of some urban mobility scenarios where residents of each patch spend daily the 100% (no human movement between patches), 75% and 50% of their day at their place of residence. For the north side, estimates of transmission rates do not vary and it is more likely that the local basic reproductive number to be greater than one for all three different scenarios. On the contrary, transmission rates of the south side have more weight in lower values when consider the human movement between patches compared to the uncoupled case. In fact, local basic reproductive numbers less than 1 are not negligible for the south side. If information about commuting is known, this work might be useful to obtain better estimates of some contagion local properties of a patch, such as the basic reproductive number.

Effect of daily human movement on some characteristics of dengue dynamics

Human movement is a key factor in infectious diseases spread such as dengue. Here, we explore a mathematical modeling approach based on a system of ordinary differential equations to study the effect of human movement on characteristics of dengue dynamics such as the existence of endemic equilibria, and the start, duration, and amplitude of the outbreak. The model considers that every day is divided into two periods: high-activity and low-activity. Periodic human movement between patches occurs in discrete times. Based on numerical simulations, we show unexpected scenarios such as the disease extinction in regions where the local basic reproductive number is greater than 1. In the same way, we obtain scenarios where outbreaks appear despite the fact that the local basic reproductive numbers in these regions are less than 1 and the outbreak size depends on the length of high-activity and low-activity periods.

Mathematical modelling for malaria under resistance and population movement

In this work, two mathematical models for malaria under resistance are presented. More precisely, the first model shows the interaction between humans and mosquitoes inside a patch under infection of malaria when the human population is resistant to antimalarial drug and mosquitoes population is resistant to insecticides. For the second model, human–mosquitoes population movements in two patches is analyzed under the same malaria transmission dynamic established in a patch. For a single patch, existence and stability conditions for the equilibrium solutions in terms of the local basic reproductive number are developed. These results reveal the existence of a forward bifurcation and the global stability of disease–free equilibrium. In the case of two patches, a theoretical and numerical framework on sensitivity analysis of parameters is presented. After that, the use of antimalarial drugs and insecticides are incorporated as control strategies and an optimal control problem is formulated. Numerical experiments are carried out in both models to show the feasibility of our theoretical results.

Dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate

This paper is concerned with the dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate. We first establish the well-posedness of this model. Then we clarify the relationship between the local basic reproduction number R̃ and the basic reproduction number R0. It could be seen that R0 plays an important role in determining the global dynamics of this model. In fact, we show that the disease-free equilibrium is globally asymptotically stable when R0<1. If R0=1, then the disease-free equilibrium is globally asymptotically stable under some assumptions. In addition, the phenomena of uniform persistence occurs when R0>1. We also consider the local and global stability of endemic equilibrium when all the parameters of this model are constant. In the case R0>1, we further establish the existence of traveling wave solutions of this model. Moreover, we provide an example and numerical simulations to support our theoretical results. Our model extended some known results.

Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate

This work is concerned with the dynamics of a diffusive SIRI epidemic model with nonlinear incidence rate. We first establish the well-posedness of the model. Then we show the basic reproduction number R0 is a threshold parameter for the stability of the model. In fact, the disease-free equilibrium is globally asymptotically stable when R0<1, while the phenomena of uniform persistence occurs when R0>1. If R0=1, the disease-free equilibrium is globally asymptotically stable under some assumptions. We also clarify the relationship between R0 and the local basic reproduction number. Moreover, we establish the local and global stability of the endemic equilibrium provided that all parameters of this model are constants.

