The control of vector-borne disease epidemics

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, R₀, the measure of long term endemicity. Despite its widespread application, R₀ does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, E₀, which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on E₀ and R₀ may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector-host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index E₀ to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.     

Spatial Heterogeneity,Host Movement and Mosquito-Borne Disease Transmission

Mosquito-borne diseases are a global health priority disproportionately affecting low-income populations in tropical and sub-tropical countries. These pathogens live in mosquitoes and hosts that interact in spatially heterogeneous environments where hosts move between regions of varying transmission intensity. Although there is increasing interest in the implications of spatial processes for mosquito-borne disease dynamics, most of our understanding derives from models that assume spatially homogeneous transmission. Spatial variation in contact rates can influence transmission and the risk of epidemics, yet the interaction between spatial heterogeneity and movement of hosts remains relatively unexplored. Here we explore, analytically and through numerical simulations, how human mobility connects spatially heterogeneous mosquito populations, thereby influencing disease persistence (determined by the basic reproduction number R ₀), prevalence and their relationship. We show that, when local transmission rates are highly heterogeneous, R ₀ declines asymptotically as human mobility increases, but infection prevalence peaks at low to intermediate rates of movement and decreases asymptotically after this peak. Movement can reduce heterogeneity in exposure to mosquito biting. As a result, if biting intensity is high but uneven, infection prevalence increases with mobility despite reductions in R ₀. This increase in prevalence decreases with further increase in mobility because individuals do not spend enough time in high transmission patches, hence decreasing the number of new infections and overall prevalence. These results provide a better basis for understanding the interplay between spatial transmission heterogeneity and human mobility, and their combined influence on prevalence and R ₀.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

Physiological analysis on oscillatory behavior of glucose-insulin regulation by model with delays

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R₀, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R₀ is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?₀ can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ? _j , j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

The Effects of Vector Movement and Distribution in a Mathematical Model of Dengue Transmission

Background

Mathematical models have been used to study the dynamics of infectious disease outbreaks and predict the effectiveness of potential mass vaccination campaigns. However, models depend on simplifying assumptions to be tractable, and the consequences of making such assumptions need to be studied. Two assumptions usually incorporated by mathematical models of vector-borne disease transmission is homogeneous mixing among the hosts and vectors and homogeneous distribution of the vectors.

Methodology/Principal Findings

We explored the effects of mosquito movement and distribution in an individual-based model of dengue transmission in which humans and mosquitoes are explicitly represented in a spatial environment. We found that the limited flight range of the vector in the model greatly reduced its ability to transmit dengue among humans. A model that does not assume a limited flight range could yield similar attack rates when transmissibility of dengue was reduced by 39%. A model in which mosquitoes are distributed uniformly across locations behaves similarly to one in which the number of mosquitoes per location is drawn from an exponential distribution with a slightly higher mean number of mosquitoes per location. When the models with different assumptions were calibrated to have similar human infection attack rates, mass vaccination had nearly identical effects.

Conclusions/Significance

Small changes in assumptions in a mathematical model of dengue transmission can greatly change its behavior, but estimates of the effectiveness of mass dengue vaccination are robust to some simplifying assumptions typically made in mathematical models of vector-borne disease.     

Multiple sources and routes of information transmission: Implications for epidemic dynamics

In a recent paper [20], we proposed and analyzed a compartmental ODE-based model describing the dynamics of an infectious disease where the presence of the pathogen also triggers the diffusion of information about the disease. In this paper, we extend this previous work by presenting results based on pairwise and simulation models that are better suited for capturing the population contact structure at a local level. We use the pairwise model to examine the potential of different information generating mechanisms and routes of information transmission to stop disease spread or to minimize the impact of an epidemic. The individual-based simulation is used to better differentiate between the networks of disease and information transmission and to investigate the impact of different basic network topologies and network overlap on epidemic dynamics. The paper concludes with an individual-based semi-analytic calculation of R₀ at the non-trivial disease free equilibrium.     

