The type-reproduction number T in models for infectious disease control

A ubiquitous quantity in epidemic modelling is the basic reproduction number R(0). This became so popular in the 1990s that 'All you need know is R(0)!' became a familiar catch-phrase. The value of R(0) defines, among other things, the control effort needed to eliminate the infection from a homogeneous host population, but can be misleading when applied to a heterogeneous population for the same purpose. We have defined the type-reproduction number T for an infectious disease, and shown that this not only has the required threshold behaviour, but also correctly determines the critical control effort for heterogeneous populations. The two quantities coincide for homogeneous populations. In this paper we further develop the new threshold quantity as an indicator of control effort required in a system where multiple types of individuals are recognised when control targets a specific type.     

The basic depression ratio of the host: the evolution of host resistance to microparasites

The basic reproduction ratio R0 occupies a central position in the theory of host pathogen interactions. However, this quantity stresses the role of the pathogen. This paper proposes an additional, more host-centred char acterization using the basic depression ratio D0. This quantity is the number of host individuals per infected by which the infected host population is depressed below its uninfected level. This paper shows that a baseline criterion for the evolution of host resistance to microparasites is that resistance evolves to minimize D0. This parallels the result for pathogen virulence where R0 is maximized. The tension between these two criteria is noted. The framework established allows a discussion of trade-offs between aspects of the pathogen-free host biology and the host pathogen interaction. For certain linear and convex trade-offs it is shown that the strain with the lowest transmission parameter beta wins (despite the fact that it has the lowest intrinsic birth rate a). For corresponding concave trade-offs, either the strain with minimum beta and a or the strain with maximum beta and a wins. Finally the connection with the techniques of adaptive dynamics is made. Evolutionary singular points are shown to occur at extrema of D0. The evolutionary attainment of the results is discussed.     

Evolutionary implications for interactions between multiple strains of host and parasite

The interaction between multiple parasite strains within different host types may influence the evolutionary trajectories of parasites. In this article, we formulate a deterministic model with two strains of parasites and two host types in order to investigate how heterogeneities in parasite virulence and host life-history may affect the persistence and spread of diseases in natural systems. We compute the reproductive number of strain i (R(i)) independently, as well as the (conditional) "invasion" reproductive number for strains i (R(i)(j), j not equal i) when strain j is at a positive equilibrium. We show that the disease-free equilibrium is locally asymptotically stable if R(i)<1 for both strains and is unstable if R(i)>1 for one stain. We establish the criterion R(i)(j)>1 for strain i to invade strain j. Subthreshold coexistence driven by coinfection is possible even when R(i) of one strain is below 1. We identify conditions that determine the evolution of parasite specialism or generalism based on the life-history strategies employed by hosts, and investigate how host strains may influence parasite persistence.     

A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host

The main purpose of this article is to formulate a deterministic mathematical model for the transmission of malaria that considers two host types in the human population. The first type is called "non-immune" comprising all humans who have never acquired immunity against malaria and the second type is called "semi-immune". Non-immune are divided into susceptible, exposed and infectious and semi-immune are divided into susceptible, exposed, infectious and immune. We obtain an explicit formula for the reproductive number, R(0) which is a function of the weight of the transmission semi-immune-mosquito-semi-immune, R(0a), and the weight of the transmission non-immune-mosquito-non-immune, R(0e). Then, we study the existence of endemic equilibria by using bifurcation analysis. We give a simple criterion when R(0) crosses one for forward and backward bifurcation. We explore the possibility of a control for malaria through a specific sub-group such as non-immune or semi-immune or mosquitoes.     

On the definition and the computation of the type-reproduction number T for structured populations in heterogeneous environments

In the context of mathematical epidemiology, the type-reproduction number (TRN) for a specific host type is interpreted as the average number of secondary cases of that type produced by the primary cases of the same host type during the entire course of infection. Here, it must be noted that T takes into account not only the secondary cases directly transmitted from the specific host but also the cases indirectly transmitted by way of other types, who were infected from the primary cases of the specific host with no intermediate cases of the target host. Roberts and Heesterbeek (Proc R Soc Lond B 270:1359–1364, 2003) have shown that T is a useful measure when a particular single host type is targeted in the disease control effort in a community with various types of host, based on the fact that the sign relation sign(R ₀ ? 1) = sign(T ? 1) holds between the basic reproduction number R ₀ and T. In fact, T can be seen as an extension of R ₀ in a sense that the threshold condition of the total population growth can be formulated by the reproduction process of the target type only. However, the original formulation is limited to populations with discrete state space in constant environments. In this paper, based on a new perspective of R ₀ in heterogeneous environments (Inaba in J Math Biol 2011), we give a general definition of the TRN for continuously structured populations in heterogeneous environments and show some examples of its computation and applications.     