An outbreak vector-host epidemic model with spatial structure: the 2015\u20132016 Zika outbreak in Rio De Janeiro

BackgroundA deterministic model is developed for the spatial spread of an epidemic disease in a geographical setting. The disease is borne by vectors tosusceptible hosts through criss-cross dynamics. The model is focused on an outbreak that arises from a small number of infected hosts imported into a subregion of the geographical setting. The goal is to understand how spatial heterogeneity of the vector and host populations influences the dynamics of the outbreak, in both the geographical spread and the final size of the epidemic.MethodsPartial differential equations are formulated to describe the spatial interaction of the hosts and vectors. The partial differential equations have reaction-diffusion terms to describe the criss-cross interactions of hosts and vectors. The partial differential equations of the model are analyzed and proven to be well-posed. A local basic reproduction number for the epidemic is analyzed.ResultsThe epidemic outcomes of the model are correlated to the spatially dependent parameters and initial conditions of the model. The partial differential equations of the model are adapted to seasonality of the vector population, and applied to the 2015–2016 Zika seasonal outbreak in Rio de Janeiro Municipality in Brazil.ConclusionsThe results for the model simulations of the 2015–2016 Zika seasonal outbreak in Rio de Janeiro Municipality indicate that the spatial distribution and final size of the epidemic at the end of the season are strongly dependent on the location and magnitude of local outbreaks at the beginning of the season. The application of the model to the Rio de Janeiro Municipality Zika 2015–2016 outbreak is limited by incompleteness of the epidemic data and by uncertainties in the parametric assumptions of the model.

Effects of human dynamics on epidemic spreading in Côte d’Ivoire

Understanding and predicting outbreaks of contagious diseases are crucial to the development of society and public health, especially for underdeveloped countries. However, challenging problems are encountered because of complex epidemic spreading dynamics influenced by spatial structure and human dynamics (including both human mobility and human interaction intensity). We propose a systematical model to depict nationwide epidemic spreading in C\^{o}te d'Ivoire, which integrates multiple factors, such as human mobility, human interaction intensity, and demographic features. We provide insights to aid in modeling and predicting the epidemic spreading process by data-driven simulation and theoretical analysis, which is otherwise beyond the scope of local evaluation and geometrical views. We show that the requirement that the average local basic reproductive number to be greater than unity is not necessary for outbreaks of epidemics. The observed spreading phenomenon can be roughly explained as a heterogeneous diffusion-reaction process by redefining mobility distance according to the human mobility volume between nodes, which is beyond the geometrical viewpoint. However, the heterogeneity of human dynamics still poses challenges to precise prediction.

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

We consider the dynamics of a mosquito-transmitted pathogen in a multi-patch Ross–Macdonald malaria model with mobile human hosts, mobile vectors, and a heterogeneous environment. We show the existence of a globally stable steady state, and a threshold that determines whether a pathogen is either absent from all patches, or endemic and present at some level in all patches. Each patch is characterized by a local basic reproduction number, whose value predicts whether the disease is cleared or not when the patch is isolated: patches are known as “demographic sinks” if they have a local basic reproduction number less than one, and hence would clear the disease if isolated; patches with a basic reproduction number above one would sustain endemic infection in isolation, and become “demographic sources” of parasites when connected to other patches. Sources are also considered focal areas of transmission for the larger landscape, as they export excess parasites to other areas and can sustain parasite populations. We show how to determine the various basic reproduction numbers from steady state estimates in the patched network and knowledge of additional model parameters, hereby identifying parasite sources in the process. This is useful in the context of control of the infection on natural landscapes, because a commonly suggested strategy is to target focal areas, in order to make their corresponding basic reproduction numbers less than one, effectively turning them into sinks. We show that this is indeed a successful control strategy—albeit a conservative and possibly expensive one—in case either the human host, or the vector does not move. However, we also show that when both humans and vectors move, this strategy may fail, depending on the specific movement patterns exhibited by hosts and vectors.

