A new method for estimating the effort required to control an infectious disease

Trypanosoma cruzi in domestic and wild mammals in the northeast region of Colombia

Trypanosoma cruzi hosts can serve as a source of infection for animals, vectors, and humans, contributing to the establishment of Chagas disease (CD) in a given area. Traditionally, the Department of Córdoba has not been considered a transmission area for CD; however, the report of several acute cases of Chagas disease highlights the importance of studying the dynamics of disease transmission in this region. This study aimed to detect T. cruzi in domestic and wild mammals in the department of Córdoba. In 2017, a cross-sectional descriptive study was conducted in six villages in two municipalities in the department of Córdoba. Blood samples from dogs living in the zones were collected in EDTA vacutainer tubes for domestic mammals. Wild mammals were collected using Sherman and Tomahawk traps and mist nets in crops and peridomiciles. T. cruzi DNA was detected using the kinetoplast DNA (kDNA) variable region and the tandem repeat satellite region of T. cruzi as molecular targets. We sampled 168 dogs and 146 wild mammals. The detected prevalence of T. cruzi was 6.37%; the TcI lineage was found in D. marsupialis, H. anomalus, and one canine. A specimen of D. marsupialis with TcI and TcII lineages was also identified. T. cruzi DNA was detected in domestic and wild animals in the study area, indicating the circulation of the parasite in peridomestic environments. D. marsupialis may represent an important host in maintaining this region's wild and domestic cycle.

A Scoping Review of Mathematical Models Used to Investigate the Role of Dogs in Chagas Disease Transmission

Chagas disease is a zoonotic vector-borne disease caused by the parasite Trypanosoma cruzi, which affects a variety of mammalian species across the Americas, including humans and dogs. Mathematical modeling has been widely used to investigate the transmission dynamics and control of vector-borne diseases. We performed a scoping review of mathematical models that investigated the role of dogs in T. cruzi transmission. We identified ten peer-reviewed papers that have explicitly modeled the role of dogs in Chagas transmission dynamics. We discuss the different methods employed in these studies, the different transmission metrics, disease transmission routes, and disease control strategies that have been considered and evaluated. In general, mathematical modeling studies have shown that dogs are not only at high risk of T. cruzi infection but are also major contributors to T. cruzi transmission to humans. Moreover, eliminating infected dogs from households or frequent use of insecticide was shown to be effective for curtailing T. cruzi transmission in both humans and dogs. However, when insecticide spraying is discontinued, T. cruzi infections in dogs were shown to return to their pre-spraying levels. We discuss the challenges and opportunities for future modeling studies to improve our understanding of Chagas disease transmission dynamics and control.

Understanding the Role of Environmental Transmission on COVID-19 Herd Immunity and Invasion Potential

COVID-19 is caused by the SARS-CoV-2 virus, which is mainly transmitted directly between humans. However, it is observed that this disease can also be transmitted through an indirect route via environmental fomites. The development of appropriate and effective vaccines has allowed us to target and anticipate herd immunity. Understanding of the transmission dynamics and the persistence of the virus on environmental fomites and their resistive role on indirect transmission of the virus is an important scientific and public health challenge because it is essential to consider all possible transmission routes and route specific transmission strength to accurately quantify the herd immunity threshold. In this paper, we present a mathematical model that considers both direct and indirect transmission modes. Our analysis focuses on establishing the disease invasion threshold, investigating its sensitivity to both transmission routes and isolate route-specific transmission rate. Using the tau-leap algorithm, we perform a stochastic model simulation to address the invasion potential of both transmission routes. Our analysis shows that direct transmission has a higher invasion potential than that of the indirect transmission. As a proof of this concept, we fitted our model with early epidemic data from several countries to uniquely estimate the reproduction numbers associated with direct and indirect transmission upon confirming the identifiability of the parameters. As the indirect transmission possess lower invasion potential than direct transmission, proper estimation and necessary steps toward mitigating it would help reduce vaccination requirement.

Sensitivity analysis on the declining population in Japan: Effects of prefecture-specific fertility and interregional migration

Japan has been facing a population decline since 2010 due to low birth rates, interregional migration, and regional traits. In this study, we modeled the demographic dynamics of Japan using a transition matrix model. Then, from the mathematical structure of the model, we quantitatively evaluated the domestic factors of population decline. To achieve this, we constructed a multi-regional Leslie matrix model and developed a method for representing the reproductive value and stable age distribution using matrix entries. Our method enabled us to interpret the mathematical indices using the genealogies of the migration history of individuals and their ancestors. Furthermore, by combining our method with sensitivity analysis, we analyzed the effect of region-specific fertility rates and interregional migration rates on the population decline in Japan. We found that the sensitivity of the population growth rate to the migration rate from urban areas with large populations to prefectures with high fertility rates was greatest for people aged under 30. In addition, compared to other areas, the fertility rates of urban areas exhibited higher sensitivity for people aged over 30. Because this feature is robust in comparison with those in 2010 and 2015, it can be said to be a unique structure in Japan in recent years. We also established a method to represent the reproductive value and stable age distribution in an irreducible non-negative matrix population model by using the matrix entries. Furthermore, we show the effects of fertility and migration rates numerically in urban and non-urban areas on the population growth rates for each age group in a society with a declining population.

Why the Spectral Radius? An intuition-building introduction to the basic reproduction number

The basic reproduction number [Formula: see text] is a fundamental concept in mathematical epidemiology and infectious disease modeling. Loosely speaking, it describes the number of people that an infectious person is expected to infect. The basic reproduction number has profound implications for epidemic trajectories and disease control strategies. It is well known that the basic reproduction number can be calculated as the spectral radius of the next generation matrix, but why this is the case may not be intuitively obvious. Here, we walk through how the discrete, next generation process connects to the ordinary differential equation disease system of interest, linearized at the disease-free equilibrium. Then, we use linear algebra to develop a geometric explanation of why the spectral radius of the next generation matrix is an epidemic threshold. Finally, we work through a series of examples that help to build familiarity with the kinds of patterns that arise in parameter combinations produced by the next generation method. This article is intended to help new infectious disease modelers develop intuition for the form and interpretation of the basic reproduction number in their disease systems of interest.