Threshold conditions for infection persistence in complex host-vectors interactions

As classically defined by Macdonald in the early 1950s, for the case of diseases with one vector and one host, the Basic Reproduction Number, R0, is defined as the number of secondary infections caused by a single infective of the same type (vector or host) during its infectiousness period in an entirely susceptible population. In the case of a disease which has one vector and one host, it is easy to show that R0 coincides with the threshold for the establishment of an endemic state: if R0 > 1 (< 1), the disease can invade (cannot invade) the host population. In this paper we examine various epidemic situations in which there are more than one vector and/or host. We show that in those more complex systems it is not possible to deduce a single R0 but rather a threshold for infection persistence which is a composite of several quantities closely related to the classical expression of R0. Another definition of R0 given by Diekmann, Heesterbeek and Metz, and denoted in this paper R0NGO is discussed and applied as an alternative to calculate the thresholds for infection establishment.     

Early real-time estimation of the basic reproduction number of emerging or reemerging infectious diseases in a community with heterogeneous contact pattern: Using data from Hong Kong 2009 H1N1 Pandemic Influenza as an illustrative example

Emerging and re-emerging infections such as SARS (2003) and pandemic H1N1 (2009) have caused concern for public health researchers and policy makers due to the increased burden of these diseases on health care systems. This concern has prompted the use of mathematical models to evaluate strategies to control disease spread, making these models invaluable tools to identify optimal intervention strategies. A particularly important quantity in infectious disease epidemiology is the basic reproduction number, R_0. Estimation of this quantity is crucial for effective control responses in the early phase of an epidemic. In our previous study, an approach for estimating the basic reproduction number in real time was developed. This approach uses case notification data and the structure of potential transmission contacts to accurately estimate R₀ from the limited amount of information available at the early stage of an outbreak. Based on this approach, we extend the existing methodology; the most recent method features intra- and inter-age groups contact heterogeneity. Given the number of newly reported cases at the early stage of the outbreak, with parsimony assumptions on removal distribution and infectivity profile of the diseases, experiments to estimate real time R₀ under different levels of intra- and inter-group contact heterogeneity using two age groups are presented. We show that the new method converges more quickly to the actual value of R₀ than the previous one, in particular when there is high-level intra-group and inter-group contact heterogeneity. With the age specific contact patterns, number of newly reported cases, removal distribution, and information about the natural history of the 2009 pandemic influenza in Hong Kong, we also use the extended model to estimate R₀ and age-specific R₀.     

Extending the type reproduction number to infectious disease control targeting contacts between types

A new quantity called the target reproduction number is defined to measure control strategies for infectious diseases with multiple host types such as waterborne, vector-borne and zoonotic diseases. The target reproduction number includes as a special case and extends the type reproduction number to allow disease control targeting contacts between types. Relationships among the basic, type and target reproduction numbers are established. Examples of infectious disease models from the literature are given to illustrate the use of the target reproduction number.     

On the capacity of macroparasites to control insect populations

A graphical model of the population dynamics of macroparasites and their hosts is developed. Three principal means by which the parasites can be regulated are considered: reduction in host density as a result of parasite-induced host mortality, reduction in host density as a result of parasite-induced host sterility, and competition among parasites within multiply-infected hosts. The means by which parasites are regulated has a major effect on the degree to which they can depress host population densities. In particular, a parasite that sterilizes its host is expected to reduce host density more than one that causes an equivalent decline in host fitness through increased mortality. A special case of the model is developed for herbivorous insects that, in the absence of parasites, are limited by larval food resources. Parasites that are regulated via parasite-induced host sterility will control the insect populations below the level set by larval resources if the threshold host density for the parasites (N(T)) is less than the ratio of carrying capacity to net reproductive rate of the insects (K/R). Data are presented showing that all three means of parasite regulation, but especially parasite-induced host sterility, can operate in Howardula aoronymphium, a nematode parasite of mycophagous Drosophila flies. Data from a field cage experiment show that, if these nematodes are regulated primarily via reductions in host density due to this sterility, the parameters N(T), K, and R are such that Howardula is likely to play an important role in controlling Drosophila populations. However, this conclusion must be tempered by the fact that these nematodes also cause increased host mortality and experience within-host competition, making the conditions for parasite control of the flies more stringent.     

