A new approach for designing disease intervention strategies in metapopulation models

We describe a new approach for investigating the control strategies of compartmental disease transmission models. The method rests on the construction of various alternative next-generation matrices, and makes use of the type reproduction number and the target reproduction number. A general metapopulation SIRS (susceptible–infected–recovered–susceptible) model is given to illustrate the application of the method. Such model is useful to study a wide variety of diseases where the population is distributed over geographically separated regions. Considering various control measures such as vaccination, social distancing, and travel restrictions, the procedure allows us to precisely describe in terms of the model parameters, how control methods should be implemented in the SIRS model to ensure disease elimination. In particular, we characterize cases where changing only the travel rates between the regions is sufficient to prevent an outbreak.     

Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings

The dengue virus is a vector-borne disease transmitted by mosquito Aedes aegypti and the incidence is strongly influenced by temperature and humidity which vary seasonally. To assess the effects of temperature on dengue transmission, mathematical models are developed based on the population dynamics theory. However, depending on the hypotheses of the modelling, different outcomes regarding to the risk of epidemics are obtained. We address this question comparing two simple models supplied with model's parameters estimated from temperature-controlled experiments, especially the entomological parameters regarded to the mosquito's life cycle in different temperatures. Once obtained the mortality and transition rates of different stages comprising the life cycle of mosquito and the oviposition rate, we compare the capacity of vector reproduction (the basic offspring number) and the risk of infection (basic reproduction number) provided by two models. The extended model, which is more realistic, showed that both mosquito population and dengue risk are situated at higher values than the simplified model, even that the basic offspring number is lower.     

The backward bifurcation in compartmental models for West Nile virus

In all of the West Nile virus (WNV) compartmental models in the literature, the basic reproduction number serves as a crucial control threshold for the eradication of the virus. However, our study suggests that backward bifurcation is a common property shared by the available compartmental models with a logistic type of growth for the population of host birds. There exists a subthreshold condition for the outbreak of the virus due to the existence of backward bifurcation. In this paper, we first review and give a comparison study of the four available compartmental models for the virus, and focus on the analysis of the model proposed by Cruz-Pacheco et al. to explore the backward bifurcation in the model. Our comparison study suggests that the mosquito population dynamics itself cannot explain the occurrence of the backward bifurcation, it is the higher mortality rate of the avian host due to the infection that determines the existence of backward bifurcation.     

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 \({\textit{SIRS}}\) 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.     

A model for influenza with vaccination and antiviral treatment

Compartmental models for influenza that include control by vaccination and antiviral treatment are formulated. Analytic expressions for the basic reproduction number, control reproduction number and the final size of the epidemic are derived for this general class of disease transmission models. Sensitivity and uncertainty analyses of the dependence of the control reproduction number on the parameters of the model give a comparison of the various intervention strategies. Numerical computations of the deterministic models are compared with those of recent stochastic simulation influenza models. Predictions of the deterministic compartmental models are in general agreement with those of the stochastic simulation models.     

Linkage disequilibrium under skewed offspring distribution among individuals in a population

Correlations in coalescence times between two loci are derived under selectively neutral population models in which the offspring of an individual can number on the order of the population size. The correlations depend on the rates of recombination and random drift and are shown to be functions of the parameters controlling the size and frequency of these large reproduction events. Since a prediction of linkage disequilibrium can be written in terms of correlations in coalescence times, it follows that the prediction of linkage disequilibrium is a function not only of the rate of recombination but also of the reproduction parameters. Low linkage disequilibrium is predicted if the offspring of a single individual frequently replace almost the entire population. However, high linkage disequilibrium can be predicted if the offspring of a single individual replace an intermediate fraction of the population. In some cases the model reproduces the standard Wright-Fisher predictions. Contrary to common intuition, high linkage disequilibrium can be predicted despite frequent recombination, and low linkage disequilibrium under infrequent recombination. Simulations support the analytical results but show that the variance of linkage disequilibrium is very large.     