Global stability of an age-structured infection model in vivo with two compartments and two routes

In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio $ R_0 $ gives the threshold of the stability. If $ R_0 > 1 $, the interior equilibrium is unique and globally stable, and if $ R_0 \le 1 $, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model.

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

Understanding the incidence and timing of rabies cases in domestic animals and wildlife in south-east Tanzania in the presence of widespread domestic dog vaccination campaigns

The "Zero by 30" strategic plan aims to eliminate human deaths from dog-mediated rabies by 2030 and domestic dog vaccination is a vital component of this strategic plan. In areas where domestic dog vaccination has been implemented, it is important to assess the impact of this intervention. Additionally, understanding temporal and seasonal trends in the incidence of animal rabies cases may assist in optimizing such interventions. Data on the incidence of probable rabies cases in domestic and wild animals were collected between January 2011 and December 2018 in thirteen districts of south-east Tanzania where jackals comprise over 40% of reported rabies cases. Vaccination coverage was estimated over this period, as five domestic dog vaccination campaigns took place in all thirteen districts between 2011 and 2016. Negative binomial generalized linear models were used to explore the impact of domestic dog vaccination on the annual incidence of animal rabies cases, whilst generalized additive models were used to investigate the presence of temporal and/or seasonal trends. Increases in domestic dog vaccination coverage were significantly associated with a decreased incidence of rabies cases in both domestic dogs and jackals. A 35% increase in vaccination coverage was associated with a reduction in the incidence of probable dog rabies cases of between 78.0 and 85.5% (95% confidence intervals ranged from 61.2 to 92.2%) and a reduction in the incidence of probable jackal rabies cases of between 75.3 and 91.2% (95% confidence intervals ranged from 53.0 to 96.1%). A statistically significant common seasonality was identified in the monthly incidence of probable rabies cases in both domestic dogs and jackals with the highest incidence from February to August and lowest incidence from September to January. These results align with evidence supporting the use of domestic dog vaccination as part of control strategies aimed at reducing animal rabies cases in both domestic dogs and jackals in this region. The presence of a common seasonal trend requires further investigation but may have implications for the timing of future vaccination campaigns.

Reservoir dynamics of rabies in south-east Tanzania and the roles of cross-species transmission and domestic dog vaccination

Understanding the role of different species in the transmission of multi-host pathogens, such as rabies virus, is vital for effective control strategies. Across most of sub-Saharan Africa domestic dogs Canis familiaris are considered the reservoir for rabies, but the role of wildlife has been long debated. Here we explore the multi-host transmission dynamics of rabies across south-east Tanzania.Between January 2011 and July 2019, data on probable rabies cases were collected in the regions of Lindi and Mtwara. Hospital records of animal-bite patients presenting to healthcare facilities were used as sentinels for animal contact tracing. The timing, location and species of probable rabid animals were used to reconstruct transmission trees to infer who infected whom and the relative frequencies of within- and between-species transmission.During the study, 688 probable human rabies exposures were identified, resulting in 47 deaths. Of these exposures, 389 were from domestic dogs (56.5%) and 262 from jackals (38.1%). Over the same period, 549 probable animal rabies cases were traced: 303 in domestic dogs (55.2%) and 221 in jackals (40.3%), with the remainder in domestic cats and other wildlife species.Although dog-to-dog transmission was most commonly inferred (40.5% of transmission events), a third of inferred events involved wildlife-to-wildlife transmission (32.6%), and evidence suggested some sustained transmission chains within jackal populations.A steady decline in probable rabies cases in both humans and animals coincided with the implementation of widespread domestic dog vaccination during the first 6 years of the study. Following the lapse of this program, dog rabies cases began to increase in one of the northernmost districts. Synthesis and applications. In south-east Tanzania, despite a relatively high incidence of rabies in wildlife and evidence of wildlife-to-wildlife transmission, domestic dogs remain essential to the reservoir of infection. Continued dog vaccination alongside improved surveillance would allow a fuller understanding of the role of wildlife in maintaining transmission in this area. Nonetheless, dog vaccination clearly suppressed rabies in both domestic dog and wildlife populations, reducing both public health and conservation risks and, if sustained, has potential to eliminate rabies from this region.

Spillover, hybridization, and persistence in schistosome transmission dynamics at the human-animal interface

Zoonotic spillover and hybridization of parasites are major emerging public and veterinary health concerns at the interface of infectious disease biology, evolution, and control. Schistosomiasis is a neglected tropical disease of global importance caused by parasites of the Schistosoma genus, and the Schistosoma spp. system within Africa represents a key example of a system where spillover of animal parasites into human populations has enabled formation of hybrids. Combining model-based approaches and analyses of parasitological, molecular, and epidemiological data from northern Senegal, a region with a high prevalence of schistosome hybrids, we aimed to unravel the transmission dynamics of this complex multihost, multiparasite system. Using Bayesian methods and by estimating the basic reproduction number (R₀ ), we evaluate the frequency of zoonotic spillover of Schistosoma bovis from livestock and the potential for onward transmission of hybrid S. bovis × S. haematobium offspring within human populations. We estimate R₀ of hybrid schistosomes to be greater than the critical threshold of one (1.76; 95% CI 1.59 to 1.99), demonstrating the potential for hybridization to facilitate spread and establishment of schistosomiasis beyond its original geographical boundaries. We estimate R₀ for S. bovis to be greater than one in cattle (1.43; 95% CI 1.24 to 1.85) but not in other ruminants, confirming cattle as the primary zoonotic reservoir. Through longitudinal simulations, we also show that where S. bovis and S. haematobium are coendemic (in livestock and humans respectively), the relative importance of zoonotic transmission is predicted to increase as the disease in humans nears elimination.