Fine-scale estimation of effective reproduction numbers for dengue surveillance

The effective reproduction number R_t is an epidemiological quantity that provides an instantaneous measure of transmission potential of an infectious disease. While dengue is an increasingly important vector-borne disease, few have used R_t as a measure to inform public health operations and policy for dengue. This study demonstrates the utility of R_t for real time dengue surveillance. Using nationally representative, geo-located dengue case data from Singapore over 2010–2020, we estimated R_t by modifying methods from Bayesian (EpiEstim) and filtering (EpiFilter) approaches, at both the national and local levels. We conducted model assessment of R_t from each proposed method and determined exogenous temporal and spatial drivers for R_t in relation to a wide range of environmental and anthropogenic factors. At the national level, both methods achieved satisfactory model performance (R²_EpiEstim = 0.95, R²_EpiFilter = 0.97), but disparities in performance were large at finer spatial scales when case counts are low (MASE _EpiEstim = 1.23, MASE_EpiFilter = 0.59). Impervious surfaces and vegetation with structure dominated by human management (without tree canopy) were positively associated with increased transmission intensity. Vegetation with structure dominated by human management (with tree canopy), on the other hand, was associated with lower dengue transmission intensity. We showed that dengue outbreaks were preceded by sustained periods of high transmissibility, demonstrating the potential of R_t as a dengue surveillance tool for detecting large rises in dengue cases. Real time estimation of R_t at the fine scale can assist public health agencies in identifying high transmission risk areas and facilitating localised outbreak preparedness and response.     

Demonstrating frequency-dependent transmission of sarcoptic mange in red foxes

Understanding the relationship between disease transmission and host density is essential for predicting disease spread and control. Using long-term data on sarcoptic mange in a red fox Vulpes vulpes population, we tested long-held assumptions of density- and frequency-dependent direct disease transmission. We also assessed the role of indirect transmission. Contrary to assumptions typical of epidemiological models, mange dynamics are better explained by frequency-dependent disease transmission than by density-dependent transmission in this canid. We found no support for indirect transmission. We present the first estimates of R₀ and age-specific transmission coefficients for mange in foxes. These parameters are important for managing this poorly understood but highly contagious and economically damaging disease.     

Empirical Estimation of R 0 for Unknown Transmission Functions: The Case of Chronic Wasting Disease in Alberta

We consider the problem of estimating the basic reproduction number R ₀ from data on prevalence dynamics at the beginning of a disease outbreak. We derive discrete and continuous time models, some coefficients of which are to be fitted from data. We show that prevalence of the disease is sufficient to determine R ₀. We apply this method to chronic wasting disease spread in Alberta determining a range of possible R ₀ and their sensitivity to the probability of deer annual survival.     

The epidemic threshold of vector-borne diseases with seasonality

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R ₀ which is adapted to periodic environments. This R ₀ is estimated numerically for the epidemic in Chichaoua; 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor 3.76.     

Inferring Epidemiological Dynamics with Bayesian Coalescent Inference: The Merits of Deterministic and Stochastic Models

Estimation of epidemiological and population parameters from molecular sequence data has become central to the understanding of infectious disease dynamics. Various models have been proposed to infer details of the dynamics that describe epidemic progression. These include inference approaches derived from Kingman’s coalescent theory. Here, we use recently described coalescent theory for epidemic dynamics to develop stochastic and deterministic coalescent susceptible–infected–removed (SIR) tree priors. We implement these in a Bayesian phylogenetic inference framework to permit joint estimation of SIR epidemic parameters and the sample genealogy. We assess the performance of the two coalescent models and also juxtapose results obtained with a recently published birth–death-sampling model for epidemic inference. Comparisons are made by analyzing sets of genealogies simulated under precisely known epidemiological parameters. Additionally, we analyze influenza A (H1N1) sequence data sampled in the Canterbury region of New Zealand and HIV-1 sequence data obtained from known United Kingdom infection clusters. We show that both coalescent SIR models are effective at estimating epidemiological parameters from data with large fundamental reproductive number R₀ and large population size S₀. Furthermore, we find that the stochastic variant generally outperforms its deterministic counterpart in terms of error, bias, and highest posterior density coverage, particularly for smaller R₀ and S₀. However, each of these inference models is shown to have undesirable properties in certain circumstances, especially for epidemic outbreaks with R₀ close to one or with small effective susceptible populations.     

