Vector-borne diseases models with residence times - A Lagrangian perspective

A hybrid Lagrangian-Eulerian model for vector-borne diseases

In this paper, a multi-patch and multi-group vector-borne disease model is proposed to study the effects of host commuting (Lagrangian approach) and/or vector migration (Eulerian approach) on disease spread. We first define the basic reproduction number of the model, R 0 , which completely determines the global dynamics of the model system. Namely, ifR 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, and ifR 0 > 1 , then there exists a unique endemic equilibrium which is globally asymptotically stable. Then, we show that the basic reproduction number has lower and upper bounds which are independent of the host residence times matrix and the vector migration matrix. In particular, nonhomogeneous mixing of hosts and vectors in a homogeneous environment generally increases disease persistence and the basic reproduction number of the model attains its minimum when the distributions of hosts and vectors are proportional. Moreover, R 0 can also be estimated by the basic reproduction numbers of disconnected patches if the environment is homogeneous. The optimal vector control strategy is obtained for a special scenario. In the two-patch and two-group case, we numerically analyze the dependence of the basic reproduction number and the total number of infected people on the host residence times matrix and illustrate the optimal vector control strategy in homogeneous and heterogeneous environments.

Vector-borne disease models with Lagrangian approach

We develop a multi-group and multi-patch model to study the effects of population dispersal on the spatial spread of vector-borne diseases across a heterogeneous environment. The movement of host and/or vector is described by Lagrangian approach in which the origin or identity of each individual stays unchanged regardless of movement. The basic reproduction number [Formula: see text] of the model is defined and the strong connectivity of the host-vector network is succinctly characterized by the residence times matrices of hosts and vectors. Furthermore, the definition and criterion of the strong connectivity of general infectious disease networks are given and applied to establish the global stability of the disease-free equilibrium. The global dynamics of the model system are shown to be entirely determined by its basic reproduction number. We then obtain several biologically meaningful upper and lower bounds on the basic reproduction number which are independent or dependent of the residence times matrices. In particular, the heterogeneous mixing of hosts and vectors in a homogeneous environment always increases the basic reproduction number. There is a substantial difference on the upper bound of [Formula: see text] between Lagrangian and Eulerian modeling approaches. When only host movement between two patches is concerned, the subdivision of hosts (more host groups) can lead to a larger basic reproduction number. In addition, we numerically investigate the dependence of the basic reproduction number and the total number of infected hosts on the residence times matrix of hosts, and compare the impact of different vector control strategies on disease transmission.

Modeling impact of vaccination on COVID-19 dynamics in St. Louis

The region of St. Louis, Missouri, has displayed a high level of heterogeneity in COVID-19 cases, hospitalization, and vaccination coverage. We investigate how human mobility, vaccination, and time-varying transmission rates influenced SARS-CoV-2 transmission in five counties in the St. Louis area. A COVID-19 model with a system of ordinary differential equations was developed to illustrate the dynamics with a fully vaccinated class. Using the weekly number of vaccinations, cases, and hospitalization data from five counties in the greater St. Louis area in 2021, parameter estimation for the model was completed. The transmission coefficients for each county changed four times in that year to fit the model and the changing behaviour. We predicted the changes in disease spread under scenarios with increased vaccination coverage. SafeGraph local movement data were used to connect the forces of infection across various counties.

The effect of changing COVID-19 restrictions on the transmission rate in a veterinary clinic

With the declaration of the COVID-19 pandemic by the World Health Organization on March 11, 2020, the University of Tennessee College of Veterinary Medicine (UTCVM), like other institutions, restructured their services to reduce the potential spread of the COVID-19 virus while simultaneously providing critical and essential veterinary services. A mathematical model was developed to predict the change in the level of possible COVID-19 infections due to the increased number of potential contacts within the UTCVM hospital. A system of ordinary differential equations with different compartments for UTCVM individuals and the Knox county population was formulated to show the dynamics of transmission and the number of confirmed cases. Key transmission rates in the model were estimated using COVID-19 case data from the surrounding county and UTCVM personnel. Simulations from this model show the increasing number of COVID-19 cases among UTCVM personnel as the number of daily clients and the number of veterinary staff in the clinic are increased. We also investigate how changes within the Knox county community impact the UTCVM hospital. These scenarios show the importance of understanding the effects of re-opening scenarios in veterinary teaching hospitals.