Structure of epidemic models: toward further applications in economics

In this paper, we review the structure of various epidemic models in mathematical epidemiology for the future applications in economics. The heterogeneity of population and the generalization of nonlinear terms play important roles in making more elaborate and realistic models. The basic, effective, control and type reproduction numbers have been used to estimate the intensity of epidemic, to evaluate the effectiveness of interventions and to design appropriate interventions. The advanced epidemic models includes the age structure, seasonality, spatial diffusion, mutation and reinfection, and the theory of reproduction numbers has been generalized to them. In particular, the existence of sustained periodic solutions has attracted much interest because they can explain the recurrent waves of epidemic. Although the theory of epidemic models has been developed in decades and the development has been accelerated through COVID-19, it is still difficult to completely answer the uncertainty problem of epidemic models. We would have to mind that there is no single model that can solve all questions and build a scientific attitude to comprehensively understand the results obtained by various researchers from different backgrounds.

ARE THE BEGINNING AND ENDING PHASES OF EPIDEMICS CHARACTERIZED BY THE NEXT GENERATION MATRICES? – A CASE STUDY OF DRUG-SENSITIVE AND RESISTANT TUBERCULOSIS MODEL

In epidemiological modeling, the basic reproduction number is usually defined as being the spectral radius of the next-generation matrix evaluated at the trivial equilibrium. The global stability of the trivial equilibrium point was determined by the left eigenvector associated with that next-generation matrix. More recently, the fraction of susceptible individuals was also obtained from the next generation matrix. The gross reproduction number and the fraction of susceptible individuals were calculated by revisiting the drug-sensitive and resistant tuberculosis model. Hence, the next-generation matrices shed light on the evolution of the dynamics: the beginning of the epidemic via the basic reproduction number and approaching the epidemic’s steady-state via the susceptible individuals’ asymptotic fraction.

Understanding the Spread of COVID-19 in China: Spatial–Temporal Characteristics, Risk Analysis and the Impact of the Quarantine of Hubei Province on the Railway Transportation Network

The rapid spread of COVID-19 and its global growth constitutes an international public-health emergency, posing a serious threat to global health, safety, and social economy. In this paper, we systematically studied the temporal and spatial characteristics of COVID-19, infectivity, and the impact of Hubei province’s quarantine on the national railway system on the basis of epidemic and national train data. This study found the following: (1) The overall growth of the epidemic was exponential, and the outbreak of Hubei had a strong spread in the eastern and southern directions. The epidemic was generally more serious in the capital or developed cities in each province, and the epidemic outside Hubei was under control after the imported growth ended. (2) On the basis of analyzing the disturbance of the spread of the epidemic by traffic control, the average incubation period of COVID-19 was approximately 4 days. The ratio of the number of cured people to the number of deaths gradually increased, indicating that, given sufficient medical service, the cure rate can be greatly improved. (3) The quarantine of Hubei had greater impact on cities with higher centrality, especially in the Yangtze River Delta region, and smaller impact on the overall connectivity of the national railway network. For local people, quarantine had great impact on the outflow of local people to neighboring provinces.

Quantifying the risk of vector-borne disease transmission attributable to genetically modified vectors

The relative risk of disease transmission caused by the potential release of transgenic vectors, such as through sterile insect technique or gene drive systems, is assessed with comparison with wild-type vectors. The probabilistic risk framework is demonstrated with an assessment of the relative risk of lymphatic filariasis, malaria and o'nyong'nyong arbovirus transmission by mosquito vectors to human hosts given a released transgenic strain of Anopheles coluzzii carrying a dominant sterile male gene construct. Harm is quantified by a logarithmic loss function that depends on the causal risk ratio, which is a quotient of basic reproduction numbers derived from mathematical models of disease transmission. The basic reproduction numbers are predicted to depend on the number of generations in an insectary colony and the number of backcrosses between the transgenic and wild-type lineages. Analogous causal risk ratios for short-term exposure to a single cohort release are also derived. These causal risk ratios were parametrized by probabilistic elicitations, and updated with experimental data for adult vector mortality. For the wild-type, high numbers of insectary generations were predicted to reduce the number of infectious human cases compared with uncolonized wild-type. Transgenic strains were predicted to produce fewer infectious cases compared with the uncolonized wild-type.

Megacities as drivers of national outbreaks: The 2017 chikungunya outbreak in Dhaka, Bangladesh

BACKGROUND

Several large outbreaks of chikungunya have been reported in the Indian Ocean region in the last decade. In 2017, an outbreak occurred in Dhaka, Bangladesh, one of the largest and densest megacities in the world. Population mobility and fluctuations in population density are important drivers of epidemics. Measuring population mobility during outbreaks is challenging but is a particularly important goal in the context of rapidly growing and highly connected cities in low- and middle-income countries, which can act to amplify and spread local epidemics nationally and internationally.

METHODS

We first describe the epidemiology of the 2017 chikungunya outbreak in Dhaka and estimate incidence using a mechanistic model of chikungunya transmission parametrized with epidemiological data from a household survey. We combine the modeled dynamics of chikungunya in Dhaka, with mobility estimates derived from mobile phone data for over 4 million subscribers, to understand the role of population mobility on the spatial spread of chikungunya within and outside Dhaka during the 2017 outbreak.

RESULTS

We estimate a much higher incidence of chikungunya in Dhaka than suggested by official case counts. Vector abundance, local demographics, and population mobility were associated with spatial heterogeneities in incidence in Dhaka. The peak of the outbreak in Dhaka coincided with the annual Eid holidays, during which large numbers of people traveled from Dhaka to other parts of the country. We show that travel during Eid likely resulted in the spread of the infection to the rest of the country.

CONCLUSIONS

Our results highlight the impact of large-scale population movements, for example during holidays, on the spread of infectious diseases. These dynamics are difficult to capture using traditional approaches, and we compare our results to a standard diffusion model, to highlight the value of real-time data from mobile phones for outbreak analysis, forecasting, and surveillance.

Estimating and explaining the spread of COVID-19 at the county level in the USA

The basic reproduction number, R0, determines the rate of spread of a communicable disease and therefore gives fundamental information needed to plan public health interventions. Using mortality records, we estimated the rate of spread of COVID-19 among 160 counties and county-aggregates in the USA at the start of the epidemic. We show that most of the high among-county variance is explained by four factors (R2 = 0.70): the timing of outbreak, population size, population density, and spatial location. For predictions of future spread, population density and spatial location are important, and for the latter we show that SARS-CoV-2 strains containing the G614 mutation to the spike gene are associated with higher rates of spread. Finally, the high predictability of R0 allows extending estimates to all 3109 counties in the conterminous 48 states. The high variation of R0 argues for public health policies enacted at the county level for controlling COVID-19.