Epidemiology and disease-control under gene-for-gene plant-pathogen interaction

An introduction of disease-resistant variety of a crop plant often leads to the development of a virulent race in pathogen species that restores the pathogenicity to the resistant crop. This often makes disease control of crop plants extremely difficult. In this paper, we theoretically explore the optimal 'multiline' control, which makes use of several different resistant varieties, that minimizes the expected degree of crop damages caused by epidemic outbreaks of the pathogen. We examine both single-locus and two-locus gene-for-gene (GFG) systems for the compatibility relationship between host genotypes and pathogen genotypes, in which host haplotype has either susceptible or resistant allele in each resistance locus, and the pathogen haplotype has either avirulent or virulent allele in the corresponding virulence locus. We then study the optimal planting strategy of host resistant genotypes based on standard epidemiological dynamics with pathogen spore stages. The most striking result of our single-locus GFG model is that there exists an intermediate optimum mixing ratio for the susceptible and resistant crops that maximizes the final yield, in spite of the fact that the susceptible crop has no use to fight against either avirulent or virulent race of the pathogen. The intermediate mixture is optimum except when the initial pathogen spore population in the season consists exclusively of the virulent race. The optimal proportion of resistant crops is approximately 1/R(0), where R(0) is the basic reproductive ratio of pathogen--the rest (the vast majority if R(0) is large) of crops should be the susceptible genotype. By mixing susceptible and resistant crops, we can force the pathogen races to compete with each other for their available hosts. This competition between avirulent and virulent races prevents the fatal outbreak of the virulent race (the super-race) that can infect all the host genotypes. In the two-locus GFG control, there again exists the optimal mixing ratio for the fraction of universally susceptible genotype and the total fraction of various resistant genotypes, with the ratio close to 1/R(0).     

Rickettsia parkeri invasion of diverse host cells involves an Arp2/3 complex, WAVE complex and Rho-family GTPase-dependent pathway

Rickettsiae are obligate intracellular pathogens that are transmitted to humans by arthropod vectors and cause diseases such as spotted fever and typhus. Although rickettsiae require the host cell actin cytoskeleton for invasion, the cytoskeletal proteins that mediate this process have not been completely described. To identify the host factors important during cell invasion by Rickettsia parkeri, a member of the spotted fever group (SFG), we performed an RNAi screen targeting 105 proteins in Drosophila melanogaster S2R+ cells. The screen identified 21 core proteins important for invasion, including the GTPases Rac1 and Rac2, the WAVE nucleation-promoting factor complex and the Arp2/3 complex. In mammalian cells, including endothelial cells, the natural targets of R. parkeri, the Arp2/3 complex was also crucial for invasion, while requirements for WAVE2 as well as Rho GTPases depended on the particular cell type. We propose that R. parkeri invades S2R+ arthropod cells through a primary pathway leading to actin nucleation, whereas invasion of mammalian endothelial cells occurs via redundant pathways that converge on the host Arp2/3 complex. Our results reveal a key role for the WAVE and Arp2/3 complexes, as well as a higher degree of variation than previously appreciated in actin nucleation pathways activated during Rickettsia invasion.     