Modeling malaria transmission in Nepal: impact of imported cases through cross-border mobility

The cross-border mobility of malaria cases poses an obstacle to malaria elimination programmes in many countries, including Nepal. Here, we develop a novel mathematical model to study how the imported malaria cases through the Nepal-India open-border affect the Nepal government's goal of eliminating malaria by 2026. Mathematical analyses and numerical simulations of our model, validated by malaria case data from Nepal, indicate that eliminating malaria from Nepal is possible if strategies promoting the absence of cross-border mobility, complete protection of transmission abroad, or strict border screening and isolation are implemented. For each strategy, we establish the conditions for the elimination of malaria. We further use our model to identify the control strategies that can help maintain a low endemic level. Our results show that the ideal control strategies should be designed according to the average mosquito biting rates that may depend on the location and season.

A patchy theoretical model for the transmission dynamics of SARS-Cov-2 with optimal control

Short-term human movements play a major part in the transmission and control of COVID-19, within and between countries. Such movements are necessary to be included in mathematical models that aim to assist in understanding the transmission dynamics of COVID-19. A two-patch basic mathematical model for COVID-19 was developed and analyzed, incorporating short-term human mobility. Here, we modeled the human mobility that depended on its epidemiological status, by the Lagrangian approach. A sharp threshold for disease dynamics known as the reproduction number was computed. Particularly, we portrayed that when the disease threshold is less than unity, the disease dies out and the disease persists when the reproduction number is greater than unity. Optimal control theory was also applied to the proposed model, with the aim of investigating the cost-effectiveness strategy. The findings were further investigated through the usage of the results from the cost objective functional, the average cost-effectiveness ratio (ACER), and then the infection averted ratio (IAR).

Relating Eulerian and Lagrangian spatial models for vector-host disease dynamics through a fundamental matrix

We explore the relationship between Eulerian and Lagrangian approaches for modeling movement in vector-borne diseases for discrete space. In the Eulerian approach we account for the movement of hosts explicitly through movement rates captured by a graph Laplacian matrix L. In the Lagrangian approach we only account for the proportion of time that individuals spend in foreign patches through a mixing matrix P. We establish a relationship between an Eulerian model and a Lagrangian model for the hosts in terms of the matrices L and P. We say that the two modeling frameworks are consistent if for a given matrix P, the matrix L can be chosen so that the residence times of the matrix P and the matrix L match. We find a sufficient condition for consistency, and examine disease quantities such as the final outbreak size and basic reproduction number in both the consistent and inconsistent cases. In the special case of a two-patch model, we observe how similar values for the basic reproduction number and final outbreak size can occur even in the inconsistent case. However, there are scenarios where the final sizes in both approaches can significantly differ by means of the relationship we propose.

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.

Dynamical analysis of an SIS epidemic model with migration and residence time

Migration can be divided into temporary and permanent migration, which is related to the residence time of people in the patch, thus we consider an SIS epidemic model with migration and residence time in a patchy environment. If [Formula: see text], the disease-free equilibrium is globally asymptotically stable and the disease dies out. With the same migration rate of susceptible and infectious individuals and without disease-induced death, when [Formula: see text], the endemic equilibrium is unique and globally asymptotically stable. Numerical simulations are carried out to show the effects of residence time and the migration rate on disease prevalence.

Complexity and critical thresholds in the dynamics of visceral leishmaniasis

We study a general multi-host model of visceral leishmaniasis including both humans and animals, and where host and vector characteristics are captured via host competence along with vector biting preference. Additionally, the model accounts for spatial heterogeneity in human population and heterogeneity in biting behaviour of sandflies. We then use parameters for visceral leishmaniasis in the Indian subcontinent as an example and demonstrate that the model exhibits backward bifurcation, i.e. it has a human infection and a sandfly population threshold, characterized by a bi-stable region. These thresholds shift as a function of host competence, host population size, vector feeding preference, spatial heterogeneity, biting heterogeneity and control efforts. In particular, if control is applied through human treatment a new and lower human infection threshold is created, making elimination difficult to achieve, before eventually the human infection threshold no longer exists, making it impossible to control the disease by only reducing the infection levels below a certain threshold. A better strategy would be to reduce the human infection below a certain threshold potentially by early diagnosis, control animal population levels and keep the vector population under check. Spatial heterogeneity in human populations lowers the overall thresholds as a result of weak migration between patches.