Optimal control for COVID-19 pandemic with quarantine and antiviral therapy

In the absence of a proper cure for the disease, the recent pandemic caused by COVID-19 has been focused on isolation strategies and government measures to control the disease, such as lockdown, media coverage, and improve public hygiene. Mathematical models can help when these intervention mechanisms find some optimal strategies for controlling the spread of such diseases. We propose a set of nonlinear dynamic systems with optimal strategy including practical measures to limit the spread of the virus and to diagnose and isolate infected people while maintaining consciousness for citizens. We have used Pontryagin's maximum principle and solved our system by the finite difference method. In the end, several numerical simulations have been executed to verify the proposed model using Matlab. Also, we pursued the resilience of the parameters of control of the nonlinear dynamic systems, so that we can easily handle the pandemic situation.

Target reproduction numbers for reaction-diffusion population models

A very important population threshold quantity is the target reproduction number, which is a measure of control effort required for a target prevention, intervention or control. This concept, as a generalization of type reproduction number, was first introduced in Shuai et al. (J Math Biol 67:1067-1082, 2013) for nonnegative matrices with immediate applications to compartmental population models of ordinary differential equations. The current paper is devoted to the study of all target reproduction numbers for reaction-diffusion population models with compartmental structure. It turns out that the target reproduction number can be regarded as the basic reproduction number of a modified system, where the state of newborn individuals is limited to the target control set and the offspring from the non-target set is regarded as a part of the transition. In other words, the target reproduction number can be interpreted as the expected number of offspring in a specific target set that a primary newborn individual of the same set would produce during its lifetime. We also characterize the target reproduction number so that it can be easily computed numerically for reaction-diffusion models. At the end, we demonstrate our theoretical observations using two examples.

Modeling and Dynamics Analysis of Zika Transmission with Limited Medical Resources

Zika virus, a reemerging mosquito-borne flavivirus, posed a global public health emergency in 2016. Brazil is the most seriously affected country. Some measures have been implemented to control the Zika transmission, such as spraying mosquitoes, developing vaccines and drugs. However, because of the limited medical resources (LMRs) in the country, not every infected patient can be treated in time when infected with Zika virus. We aim to build a deterministic Zika model by introducing a piecewise smooth treatment recovery rate to research the effect of LMRs on the transmission and control of Zika. For the model without treatment, we analyze the global stability of equilibria. For the model with treatment, the model exhibits complex dynamics. We prove that the model with treatment undergoes backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2. It means that the model with LMRs is sensitive to parameters and initial conditions, which has important significance for control of Zika. We also apply the model to estimate the basic and control reproduction numbers for the Zika transmission by using the data on weekly reported accumulated Zika cases from March 25, 2016, to April 14, 2018, in Brazil.

Stability Analysis and Optimal Control for Yellow Fever Model with Vertical Transmission

In this study, a deterministic model for the transmission dynamics of yellow fever (YF) in a human-mosquito setting in the presence of control measures is constructed and rigorously analyzed. In addition to horizontal transmissions, vertical transmission within mosquito population is incorporated. Analysis of the mosquito-only component of the model shows that the reduced model has a mosquito-extinction equilibrium, which is globally-asymptotically stable whenever the basic offspring number ( N 0 ) is less than unity. The vaccinated and type reproduction numbers of the full-model are computed. Condition for global-asymptotic stability of the disease-free equilibrium of the model whenN 0 > 1 is presented. It is shown that, fractional dosing of YF vaccine does not meet YF vaccination requirements. Optimal control theory is applied to the model to characterize the controls parameters. Using Pontryagin's maximum principle and modified forward-backward sweep technique, the necessary conditions for existence of solutions to the optimal control problem is determined. Numerical simulations of the models to assess the effect of fractional vaccine dosing on the disease dynamics and global sensitivity analysis are presented.

Mathematical Analyses on the Effects of Control Measures for a Waterborne Disease Model with Socioeconomic Conditions

Waterborne diseases are present major health problems to humanity especially in rural communities where many individuals belong to the lower socioeconomic classes (SECs). The impacts of introducing waterborne disease control measures for such communities are investigated by considering a waterborne disease model. The model is extended by introducing treatment of infected individuals and water purification as control measures. The possible benefits of considering these control measures for the various SECs are investigated. Further analyses show how different degrees of control impact the rate at which waterborne diseases are spread across SECs. The disease control model is validated by using it to study the cholera outbreak in Haiti.

THE COMPUTATION OF REPRODUCTION NUMBERS FOR THE ENVIRONMENT-HOST-ENVIRONMENT CHOLERA TRANSMISSION DYNAMICS

This study presents a new model for the environment-host-environment transmission dynamics of V. cholerae in a community with an interconnected aquatic pond–river water network. For the case when the human host is the sole target of anti-cholera control and the volume of water in the pond is maximum, the disease-free equilibrium of the model is shown to be globally asymptotically stable whenever a certain epidemiological threshold, known as the basic reproduction number [Formula: see text], is less than unity. The epidemiological implication of this result is that cholera can be eliminated from the community if the control strategies implemented can bring (and maintain) [Formula: see text] to a value less than unity. Four scenarios, that represent different interpretations of the role of the V. cholerea pathogen within the environment, were studied. The corresponding basic reproduction numbers were shown to exhibit the same threshold property with respect to the value unity (i.e., if one is less (equal, greater) than unity, then the three others are also less (equal, greater) than unity. Further, it was shown that for the case where anti-cholera control is focused on the human host population, the associated type reproduction number of the model (corresponding to each of the four transmission scenarios considered) is unique. The implication of this result is that the estimate of the effort needed for disease elimination (i.e., the required herd immunity threshold) is unique, regardless of which of the four transmission scenarios is considered. However, when any of the other two bacterial population types in the aquatic environment (i.e., bacterial in the pond or river) is the focus of the control efforts, this study shows that the associated type reproduction number is not unique. Extensive numerical simulations of the model, using a realistic set of parameters from the published literature, show that the community-wide implementation of a strategy that focus on improved water quality, sanitation, and hygiene (known as WASH-only strategy), using the current estimated coverage of 50% and efficacy of 60%, is unable to lead to the elimination of the disease. Such elimination is attainable if the coverage and efficacy are increased (e.g., to 80% and 90%, respectively). Further, elimination can be achieved using a strategy that focuses on oral rehydration therapy and the use of antibiotics to treat the infected humans (i.e., treatment-only strategy) for moderate effectiveness and coverage levels. The combined hybrid WASH-treatment strategy provides far better population-level impact vis a vis disease elimination. This study ranks the three interventions in the following order of population-level effectiveness: combined WASH-treatment, followed by treatment-only and then WASH-only strategy.