Invasion and persistence of infectious agents in fragmented host populations

One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the migration of hosts. Such fragmented population structures are studied extensively with metapopulation models. Being able to define and calculate an indicator for the success of invasion and persistence of an infectious agent is essential for obtaining general qualitative insights into infection dynamics, for the comparison of prevention and control scenarios, and for quantitative insights into specific systems. For homogeneous populations, the basic reproduction ratio R(0) plays this role. For metapopulations, defining such an 'invasion indicator' is not straightforward. Some indicators have been defined for specific situations, e.g., the household reproduction number R*. However, these existing indicators often fail to account for host demography and especially host migration. Here we show how to calculate a more broadly applicable indicator R(m) for the invasion and persistence of infectious agents in a host metapopulation of equally connected patches, for a wide range of possible epidemiological models. A strong feature of our method is that it explicitly accounts for host demography and host migration. Using a simple compartmental system as an example, we illustrate how R(m) can be calculated and expressed in terms of the key determinants of epidemiological dynamics.     

Closely related parasitic plants have similar host requirements and related effects on hosts

The performance of root hemiparasites depends strongly on host species identity, but it remains unknown whether there exist general patterns in the quality of species as hosts for hemiparasites and in their sensitivity to parasitism. In a comparative approach, the model root hemiparasites Rhinanthus minor and R. alectorolophus were grown with 25 host species (grasses, forbs, and legumes) at two nutrient levels. Hosts grown without parasites served as a control. Host species identity strongly influenced parasite biomass and other traits, and both parasites grew better with legumes and grasses than with forbs. The biomass of R. alectorolophus was much higher than that of R. minor with all host plants and R. alectorolophus responded much more strongly to higher nutrient availability than R. minor. The performance of the two species of Rhinanthus with individual hosts was strongly correlated, and it was also correlated with that of R. alectorolophus and the related Odontites vulgaris in previous experiments with many of the same hosts, but only weakly with that of the less closely related Melampyrum arvense. The negative effect of R. minor on host biomass was less strong than that of R. alectorolophus, but stronger relative to its own biomass, suggesting that it is more parasitic. The impact of the two parasites on individual hosts did not depend on nutrient level and was correlated. Several legumes and grasses were tolerant of parasitism. While R. minor slightly reduced mean overall productivity, R. alectorolophus increased it with several species, indicating that the loss of host biomass was more than compensated by that of the parasite. The results show that closely related parasites have similar host requirements and correlated negative effects on individual hosts, but that there are also specific interactions between pairs of parasitic plants and their hosts.     

Duelling timescales of host movement and disease recovery determine invasion of disease in structured populations

The epidemic potential of a disease is traditionally assessed using the basic reproductive number, R ₀. However, in populations with social or spatial structure a chronic disease is more likely to invade than an acute disease with the same R ₀, because it persists longer within each group and allows for more host movement between groups. Acute diseases 'perceive' a more structured host population, and it is more important to consider host population structure in analyses of these diseases. The probability of a pandemic does not arise independently from characteristics of either the host or disease, but rather from the interaction of host movement and disease recovery timescales. The R _* statistic, a group-level equivalent of R ₀, is a better indicator of disease invasion in structured populations than the individual-level R ₀.     

Emerging infectious pathogens of wildlife

The first part of this paper surveys emerging pathogens of wildlife recorded on the ProMED Web site for a 2-year period between 1998 and 2000. The majority of pathogens recorded as causing disease outbreaks in wildlife were viral in origin. Anthropogenic activities caused the outbreaks in a significant majority of cases. The second part of the paper develops some matrix models for quantifying the basic reproductive number, R(0), for a variety of potential types of emergent pathogen that cause outbreaks in wildlife. These analyses emphasize the sensitivity of R(0) to heterogeneities created by either the spatial structure of the host population, or the ability of the pathogens to utilize multiple host species. At each stage we illustrate how the approach provides insight into the initial dynamics of emergent pathogens such as canine parvovirus, Lyme disease, and West Nile virus in the United States.     