Habitat fragmentation promotes malaria persistence

Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.

Morbi-Mortality of the Victims of Internal Conflict and Poor Population in the Risaralda Province, Colombia

This work studies the health status of two populations similar in most social and environmental interactions but one: the individuals from one population are victims of an internal armed conflict. Both populations are located in the Risaralda province, Colombia and the data for this study results from a combination of administrative records from the health system, between 2011 and 2016. We implemented a methodology based on graph theory that defines the system as a set of heterogeneous social actors, including individuals as well as organizations, embedded in a biological environment. The model of analysis uses the diagnoses in medical records to detect morbidity and mortality patterns for each individual (ego-networks), and assumes that these patterns contain relevant information about the effects of the actions of social actors, in a given environment, on the status of health. The analysis of the diagnoses and causes of specific mortality, following the Social Network Analysis framework, shows similar morbidity and mortality rates for both populations. However, the diagnoses' patterns show that victims portray broader interactions between diagnoses, including mental and behavioral disorders, due to the hardships of this population.

Optimal Strategies for Dengue Prevention and Control during Daily Commuting between Two Residential Areas

: In this paper, we report an application for the mathematical theory of dynamic optimization for design of optimal strategies that account for daily commuting of human residents, aiming to reduce vector-borne infections (dengue) among human populations. Our analysis is based on a two-patch dengue transmission model amended with control variables that represent personal protection measures aimed at reduction of the number of contacts between mosquitoes and human hosts (e.g., the use of repellents, mosquito nets, or insecticide-treated clothing). As a result, we have proposed and numerically solved an optimal control problem to minimize the costs associated with the application of control measures, while also minimizing the total number of dengue-infected people in both residential areas. Our principal goal was to identify an optimal strategy for personal protection that renders the maximal number of averted human infections per unit of invested cost, and this goal has been accomplished on the grounds of cost-effectiveness analysis.

Effects of human mobility, temperature and mosquito control on the spatiotemporal transmission of dengue

Dengue transmission exhibits evident geographic variations and seasonal differences. Such heterogeneity is caused by various impact factors, in which temperature and host/vector behaviors could drive its spatiotemporal transmission, but mosquito control could stop its progression. These factors together contribute to the observed distributions of dengue incidence from surveillance systems. To effectively and efficiently monitor and response to dengue outbreak, it would be necessary to systematically model these factors and their impacts on dengue transmission. This paper introduces a new modeling framework with consideration of multi-scale factors and surveillance data to clarify the hidden dynamics accounting for dengue spatiotemporal transmission. The model is based on compartmental system which takes into account the biting-based interactions among humans, viruses and mosquitoes, as well as the essential impacts of human mobility, temperature and mosquito control. This framework was validated with real epidemic data by applying retrospectively to the 2014 dengue epidemic in the Pearl River Delta (PRD) in southern China. The results indicated that suitable condition of temperature could be responsible for the explosive dengue outbreak in the PRD, and human mobility could be the causal factor leading to its spatial transmission across different cities. It was further found that mosquito intervention has significantly reduced dengue incidence, where a total of 52,770 (95% confidence interval [CI]: 29,231-76,308) dengue cases were prevented in the PRD in 2014. The findings can offer new insights for improving the predictability and risk assessment of dengue epidemics. The model also can be readily extended to investigate the transmission dynamics of other mosquito-borne diseases.

Challenges, Opportunities and Theoretical Epidemiology

Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade, remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks. The case of SARS 2003 in 2003, the efforts to reduce the burden of H1N1 influenza cases in 2009, and the challenges faced in reducing the number of Ebola cases in 2014 are the three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information, the impact of information along social connections, or the role of past disease outbreak experiences on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information.”