Modeling Chikungunya control strategies and Mayaro potential outbreak in the city of Rio de Janeiro

Mosquito-borne diseases have become a significant health issue in many regions around the world. For tropical countries, diseases such as Dengue, Zika, and Chikungunya, became epidemic in the last decades. Health surveillance reports during this period were crucial in providing scientific-based information to guide decision making and resources allocation to control outbreaks. In this work, we perform data analysis of the last Chikungunya epidemics in the city of Rio de Janeiro by applying a compartmental mathematical model. Sensitivity analyses were performed in order to describe the contribution of each parameter to the outbreak incidence. We estimate the "basic reproduction number" for those outbreaks and predict the potential epidemic outbreak of the Mayaro virus. We also simulated several scenarios with different public interventions to decrease the number of infected people. Such scenarios should provide insights about possible strategies to control future outbreaks.

A Next-Generation Approach to Calculate Source-Sink Dynamics in Marine Metapopulations

In marine systems, adult populations confined to isolated habitat patches can be connected by larval dispersal. Source-sink theory provides effective tools to quantify the effect of specific habitat patches on the dynamics of connected populations. In this paper, we construct the next-generation matrix for a marine metapopulation and demonstrate how it can be used to calculate the source-sink dynamics of habitat patches. We investigate the effect of environmental variables on the source-sink dynamics and demonstrate how the next-generation matrix can provide useful biological insight into transient as well as asymptotic dynamics of the model.

Modelling Vector Transmission and Epidemiology of Co-Infecting Plant Viruses

Co-infection of plant hosts by two or more viruses is common in agricultural crops and natural plant communities. A variety of models have been used to investigate the dynamics of co-infection which track only the disease status of infected and co-infected plants, and which do not explicitly track the density of inoculative vectors. Much less attention has been paid to the role of vector transmission in co-infection, that is, acquisition and inoculation and their synergistic and antagonistic interactions. In this investigation, a general epidemiological model is formulated for one vector species and one plant species with potential co-infection in the host plant by two viruses. The basic reproduction number provides conditions for successful invasion of a single virus. We derive a new invasion threshold which provides conditions for successful invasion of a second virus. These two thresholds highlight some key epidemiological parameters important in vector transmission. To illustrate the flexibility of our model, we examine numerically two special cases of viral invasion. In the first case, one virus species depends on an autonomous virus for its successful transmission and in the second case, both viruses are unable to invade alone but can co-infect the host plant when prevalence is high.

The type-reproduction number of sexually transmitted infections through heterosexual and vertical transmission

Multiple sexually transmitted infections (STIs) have threatened human health for centuries. Most STIs spread not only through sexual (horizontal) transmission but also through mother-to-child (vertical) transmission. In a previous work (Ito et al. 2019), we studied a simple model including heterosexual and mother-to-child transmission and proposed a formulation of the basic reproduction number over generations. In the present study, we improved the model to take into account some factors neglected in the previous work: adult mortality from infection, infant mortality caused by mother-to-child transmission, infertility or stillbirth caused by infection, and recovery with treatment. We showed that the addition of these factors has no essential effect on the theoretical formulation. To study the characteristics of the epidemic threshold, we derived analytical formulas for three type-reproduction numbers for adult men, adult women and juveniles. Our result indicates that if an efficient vaccine exists for a prevalent STI, vaccination of females is more effective for containment of the STI than vaccination of males, because the type-reproduction number for adult men is larger than that for adult women when they are larger than one.

Incorporating tick feeding behaviour into R0 for tick-borne pathogens

Tick-borne pathogens pose a considerable disease burden in Europe and North America, where increasing numbers of human cases and the emergence of new tick-borne pathogens has renewed interest in resolving the mechanisms underpinning their geographical distribution and abundance. For Borrelia burgdorferi and tick-borne encephalitis (TBE) virus, transmission of infection from one generation of ticks to another occurs when older nymphal ticks infect younger larval ticks feeding on the same host, either indirectly via systemic infection of the vertebrate host or directly when feeding in close proximity. Here, expressions for the basic reproduction number, R₀, and the related tick type-reproduction number, T, are derived that account for the observation that larval and nymphal ticks tend to aggregate on the same minority of hosts, a tick feeding behaviour known as co-aggregation. The pattern of tick blood meals is represented as a directed, acyclic, bipartite contact network, with individual vertebrate hosts having in-degree, k_in, and out-degree, k_out, that respectively represent cumulative counts of nymphal and larval ticks fed over the lifetime of the host. The in- and out-degree are not independent when co-aggregation occurs such that [Formula: see text] where 〈.〉 indicates expected value. When systemic infection in the vertebrate host is the dominant transmission route R₀²=T, whereas when direct transmission between ticks co-feeding on the same host is dominant then R₀=T and the effect of co-aggregation on R₀ is more pronounced. Simulations of B. burgdorferi and TBE virus transmission on theoretical tick-mouse contact networks revealed that aggregation and co-aggregation have a synergistic effect on R₀ and T, that co-aggregation always increases R₀ and T, and that aggregation only increases R₀ and T when larvae and nymphs also co-aggregate. Co-aggregation has the greatest absolute effect on R₀ and T when the mean larval burden of hosts is high, and the largest relative effect on R₀ for pathogens sustained by co-feeding transmission, e.g. TBE virus in Europe, compared with those predominantly spread by systemic infection, e.g. B. burgdorferi. For both pathogens, though, co-aggregation increases the mean number of ticks infected per infectious tick, T, and so too the likelihood of pathogen persistence.