Historic Disease Data as Epidemiological Resource: Searching for the Origin and Local Basic Reproduction Number of the 1878 Yellow Fever Epidemic in Memphis, Tennessee

Emerging and reemerging infectious diseases continue to pose considerable regional and global concerns. A vital contribution to be made by geographers is in developing an understanding of the spatial structure of these epidemics across various scales. Confidentiality concerns and a general lack of individual data from many developing world areas mean that individual or subneighborhood-scale epidemic information is often unavailable. One alternative potential source of data is historical epidemics. Although these data exist in the form of board of health reports, these should not be considered complete, and the onus is on the researcher to perform due diligence on data validation and identifying supplementary spatial and cultural context. This article presents an example of such a methodological task for the 1878 yellow fever epidemic of Memphis, which leads to exploration of two important spatial questions: the correct origin of the epidemic in the city and its associated local basic reproduction number, which is the number of ensuing cases stemming from an original. This article should be viewed as a template for a subsequent series of fine-scale historical epidemic analyses, which together can produce an important conduit into further development of spatial epidemiological theory.

A patchy model for Chikungunya-like diseases

We consider a n-patches model, to study the impact of human population movements between cities (patches) in the spread of Chikungunya or even Dengue diseases. In previous works, it was showed that the basic reproduction number can vary from place to place, but this result was obtained without taking into account human movements. We provide a theoretical study of the patchy model, and derive the basic reproduction number, which may depend on Human movement rates between the patches and on local population sizes. We show that the basic reproduction number is bounded by the maximum of local basic reproduction number. We also show that there exists a disease-free equilibrium (DFE) that is locally asymptotically stable whenever the basic reproduction number is less than 1. Under suitable assumptions, DFE is even globally asymptotically stable. We emphasize that Human movements are of particular importance to evaluate the spreading or not of Chikungunya or Dengue diseases, and thus movement rates have to be estimated very accurately. We confirm also the importance of the local basic reproduction numbers and show that even local field interventions can be of benefit to control/reduce the spread of a disease. A complete analytical study for a 2-patches model and several examples are provided to illustrate our conclusion.

Detecting Human-to-Human Transmission of Avian Influenza A (H5N1)

Highly pathogenic avian influenza A (HPAI) subtype H5N1 has caused family case clusters, mostly in Southeast Asia, that could be due to human-to-human transmission. Should this virus, or another zoonotic influenza virus, gain the ability of sustained human-to-human transmission, an influenza pandemic could result. We used statistical methods to test whether observed clusters of HPAI (H5N1) illnesses in families in northern Sumatra, Indonesia, and eastern Turkey were due to human-to-human transmission. Given that human-to-human transmission occurs, we estimate the infection secondary attack rates (SARs) and the local basic reproductive number, R0. We find statistical evidence of human-to-human transmission (p = 0.009) in Sumatra but not in Turkey (p = 0.114). For Sumatra, the estimated household SAR was 29% (95% confidence interval [CI] 15%-51%). The estimated lower limit on the local R0 was 1.14 (95% CI 0.61-2.14). Effective HPAI (H5N1) surveillance, containment response, and field evaluation are essential to monitor and contain potential pandemic strains.

Detecting Human-to-Human Transmission of Avian Influenza A (H5N1)

Highly pathogenic avian influenza A (HPAI) subtype H5N1 has caused family case clusters, mostly in Southeast Asia, that could be due to human-to-human transmission. Should this virus, or another zoonotic influenza virus, gain the ability of sustained human-to-human transmission, an influenza pandemic could result. We used statistical methods to test whether observed clusters of HPAI (H5N1) illnesses in families in northern Sumatra, Indonesia, and eastern Turkey were due to human-to-human transmission. Given that human-to-human transmission occurs, we estimate the infection secondary attack rates (SARs) and the local basic reproductive number, R0. We find statistical evidence of human-to-human transmission (p = 0.009) in Sumatra but not in Turkey (p = 0.114). For Sumatra, the estimated household SAR was 29% (95% confidence interval [CI] 15%-51%). The estimated lower limit on the local R0 was 1.14 (95% CI 0.61-2.14). Effective HPAI (H5N1) surveillance, containment response, and field evaluation are essential to monitor and contain potential pandemic strains.