Assessing the effects of daily commuting in two-patch dengue dynamics: A case study of Cali, Colombia

There are many infectious diseases that can be spread by daily commuting of people and dengue fever is one of them. The absence of vaccine and irregularities in ongoing vector control programs make this disease the most frequent and persistent in many tropical and subtropical regions of the world. This paper targets to access the effects of daily commuting on dengue transmission dynamics by using a deterministic two-patch model fitted to observed data gathered in Cali, Colombia where dengue fever is highly persistent and exhibits endemo-epidemic patterns. The two-patch dengue transmission model with daily communing of human residents between patches (that is, between the city and its suburban areas) is presented using the concept of residence times, which certainly affect the disease transmission rates by inducing variability in human population sizes and vectorial densities at each patch. The same modeling framework is applied to two primary scenarios (epidemic outbreaks and endemic persistence of the disease) and for each scenario two coupling cases (one-way and asymmetric commuting) with different inflow and outflow intensities are analyzed. The concept of effective vectorial density, introduced in this paper, allows to explain in very simple terms why the daily commuting affects quite differently the dengue morbidity among human residents in both patches. In particular, residents of the patch with a greater share of incoming than outgoing commuters may actually "benefit" from inflow of daily commuter by avoiding a considerable number of infections. However, residents of the patch with a greater share of outgoing than incoming commuters, especially those who stay at home patch, incur more risk of getting infected. Additionally, the model shows that daily commuting enhance the total number of human infections acquired in both patches and may even provoke an epidemic outbreak in one patch while moderately lowering the level of the disease persistence in another patch.

Vector-borne disease risk indexes in spatially structured populations

There are economic and physical limitations when applying prevention and control strategies for urban vector borne diseases. Consequently, there are increasing concerns and interest in designing efficient strategies and regulations that health agencies can follow in order to reduce the imminent impact of viruses like Dengue, Zika and Chikungunya. That includes fumigation, abatization, reducing the hatcheries, picking up trash, information campaigns. A basic question that arise when designing control strategies is about which and where these ones should focus. In other words, one would like to know whether preventing the contagion or decrease vector population, and in which area of the city, is more efficient. In this work, we propose risk indexes based on the idea of secondary cases from patch to patch. Thus, they take into account human mobility and indicate which patch has more chance to be a corridor for the spread of the disease and which is more vulnerable, i.e. more likely to have cases?. They can also indicate the neighborhood where hatchery control will reduce more the number of potential cases. In order to illustrate the usefulness of these indexes, we run a set of numerical simulations in a mathematical model that takes into account the urban mobility and the differences in population density among the areas of a city. If we label by i a particular neighborhood, the transmission risk index (TRi) measures the potential secondary cases caused by a host in that neighborhood. The vector transmission risk index (VTRi) measures the potential secondary cases caused by a vector. Finally, the vulnerability risk index (VRi) measures the potential secondary cases in the neighborhood. Transmission indexes can be used to give geographical priority to some neighborhoods when applying prevention and control measures. On the other hand, the vulnerability index can be useful to implement monitoring campaigns or public health investment.

Multi-patch and multi-group epidemic models: a new framework

We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number [Formula: see text] is derived. The effects of heterogeneity in groups, patches and mobility patterns on [Formula: see text] and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number.

The role of mobility and health disparities on the transmission dynamics of Tuberculosis

Background

The transmission dynamics of Tuberculosis (TB) involve complex epidemiological and socio-economical interactions between individuals living in highly distinct regional conditions. The level of exogenous reinfection and first time infection rates within high-incidence settings may influence the impact of control programs on TB prevalence. The impact that effective population size and the distribution of individuals’ residence times in different patches have on TB transmission and control are studied using selected scenarios where risk is defined by the estimated or perceive first time infection and/or exogenous re-infection rates.

Methods

This study aims at enhancing the understanding of TB dynamics, within simplified, two patch, risk-defined environments, in the presence of short term mobility and variations in reinfection and infection rates via a mathematical model. The modeling framework captures the role of individuals’ ‘daily’ dynamics within and between places of residency, work or business via the average proportion of time spent in residence and as visitors to TB-risk environments (patches). As a result, the effective population size of Patch i (home of i-residents) at time t must account for visitors and residents of Patch i, at time t.

Results

The study identifies critical social behaviors mechanisms that can facilitate or eliminate TB infection in vulnerable populations. The results suggest that short-term mobility between heterogeneous patches contributes to significant overall increases in TB prevalence when risk is considered only in terms of direct new infection transmission, compared to the effect of exogenous reinfection. Although, the role of exogenous reinfection increases the risk that come from large movement of individuals, due to catastrophes or conflict, to TB-free areas.