A Population Dynamics Model of Mosquito-Borne Disease Transmission, Focusing on Mosquitoes' Biased Distribution and Mosquito Repellent Use

We present an improved mathematical model of population dynamics of mosquito-borne disease transmission. Our model considers the effect of mosquito repellent use and the mosquito's behavior or attraction to the infected human, which cause mosquitoes' biased distribution around the human population. Our analysis of the model clearly shows the existence of thresholds for mosquito repellent efficacy and its utilization rate in the human population with respect to the elimination of mosquito-borne diseases. Further, the results imply that the suppression of mosquito-borne diseases becomes more difficult when the mosquitoes' distribution is biased to a greater extent around the human population.

Quantifying the burden of managing wildlife diseases in multiple host species

Mitigation of infectious wildlife diseases is especially challenging where pathogens affect communities of multiple host species. Although most ecological studies recognize the challenge posed by multiple-species pathogens, the implications for management are typically assessed only qualitatively. Translating the intuitive understanding that multiple host species are important into practice requires a quantitative assessment of whether and how secondary host species should also be targeted by management and the effort this will require. Using a multiple-species compartmental model, we determined analytically whether and how intensively secondary host species should be managed to prevent outbreaks in focal hosts based on the reproduction number of individual host species and between-species transmission rates. We applied the model to the invasive pathogenic fungus Batrachochytrium salamandrivorans in a 2-host system in northern Europe. Avoiding a disease outbreak in the focal host (fire salamanders [Salamandra salamandra]) was impossible unless management also heavily targeted the secondary host (alpine newts [Ichthyosaura alpestris]). Preventing an outbreak in the community required targeted removal of at least 80% of each species. This proportion increased to 90% in the presence of an environmental reservoir of B. salamandrivorans and when the proportion of individuals removed could not be adjusted for different host species (e.g., when using traps that are not species specific). We recommend the focus of disease-mitigation plans should shift from focal species to the community level and calculate explicitly the management efforts required on secondary host species to move beyond the simple intuitive understanding that multiple host species may all influence the system. Failure to do so may lead to underestimating the magnitude of the effort required and ultimately to suboptimal or futile management attempts.

Modelling the Effect of a Novel Autodissemination Trap on the Spread of Dengue in Shah Alam and Malaysia

In this paper, we will start off by introducing the classical Ross-Macdonald model for vector-borne diseases which we use to describe the transmission of dengue between humans and Aedes mosquitoes in Shah Alam, which is a city and the state capital of Selangor, Malaysia. We will focus on analysing the effect of using the Mosquito Home System (MHS), which is an example of an autodissemination trap, in reducing the number of dengue cases by changing the Ross-Macdonald model. By using the national dengue data from Malaysia, we are able to estimate λ, which represents the initial growth rate of the dengue epidemic, and this allows us to estimate the number of mosquitoes in Malaysia. A mathematical expression is also constructed which allows us to estimate the potential number of breeding sites of Aedes mosquitoes. By using the data available from the MHS trial carried out in Section 15 of Shah Alam, we included the potential effect of the MHS into the dengue model and thus modelled the impact MHS has on the spread of dengue within the trial area. We then extended our results to analyse the effect of the MHSs on reducing the number of dengue cases in the whole of Malaysia. A new model was constructed with a basic reproduction number, R _0,Mala ^MHS, which allows us to identify the required MHSs coverage needed to achieve extinction in Malaysia. Numerical simulations and tables of results were also produced to illustrate our results.

A model for epidemic dynamics in a community with visitor subpopulation

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With a model consisting of SIR and SIS models, we affirm claims in previous works.

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We derive different basic reproduction numbers looking at varying perspectives.

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We discuss the biological meanings of these basic reproduction numbers.

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All the basic reproduction numbers coincide with respect to the critical condition.

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Relevant public health policies are proposed based on our findings.

With a five dimensional system of ordinary differential equations based on the SIR and SIS models, we consider the dynamics of epidemics in a community which consists of residents and short-stay visitors. Taking different viewpoints to consider public health policies to control the disease, we derive different basic reproduction numbers and clarify their common/different mathematical natures so as to understand their meanings in the dynamics of the epidemic. From our analyses, the short-stay visitor subpopulation could become significant in determining the fate of diseases in the community. Furthermore, our arguments demonstrate that it is necessary to choose one variant of basic reproduction number in order to formulate appropriate public health policies.

Optimal control of a vectored plant disease model for a crop with continuous replanting

Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato, and cassava. Plant-vector-virus models with continuous replanting are investigated in terms of the effects of selection of cuttings, roguing, and insecticide use on disease prevalence in plants. Previous models are extended to include two replanting strategies: frequencyreplanting and abundance-replanting. In frequency-replanting, replanting of infected cuttings depends on the selection frequency parameter ε, whereas in abundance-replanting, replanting depends on plant abundance via a selection rate parameter also denoted as ε. The two models are analysed and new thresholds for disease elimination are defined for each model. Parameter values for cassava, whiteflies, and African cassava mosaic virus serve as a case study. A numerical sensitivity analysis illustrates how the equilibrium densities of healthy and infected plants vary with parameter values. Optimal control theory is used to investigate the effects of roguing and insecticide use with a goal of maximizing the healthy plants that are harvested. Differences in the control strategies in the two models are seen for large values of ε. Also, the combined strategy of roguing and insecticide use performs better than a single control.

A general theory for target reproduction numbers with applications to ecology and epidemiology

A general framework for threshold parameters in population dynamics is developed using the concept of target reproduction numbers. This framework identifies reproduction numbers and other threshold parameters in the literature in terms of their roles in population control. The framework is applied to the analysis of single and multiple control strategies in ecology and epidemiology, and this provides new biological insights.