Malaria's Missing Number: Calculating the Human Component of R0 by a Within-Host Mechanistic Model of Plasmodium falciparum Infection and Transmission

Human infection by malarial parasites of the genus Plasmodium begins with the bite of an infected Anopheles mosquito. Current estimates place malaria mortality at over 650,000 individuals each year, mostly in African children. Efforts to reduce disease burden can benefit from the development of mathematical models of disease transmission. To date, however, comprehensive modeling of the parameters defining human infectivity to mosquitoes has remained elusive. Here, we describe a mechanistic within-host model of Plasmodium falciparum infection in humans and pathogen transmission to the mosquito vector. Our model incorporates the entire parasite lifecycle, including the intra-erythrocytic asexual forms responsible for disease, the onset of symptoms, the development and maturation of intra-erythrocytic gametocytes that are transmissible to Anopheles mosquitoes, and human-to-mosquito infectivity. These model components were parameterized from malaria therapy data and other studies to simulate individual infections, and the ensemble of outputs was found to reproduce the full range of patient responses to infection. Using this model, we assessed human infectivity over the course of untreated infections and examined the effects in relation to transmission intensity, expressed by the basic reproduction number R₀ (defined as the number of secondary cases produced by a single typical infection in a completely susceptible population). Our studies predict that net human-to-mosquito infectivity from a single non-immune individual is on average equal to 32 fully infectious days. This estimate of mean infectivity is equivalent to calculating the human component of malarial R₀. We also predict that mean daily infectivity exceeds five percent for approximately 138 days. The mechanistic framework described herein, made available as stand-alone software, will enable investigators to conduct detailed studies into theories of malaria control, including the effects of drug treatment and drug resistance on transmission.     

A mathematical model for zoonotic transmission of malaria in the Atlantic Forest: Exploring the effects of variations in vector abundance and acrodendrophily

Transmission foci of autochthonous malaria caused by Plasmodium vivax-like parasites have frequently been reported in the Atlantic Forest in Southeastern and Southern Brazil. Evidence suggests that malaria is a zoonosis in these areas as human infections by simian Plasmodium species have been detected, and the main vector of malaria in the Atlantic Forest, Anopheles (Kerteszia) cruzii, can blood feed on human and simian hosts. In view of the lack of models that seek to predict the dynamics of zoonotic transmission in this part of the Atlantic Forest, the present study proposes a new deterministic mathematical model that includes a transmission compartment for non-human primates and parameters that take into account vector displacement between the upper and lower forest strata. The effects of variations in the abundance and acrodendrophily of An. cruzii on the prevalence of infected humans in the study area and the basic reproduction number (R₀) for malaria were analyzed. The model parameters are based on the literature and fitting of the empirical data. Simulations performed with the model indicate that (1) an increase in the abundance of the vector in relation to the total number of blood-seeking mosquitoes leads to an asymptotic increase in both the proportion of infected individuals at steady state and R₀; (2) the proportion of infected humans at steady state is higher when displacement of the vector mosquito between the forest strata increases; and (3) in most scenarios, Plasmodium transmission cannot be sustained only between mosquitoes and humans, which implies that non-human primates play an important role in maintaining the transmission cycle. The proposed model contributes to a better understanding of the dynamics of malaria transmission in the Atlantic Forest.     