A simple persistence condition for structured populations

The fundamental question in both basic and applied population biology of whether a species will increase in numbers is often investigated by finding the population growth rate as the largest eigenvalue of a deterministic matrix model. For a population classified only by age, and not stage or size, a simpler biologically interpretable condition can be used, namely whether R ₀, the mean number of offspring per newborn, is greater than one. However, for the many populations not easily described using only age classes, stage-structured models must be used for which there is currently no quantity like R ₀. We determine analogous quantities that must be greater than one for persistence of a general structured population model that have a similar useful biological interpretation. Our approach can be used immediately to determine the magnitude of changes and interactions that would either allow population persistence or would ensure control of an undesirable species.     

Direct and indirect effects of environmental temperature on the evolution of reproductive strategies: an information-theoretic approach

For ectotherms, environmental temperature affects the optimal size and number of offspring via multiple mechanisms. First, temperature influences the performance of offspring, which directly affects the optimal size of offspring. Second, temperature influences maternal body size, which indirectly affects the optimal size and/or number of offspring when larger females acquire more energetic resources or provide better parental care. Although traditional statistical approaches might distinguish the relative importance of these effects, an information-theoretic approach enables one to estimate effects more accurately by identifying the best evolutionary model in a set of candidate models. Here, we use the Akaike Information Criterion to calculate the likelihoods of seven path models, each derived from one or more optimality models of reproduction. Variation in reproductive traits among populations of lizards (Sceloporus undulatus) was used to quantify support for the models. Our results overwhelmingly supported a model based on an indirect effect of temperature that is mediated by maternal size. Path coefficients of this model were consistent with the hypotheses that, first, larger females can acquire more energy for reproduction and, second, the survival of offspring depends on both their size and their density. Our analyses exemplify how information theory can identify evolutionary hypotheses that merit experimental testing.     

The ESS under spatial variation with applications to sex allocation

I derive a new approximation which uses the backward Kolmogorov equation to describe evolution when individuals have variable numbers of offspring. This approximation is based on an explicit fixed population size assumption and therefore differs from previous models. I show that for individuals to accept an increase in the variance of offspring number, they must be compensated by an increase in mean offspring number. Based on this model and any given set of feasible alleles, an evolutionary stable strategy (ESS) can be found. Four types of ESS are possible and can be discriminated by graphical methods. These ESS values depend on population size, but population size can be reinterpreted as deme size in a structured population. I adapt this theory to the problem of sex allocation under variable returns to male and female function and derive the ESS sex allocation strategy. I show that allocation to the more variable sexual function should be reduced, but that this effect decreases as population size increases and as variability decreases. These results are compared with results from exact matrix models and computer simulations, all of which show strong congruence.     

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

A precise definition of the basic reproduction number, , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if , then the disease free equilibrium is locally asymptotically stable; whereas if , then it is unstable. Thus, is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for near one. This criterion, together with the definition of , is illustrated by treatment, multigroup, staged progression, multistrain and vector–host models and can be applied to more complex models. The results are significant for disease control.     

Modeling scenarios for mitigating outbreaks in congregate settings

The explosive outbreaks of COVID-19 seen in congregate settings such as prisons and nursing homes, has highlighted a critical need for effective outbreak prevention and mitigation strategies for these settings. Here we consider how different types of control interventions impact the expected number of symptomatic infections due to outbreaks. Introduction of disease into the resident population from the community is modeled as a stochastic point process coupled to a branching process, while spread between residents is modeled via a deterministic compartmental model that accounts for depletion of susceptible individuals. Control is modeled as a proportional decrease in the number of susceptible residents, the reproduction number, and/or the proportion of symptomatic infections. This permits a range of assumptions about the density dependence of transmission and modes of protection by vaccination, depopulation and other types of control. We find that vaccination or depopulation can have a greater than linear effect on the expected number of cases. For example, assuming a reproduction number of 3.0 with density-dependent transmission, we find that preemptively reducing the size of the susceptible population by 20% reduced overall disease burden by 47%. In some circumstances, it may be possible to reduce the risk and burden of disease outbreaks by optimizing the way a group of residents are apportioned into distinct residential units. The optimal apportionment may be different depending on whether the goal is to reduce the probability of an outbreak occurring, or the expected number of cases from outbreak dynamics. In other circumstances there may be an opportunity to implement reactive disease control measures in which the number of susceptible individuals is rapidly reduced once an outbreak has been detected to occur. Reactive control is most effective when the reproduction number is not too high, and there is minimal delay in implementing control. We highlight the California state prison system as an example for how these findings provide a quantitative framework for understanding disease transmission in congregate settings. Our approach and accompanying interactive website (https://phoebelu.shinyapps.io/DepopulationModels/) provides a quantitative framework to evaluate the potential impact of policy decisions governing infection control in outbreak settings.     