Comparison of stochastic and deterministic frameworks in dengue modelling

We perform estimations of compartment models for dengue transmission in rural Cambodia with increasing complexity regarding both model structure and the account for stochasticity. On the one hand, we successively account for three embedded sources of stochasticity: observation noise, demographic variability and environmental hazard. On the other hand, complexity in the model structure is increased by introducing vector-borne transmission, explicit asymptomatic infections and interacting virus serotypes. Using two sources of case data from dengue epidemics in Kampong Cham (Cambodia), models are estimated in the bayesian framework, with Markov Chain Monte Carlo and Particle Markov Chain Monte Carlo. We highlight the advantages and drawbacks of the different formulations in a practical setting. Although in this case the deterministic models provide a good approximation of the mean trajectory for a low computational cost, the stochastic frameworks better reflect and account for parameter and simulation uncertainty.

Evolution of self-limited cell division of symbionts

In mutualism between unicellular hosts and their endosymbionts, symbiont's cell division is often synchronized with its host's, ensuring the permanent relationship between endosymbionts and their hosts. The evolution of synchronized cell division thus has been considered to be an essential step in the evolutionary transition from symbionts to organelles. However, if symbionts would accelerate their cell division without regard for the synchronization with the host, they would proliferate more efficiently. Thus, it is paradoxical that symbionts evolve to limit their own division for synchronized cell division. Here, we theoretically explore the condition for the evolution of self-limited cell division of symbionts, by assuming that symbionts control their division rate and that hosts control symbionts' death rate by intracellular digestion and nutrient supply. Our analysis shows that symbionts can evolve to limit their own cell division. Such evolution occurs if not only symbiont's but also host's benefit through symbiosis is large. Moreover, the coevolution of hosts and symbionts leads to either permanent symbiosis where symbionts proliferate to keep pace with their host, or the arms race between symbionts that behave as lytic parasites and hosts that resist them by rapid digestion.

A model for leptospire dynamics and control in the Norway rat (Rattus norvegicus) the reservoir host in urban slum environments

Leptospirosis is a zoonosis that humans can contract via contact with animal reservoirs directly or with water contaminated with their urine. The primary reservoir of pathogenic leptospires within urban slum environments is the Norway rat (Rattus norvegicus). Motivated by the annual outbreaks of human leptospirosis in slum urban settings, the within population infection dynamics of the Norway rat were investigated in Pau da Lima, an community in Salvador, Brazil. A mechanistic model of the dynamics of leptospire infection was informed by extensive field and laboratory data was developed and explored analytically. To identify the intraspecific transmission route of most importance, a global sensitivity analysis of the basic reproduction number to its components was performed. In addition, different methods of rodent control were investigated by calculating target reproduction numbers. Our results suggest environmental transmission plays an important role in the maintenance of infection in the rodent population. To control numbers of wild Norway rats, combinations of controls are recommended but environmental control should also be investigated to reduce prevalence of infection in rats.

Monitoring the age-specificity of measles transmissions during 2009-2016 in Southern China

Background

Despite several immunization efforts, China saw a resurgence of measles in 2012. Monitoring of transmissions of individuals from different age groups could offer information that would be valuable for planning adequate disease control strategies. We compared the age-specific effective reproductive numbers (R) of measles during 2009–2016 in Guangdong, China.

Methods

We estimated the age-specific R values for 7 age groups: 0–8 months, 9–18 months, 19 months to 6 years, 7–15 years, 16–25 years, 26–45 years, and ≥46 years adapting the contact matrix of China. The daily numbers of laboratory and clinically confirmed cases reported to the Center for Disease Control and Prevention of Guangdong were used.

Results

The peak R values of the entire population were above unity from 2012 to 2016, indicating the persistence of measles in the population. In general, children aged 0–6 years and adults aged 26–45 years had larger values of R when comparing with other age groups after 2012. While the peaks of R values for children aged 0–6 years dropped steadily after 2013, the peaks of R values for adults aged 26–45 years kept at a high range every year.

Conclusions

Although the provincial supplementary immunization activities (SIAs) conducted in 2009 and 2010 were able to reduce the transmissions from 2009 to 2011, larger values of R for children aged 0–6 years were observed after 2012, indicating that the benefits of the SIAs were short-lived. In addition, the transmissions from adults aged between 26 and 45 years increased over time. Disease control strategies should target children and adult groups that carry high potential for measles transmission.

Control Strategies in Multigroup Models: The Case of the Star Network Topology

In this paper, we propose control strategies for multigroup epidemic models. We use compartmental [Formula: see text] models to study the dynamics of n host groups sharing the same source of infection in addition to the transmission among members of the same group. In particular, we consider a model for infectious diseases with free-living pathogens in the environment and a metapopulation model with a central patch. We give the detailed derivation of the target reproduction number under three public health interventions and provide the corresponding biological insights. Moreover, using the next-generation approach, we calculate the basic reproduction numbers associated with subsystems of our models and determine algebraic connections to the target reproduction number of the complete model. The analysis presented here illustrates that understanding the topological structure of the infection process and partitioning it into simple cycles is useful to design and evaluate the control strategies.

The effect of delay in viral production in within-host models during early infection

Delay in viral production may have a significant impact on the early stages of infection. During the eclipse phase, the time from viral entry until active production of viral particles, no viruses are produced. This delay affects the probability that a viral infection becomes established and timing of the peak viral load. Deterministic and stochastic models are formulated with either multiple latent stages or a fixed delay for the eclipse phase. The deterministic model with multiple latent stages approaches in the limit the model with a fixed delay as the number of stages approaches infinity. The deterministic model framework is used to formulate continuous-time Markov chain and stochastic differential equation models. The probability of a minor infection with rapid viral clearance as opposed to a major full-blown infection with a high viral load is estimated from a branching process approximation of the Markov chain model and the results are confirmed through numerical simulations. In addition, parameter values for influenza A are used to numerically estimate the time to peak viral infection and peak viral load for the deterministic and stochastic models. Although the average length of the eclipse phase is the same in each of the models, as the number of latent stages increases, the numerical results show that the time to viral peak and the peak viral load increase.