High diversity and host specificity observed among symbiotic dinoflagellates in reef coral communities from Hawaii

The Hawaiian Islands represent one of the most geographically remote locations in the Indo-Pacific, and are a refuge for rare, endemic life. The diversity of symbiotic dinoflagellates (Symbiodinium sp.) inhabiting zooxanthellate corals and other symbiotic cnidarians from the High Islands region was surveyed. From the 18 host genera examined, there were 20 genetically distinct symbiont types (17 in clade C, 1 in clade A, 1 in clade B, and 1 in clade D) distinguished by internal transcribed spacer region 2 sequences. Most types were found to associate with a particular host genus or species and nearly half of them have not been identified in surveys of Western and Eastern Pacific hosts. A clear dominant generalist symbiont is lacking among Hawaiian cnidarians. This is in marked contrast with the symbiont community structures of the western Pacific and Caribbean, which are dominated by a few prevalent generalist symbionts inhabiting numerous host taxa. Geographic isolation, low host diversity, and a high proportion of coral species that directly transmit their symbionts from generation to generation are implicated in the formation of a coral reef community exhibiting high symbiont diversity and specificity.Communicated by H.R. Lasker     

The Rhizobium meliloti region located downstream of the nod box n6 is involved in the specific nodulation of Medicago lupulina

Abstract Six nod box regulatory sequences are present in the Rhizobium meliloti genome. We have analysed the DNA region located downstream of nod box n6, and identified three open reading frames, designated nolQa, nolQb and nolS . LacZ fusions in these ORFs are not induced by classical nod gene inducers, which indicates that their expression either is not under the control of the nod box, or involves another regulatory mechanism acting in conjunction with the NodD/nod box regulatory circuit. Mutations in this n6 locus result in a delay in nodule formation on a particular host, Medicago lupulina . As this region is not strictly conserved among different R. meliloti strains, nolQa, nolQb and nolS may constitute auxiliary nodulation genes, for which the selection pressure is limited to particular host plants.     

Characterization and growth of polymorphic Rickettsia felis in a tick cell line

Morphological differentiation in some arthropod-borne bacteria is correlated with increased bacterial virulence, transmission potential, and/or as a response to environmental stress. In the current study, we utilized an in vitro model to examine Rickettsia felis morphology and growth under various culture conditions and bacterial densities to identify potential factors that contribute to polymorphism in rickettsiae. We utilized microscopy (electron microscopy and immunofluorescence), genomic (PCR amplification and DNA sequencing of rickettsial genes), and proteomic (Western blotting and liquid chromatography-tandem mass spectrometry) techniques to identify and characterize morphologically distinct, long-form R. felis. Without exchange of host cell growth medium, polymorphic R. felis was detected at 12 days postinoculation when rickettsiae were seeded at a multiplicity of infection (MOI) of 5 and 50. Compared to short-form R. felis organisms, no change in membrane ultrastructure in long-form polymorphic rickettsiae was observed, and rickettsiae were up to six times the length of typical short-form rickettsiae. In vitro assays demonstrated that short-form R. felis entered into and replicated in host cells faster than long-form R. felis. However, when both short- and long-form R. felis organisms were maintained in cell-free medium for 12 days, the infectivity of short-form R. felis was decreased compared to long-form R. felis organisms, which were capable of entering host cells, suggesting that long-form R. felis is more stable outside the host cell. The relationship between rickettsial polymorphism and rickettsial survivorship should be examined further as the yet undetermined route of horizontal transmission of R. felis may utilize metabolically and morphologically distinct forms for successful transmission.     

The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model

In this paper, we develop the theory of a state-reproduction number for a multistate class age structured epidemic system and apply it to examine the asymptomatic transmission model. We formulate a renewal integral equation system to describe the invasion of infectious diseases into a multistate class age structured host population. We define the state-reproduction number for a class age structured system, which is the net reproduction number of a specific host type and which plays an analogous role to the type-reproduction number [M.G. Roberts, J.A.P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond. B 270 (2003) 1359; J.A.P. Heesterbeek, M.G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci. 206 (2007) 3] in discussing the critical level of public health intervention. The renewal equation formulation permits computations not only of the state-reproduction number, but also of the generation time and the intrinsic growth rate of infectious diseases.Subsequently, the basic theory is applied to capture the dynamics of a directly transmitted disease within two types of infected populations, i.e., asymptomatic and symptomatic individuals, in which the symptomatic class is observable and hence a target host of the majority of interventions. The state-reproduction number of the symptomatic host is derived and expressed as a measurable quantity, leading to discussion on the critical level of case isolation. The serial interval and other epidemiologic indices are computed, clarifying the parameters on which these indices depend. As a practical example, we illustrate the eradication threshold for case isolation of smallpox. The generation time and serial interval are comparatively examined for pandemic influenza.