Fitness of alternative modes of reproduction: developmental constraints and the evolutionary maintenance of sex

Parthenogenesis, including facultative parthenogenesis, is common among orthopteroid insects. We investigated the fitness associated with sexual and asexual reproduction within a population of the facultatively parthenogenetic cockroach Nauphoeta cinerea. There is significantly reduced fitness for females reproducing parthenogenetically compared to sexually. Fewer than half of all females can reproduce parthenogenetically. In addition, tenfold fewer offspring are produced by parthenogenesis due to reductions in both the number of offspring produced per clutch and the number of clutches produced. Development and brooding of sexually or parthenogenetically produced first instar nymphs does not differ, although the production of the first parthenogenetic clutch is delayed relative to the first sexually produced clutch. The fitness of parthenogens is also lower than the fitness of sexually produced offspring. Parthenogens are less viable than sexually produced offspring even in the benign conditions of the laboratory. Development to adulthood of parthenogens is slower. Fewer parthenogens survive to adulthood and the adult life span of parthenogens is reduced. Individuals produced by parthenogenetic reproduction are unlikely to reproduce parthenogenetically themselves. Finally, parthenogenetically produced females produce fewer offspring by sexual reproduction than do sexually produced females. Since parthenogenetic reproduction is apomictic in N. cinerea and parthenogens are diploid, we suggest that asexual reproduction is developmentally constrained. Once meiosis has evolved, returning to a mitotic mode of reproduction may be difficult. Nauphoeta cinerea offers a system for testing how asexuality is constrained as modes of reproduction can be compared within a facultative parthenogen.     

Reproductive Trade-Offs May Moderate the Impact of Gyrodactylus salaris in Warmer Climates

Gyrodactylus salaris is a notifiable freshwater ectoparasite of salmonids. Its primary host is Atlantic salmon (Salmo salar), upon which infections can cause death, and have led to massive declines in salmon numbers in Norway, where the parasite is widespread. Different strains of S. salar vary in their susceptibility, with Atlantic strains (such as those found in Norway) exhibiting no resistance to the parasite, and Baltic strains demonstrating an innate resistance sufficient to regulate parasite numbers on the host causing it to either die out or persist at a low level. In this study, Leslie matrix and compartmental models were used to generate data that demonstrated the population growth of G. salaris on an individual host is dependent on the total number of offspring per parasite, its longevity and the timing of its births. The data demonstrated that the key factor determining the rate of G. salaris population growth is the time at which the parasite first gives birth, with rapid birth rate giving rise to large population size. Furthermore, it was shown that though the parasite can give birth up to four times, only two births are required for the population to persist as long as the first birth occurs before a parasite is three days old. As temperature is known to influence the timing of the parasite''s first birth, greater impact may be predicted if introduced to countries with warmer climates than Norway, such as the UK and Ireland which are currently recognised to be free of G. salaris. However, the outputs from the models developed in this study suggest that temperature induced trade-offs between the total number of offspring the parasite gives birth to and the first birth timing may prevent increased population growth rates over those observed in Norway.     