Two-Host,Two-Vector Basic Reproduction Ratio (R 0) for Bluetongue

Mathematical formulations for the basic reproduction ratio (R ₀) exist for several vector-borne diseases. Generally, these are based on models of one-host, one-vector systems or two-host, one-vector systems. For many vector borne diseases, however, two or more vector species often co-occur and, therefore, there is a need for more complex formulations. Here we derive a two-host, two-vector formulation for the R ₀ of bluetongue, a vector-borne infection of ruminants that can have serious economic consequences; since 1998 for example, it has led to the deaths of well over 1 million sheep in Europe alone. We illustrate our results by considering the situation in South Africa, where there are two major hosts (sheep, cattle) and two vector species with differing ecologies and competencies as vectors, for which good data exist. We investigate the effects on R ₀ of differences in vector abundance, vector competence and vector host preference between vector species. Our results indicate that R ₀ can be underestimated if we assume that there is only one vector transmitting the infection (when there are in fact two or more) and/or vector host preferences are overlooked (unless the preferred host is less beneficial or more abundant). The two-host, one-vector formula provides a good approximation when the level of cross-infection between vector species is very small. As this approaches the level of intraspecies infection, a combination of the two-host, one-vector R ₀ for each vector species becomes a better estimate. Otherwise, particularly when the level of cross-infection is high, the two-host, two-vector formula is required for accurate estimation of R ₀. Our results are equally relevant to Europe, where at least two vector species, which co-occur in parts of the south, have been implicated in the recent epizootic of bluetongue.     

A model of bi-mode transmission dynamics of hepatitis C with optimal control

In this paper, we present a rigorous mathematical analysis of a deterministic model for the transmission dynamics of hepatitis C. The model is suitable for populations where two frequent modes of transmission of hepatitis C virus, namely unsafe blood transfusions and intravenous drug use, are dominant. The susceptible population is divided into two distinct compartments, the intravenous drug users and individuals undergoing unsafe blood transfusions. Individuals belonging to each compartment may develop acute and then possibly chronic infections. Chronically infected individuals may be quarantined. The analysis indicates that the eradication and persistence of the disease is completely determined by the magnitude of basic reproduction number R _c. It is shown that for the basic reproduction number R _c < 1, the disease-free equilibrium is locally and globally asymptotically stable. For R _c > 1, an endemic equilibrium exists and the disease is uniformly persistent. In addition, we present the uncertainty and sensitivity analyses to investigate the influence of different important model parameters on the disease prevalence. When the infected population persists, we have designed a time-dependent optimal quarantine strategy to minimize it. The Pontryagin’s Maximum Principle is used to characterize the optimal control in terms of an optimality system which is solved numerically. Numerical results for the optimal control are compared against the constant controls and their efficiency is discussed.     

Contact rate calculation for a basic epidemic model

Mass-action epidemic models are the foundation of the majority of studies of disease dynamics in human and animal populations. Here, a kinetic model of mobile susceptible and infective individuals in a two-dimensional domain is introduced, and an examination of the contact process results in a mass-action-like term for the generation of new infectives. The conditions under which density dependent and frequency dependent transmission terms emerge are clarified. Moreover, this model suggests that epidemics in large mobile spatially distributed populations can be well described by homogeneously mixing mass-action models. The analysis generates an analytic formula for the contact rate (β) and the basic reproductive ratio (R₀) of an infectious pathogen, which contains a mixture of demographic and epidemiological parameters. The analytic results are compared with a simulation and are shown to give good agreement. The simulation permits the exploration of more realistic movement strategies and their consequent effect on epidemic dynamics.     

Global dynamics of cholera models with differential infectivity

A general compartmental model for cholera is formulated that incorporates two pathways of transmission, namely direct and indirect via contaminated water. Non-linear incidence, multiple stages of infection and multiple states of the pathogen are included, thus the model includes and extends cholera models in the literature. The model is analyzed by determining a basic reproduction number R₀ and proving, by using Lyapunov functions and a graph-theoretic result based on Kirchhoff’s Matrix Tree Theorem, that it determines a sharp threshold. If R₀?1, then cholera dies out; whereas if R₀>1, then the disease tends to a unique endemic equilibrium. When input and death are neglected, the model is used to determine a final size equation or inequality, and simulations illustrate how assumptions on cholera transmission affect the final size of an epidemic.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

A generalized cholera model and epidemic–endemic analysis

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R ₀=1 for the combined human–environment epidemiological model under biologically reasonable conditions.