The effect of sexual transmission on Zika virus dynamics

Zika virus is a human disease that may lead to neurological disorders in affected individuals, and may be transmitted vectorially (by mosquitoes) or sexually. A mathematical model of Zika virus transmission is formulated, taking into account mosquitoes, sexually active males and females, inactive individuals, and considering both vector transmission and sexual transmission from infectious males to susceptible females. Basic reproduction numbers are computed, and disease control strategies are evaluated. The effect of the incidence function used to model sexual transmission from infectious males to susceptible females is investigated. It is proved that for such functions that are sublinear, if the basic reproduction [Formula: see text], then the disease dies out and [Formula: see text] is a sharp threshold. Moreover, under certain conditions on model parameters and assuming mass action incidence for sexual transmission, it is proved that if [Formula: see text], there exists a unique endemic equilibrium that is globally asymptotically stable. However, under nonlinear incidence, it is shown that for certain functions backward bifurcation and Hopf bifurcation may occur, giving rise to subthreshold equilibria and periodic solutions, respectively. Numerical simulations for various parameter values are displayed to illustrate these behaviours.

Variation in Wolbachia effects on Aedes mosquitoes as a determinant of invasiveness and vectorial capacity

Wolbachia has been introduced into Aedes aegypti mosquitoes to control the spread of arboviruses, such as dengue, chikungunya and Zika. Studies showed that certain Wolbachia strains (such as wMel) reduce replication of dengue viruses in the laboratory, prompting the release of mosquitoes carrying the bacterium into the field, where vectorial capacity can be realistically assessed in relation to native non-carriers. Here we apply a new analysis to two published datasets, and show that wMel increases the mean and the variance in Ae. aegypti susceptibility to dengue infection when introgressed into Brazil and Vietnam genetic backgrounds. In the absence of other processes, higher mean susceptibility should lead to enhanced viral transmission. The increase in variance, however, widens the basis for selection imposed by unexplored natural forces, retaining the potential for reducing transmission overall.

Wolbachia infection in mosquitoes reduces dengue virus spread under specific lab conditions, prompting its use in disease control. Here, King et al. show that Wolbachia increases mean and variance in mosquito susceptibility and explain how this affects Wolbachia invasion and dengue transmission.

Exploring contacts facilitating transmission of influenza A(H5N1) virus between poultry farms in West Java, Indonesia: A major role for backyard farms?

Highly pathogenic avian influenza virus (HPAIV) H5N1 has been reported in Asia, including Indonesia since 2003. Although several risk factors related to the HPAIV outbreaks in poultry in Indonesia have been identified, little is known of the contact structure of farms of different poultry production types (backyard chickens, broilers, layers, and ducks). This study aims to quantify the contact rates associated with the movement of people, and movements of live birds and products and equipment that affect the risk of HPAIV H5N1 transmission between poultry farms in Indonesia. On 124 poultry farms in 6 districts in West Java, logbooks were distributed to record the movements of farmers/staff and visitors and their poultry contacts. Most movements in backyard chicken, commercial native chicken, broiler and duck farms were visits to and from other poultry farms, whilst in layer farms visits to and from poultry companies, visits to egg collection houses and visit from other poultry farms were most frequent. Over 75% of persons visiting backyard chicken and duck farms had previously visited other poultry farms on the same day. Visitors of backyard chicken farms had the highest average contact rate, either direct contact with poultry on other farms before the visits (1.35 contact/day) or contact during their visits in the farms (10.03 contact/day). These results suggest that backyard chicken farms are most at risk for transmission of HPAIV compared to farms of the other poultry production types. Since visits of farm-to-farm were high, backyard farms could also a potential source for HPAIV transmission to commercial poultry farms.

Dynamics and control of infections on social networks of population types

Random mixing in host populations has been a convenient simplifying assumption in the study of epidemics, but neglects important differences in contact rates within and between population groups. For HIV/AIDS, the assumption of random mixing is inappropriate for epidemics that are concentrated in groups of people at high risk, including female sex workers (FSW) and their male clients (MCF), injecting drug users (IDU) and men who have sex with men (MSM). To find out who transmits infection to whom and how that affects the spread and containment of infection remains a major empirical challenge in the epidemiology of HIV/AIDS. Here we develop a technique, based on the routine sampling of infection in linked population groups (a social network of population types), which shows how an HIV/AIDS epidemic in Can Tho Province of Vietnam began in FSW, was propagated mainly by IDU, and ultimately generated most cases among the female partners of MCF (FPM). Calculation of the case reproduction numbers within and between groups, and for the whole network, provides insights into control that cannot be deduced simply from observations on the prevalence of infection. Specifically, the per capita rate of HIV transmission was highest from FSW to MCF, and most HIV infections occurred in FPM, but the number of infections in the whole network is best reduced by interrupting transmission to and from IDU. This analysis can be used to guide HIV/AIDS interventions using needle and syringe exchange, condom distribution and antiretroviral therapy. The method requires only routine data and could be applied to infections in other populations.

Reproduction numbers of infectious disease models

This primer article focuses on the basic reproduction number, ℛ 0 , for infectious diseases, and other reproduction numbers related to ℛ 0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of ℛ 0 determining whether or not the disease dies out. The next generation matrix method of calculating ℛ 0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate ℛ 0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.

Comparison between chikungunya and dengue viruses transmission based on a mathematical model

Chikungunya and dengue viruses are transmitted by mosquitoes of genus Aedes. Based on a mathematical model dealing with arboviruses transmission that encompasses human and mosquito populations, the risks of dengue and chikungunya infections are compared. By the fact that chikungunya virus attains high viral load earlier than dengue virus in both humans and mosquitoes, the potential risk of chikungunya could be higher than the dengue infection. The risk of arboviruses infections is assessed by the reproduction number [Formula: see text], which is obtained by the next generation matrix method and Routh–Hurwitz criteria.

The basic reproduction number of vector-borne plant virus epidemics

The basic reproduction number R₀ is a key parameter in plant disease epidemiology, which largely determines whether or not an epidemic will occur in a plant population. The next generation matrix approach to deriving and calculating the basic reproduction number of a plant virus epidemic is described. The approach is illustrated through a series of examples of increasing complexity, ranging from the simplest case of one vector transmitting one virus to a single host, to the case of multiple vectors, to combined horizontal (vector) and vertical (seed) transmission, and where vector control using insecticides is practised. The importance of parameters representing host and vector population dynamics and their interaction in the absence of disease is stressed, and the constraints these place on the calculation of the basic reproduction number. Finally, mention is made of further elaborations to the approach that could prove useful in plant virus epidemiology.