Conclusions

The study highlights that allowing infected individuals to move from high to low TB prevalence areas (for example via the sharing of treatment and isolation facilities) may lead to a reduction in the total TB prevalence in the overall population. The higher the population size heterogeneity between distinct risk patches, the larger the benefit (low overall prevalence) under the same “traveling” patterns. Policies need to account for population specific factors (such as risks that are inherent with high levels of migration, local and regional mobility patterns, and first time infection rates) in order to be long lasting, effective and results in low number of drug resistant cases.

Perspectives on the role of mobility, behavior, and time scales in the spread of diseases

The dynamics, control, and evolution of communicable and vector-borne diseases are intimately connected to the joint dynamics of epidemiological, behavioral, and mobility processes that operate across multiple spatial, temporal, and organizational scales. The identification of a theoretical explanatory framework that accounts for the pattern regularity exhibited by a large number of host-parasite systems, including those sustained by host-vector epidemiological dynamics, is but one of the challenges facing the coevolving fields of computational, evolutionary, and theoretical epidemiology. Host-parasite epidemiological patterns, including epidemic outbreaks and endemic recurrent dynamics, are characteristic to well-identified regions of the world; the result of processes and constraints such as strain competition, host and vector mobility, and population structure operating over multiple scales in response to recurrent disturbances (like El Niño) and climatological and environmental perturbations over thousands of years. It is therefore important to identify and quantify the processes responsible for observed epidemiological macroscopic patterns: the result of individual interactions in changing social and ecological landscapes. In this perspective, we touch on some of the issues calling for the identification of an encompassing theoretical explanatory framework by identifying some of the limitations of existing theory, in the context of particular epidemiological systems. Fostering the reenergizing of research that aims at disentangling the role of epidemiological and socioeconomic forces on disease dynamics, better understood as complex adaptive systems, is a key aim of this perspective.

Role of short-term dispersal on the dynamics of Zika virus in an extreme idealized environment

In November 2015, El Salvador reported their first case of Zika virus (ZIKV) infection, an event followed by an explosive outbreak that generated over 6000 suspected cases in a period of two months. National agencies began implementing control measures that included vector control and recommending an increased use of repellents. Further, in response to the alarming and growing number of microcephaly cases in Brazil, the importance of avoiding pregnancies for two years was stressed. In this paper, we explore the role of mobility within communities characterized by extreme poverty, crime and violence. Specifically, the role of short term mobility between two idealized interconnected highly distinct communities is explored in the context of ZIKV outbreaks. We make use of a Lagrangian modeling approach within a two-patch setting in order to highlight the possible effects that short-term mobility, within highly distinct environments, may have on the dynamics of ZIKV outbreak when the overall goal is to reduce the number of cases not just in the most affluent areas but everywhere. Outcomes depend on existing mobility patterns, levels of disease risk, and the ability of federal or state public health services to invest in resource limited areas, particularly in those where violence is systemic. The results of simulations in highly polarized and simplified scenarios are used to assess the role of mobility. It quickly became evident that matching observed patterns of ZIKV outbreaks could not be captured without incorporating increasing levels of heterogeneity. The number of distinct patches and variations on patch connectivity structure required to match ZIKV patterns could not be met within the highly aggregated model that is used in the simulations.

Day-to-Day Population Movement and the Management of Dengue Epidemics

Dengue is a growing public health problem in tropical and subtropical cities. It is transmitted by mosquitoes, and the main strategy for epidemic prevention and control is insecticide fumigation. Effective management is, however, proving elusive. People’s day-to-day movement about the city is believed to be an important factor in the epidemiological dynamics. We use a simple model to examine the fundamental roles of broad demographic and spatial structures in epidemic initiation, growth and control. We show that the key factors are local dilution, characterised by the vector–host ratio, and spatial connectivity, characterised by the extent of habitually variable movement patterns. Epidemic risk in the population is driven by the demographic groups that frequent the areas with the highest vector–host ratio, even if they only spend some of their time there. Synchronisation of epidemic trajectories in different demographic groups is governed by the vector–host ratios to which they are exposed and the strength of connectivity. Strategies for epidemic prevention and management may be made more effective if they take into account the fluctuating landscape of transmission intensity associated with spatial heterogeneity in the vector–host ratio and people’s day-to-day movement patterns.