Offspring size–number tradeoffs and food quality feedbacks impact population dynamics in a Daphnia–algae system

Population fluctuations can be affected by both extrinsic (e.g. weather patterns, food availability) and intrinsic (e.g. life‐history) factors. A key life‐history tradeoff is the production of offspring size versus number, ranging from many small offspring to few large offspring. Models show that this life‐history tradeoff in offspring size and number, through maturation time, can have significant impacts on population dynamics. However, few manipulative experiments have been conducted that can isolate life‐history effects from impacts of extrinsic factors in consumer–resource systems. We experimentally tested the effect of an offspring size–number tradeoff on population stability and food availability in a consumer–resource system. Using Daphnia pulex, we created a shift from many, small offspring being produced to fewer, larger offspring. Two sets of experiments were performed to examine the interaction of an extrinsic factor (light levels) and intrinsic population structure on dynamics, and we controlled for the ingestion pressure on algal prey at the time of the manipulation. We predicted that the tradeoff would impact internal consumer population characteristics, including biasing the stage structure towards adults, increasing adult size, and increasing average population‐level reproduction. This adult‐dominated stage structure was predicted to then lead to instability and a low quantity–high quality food state. Under all light levels, the manipulated populations became dominated by large adults. Contrary to predictions, the amplitudes of fluctuations in Daphnia biomass were lower in populations shifted to few–large offspring, representing higher stability in these populations. Furthermore, in high light conditions, a stable low Daphnia – high algae biomass (low food quality) state was observed in few–large offspring treatments but not in control (many–small offspring) treatments. Our results show a strong link between light levels as an extrinsic factor and the life‐history tradeoff of consumer offspring size versus number that impacts consumer–resource population dynamics through feedbacks with resource quality.     

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.     

Lifetime reproductive success in reproductively suppressed female voles

A population of the grey red-backed vole, Clethrionomys rufocanus bedfordiae, was investigated on a 1 ha control grid and a 1 ha grid on which the voles were fed within a 2.1 ha outdoor enclosure in Hokkaido, Japan by live trapping from 1984 to 1986, for testing the Reproductive Suppression Model of Wasser and Barash (1983)-females can optimize their lifetime reproductive success by suppressing reproduction when future conditions for the survival of offspring are likely to be sufficiently better than present ones as to exceed the costs of the suppression itself. Age at the first pregnancy more varied in a higher density population on the experimental grid and females could be classified into the early and the late reproductive type in two generations (A: females born from February to June 1985; B: females born from September to November 1985). Lifetime reproductive success (the number of pregnancies, the number of successful litters, and the number of offspring) was not different between the early and the late reproducing females. The late reproducing females lived for longer periods than the early reproducing females, so that the loss by delayed start of reproduction was compensated for by a longer life span. Life span was not different between offspring of the early and the late reproducing females. These facts supported the Reproductive Suppression Model.     

A graph-theoretic method for the basic reproduction number in continuous time epidemiological models

In epidemiological models of infectious diseases the basic reproduction number is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for can be obtained from the digraphs in the reduction process.     

The effects of density-dependent offspring mortality on the synchrony of reproduction

Mortality rates often depend on the size of a population. Using ideal free theory to model the optimal timing of reproduction in model populations, I considered how the specific relationship between density-dependent offspring mortality and population size affects the optimal temporal distribution of reproduction. The results suggest that the specific form of the relationship between density-dependent mortality and the number of offspring produced determines the degree to which reproduction within a population is synchronous. Specifically, reproductive synchrony decreases as density-dependent mortality becomes increasingly inversely related to the number of offspring produced and is highest when density-dependent mortality is directly density-dependent. These findings support the suggestion that predation pressure selects for greater reproductive synchrony in species where mortality is directly density-dependent, but does not affect the timing of reproduction in species with density-independent rates of mortality. This revised version was published online in July 2006 with corrections to the Cover Date.     

Disease regulation of age-structured host populations

A lethal, contagious disease can generate a density-dependent regulation of its host, provided the hosts' contact rate grows with population size. The condition for disease-induced population control is that the expected number of offspring of an infected newborn be less than one. In vertebrates that acquired immunity if they survive infection, the disease changes the age structure of its host population. The steady-state age structure of a disease-regulated host with age-dependent fecundity is computed. Local stability analysis indicates that the equilibrium age structure is always stable. However, when the usual exponentially distributed duration of the disease is replaced by a constant duration, the population can exhibit oscillations with a long period.