The dispersion of age differences between partners and the asymptotic dynamics of the HIV epidemic

In this paper, the effect of a change in the distribution of age differences between sexual partners on the dynamics of the HIV epidemic is studied. In a gender- and age-structured compartmental model, it is shown that if the variance of the distribution is small enough, an increase in this variance strongly increases the basic reproduction number. Moreover, if the variance is large enough, the mean age difference barely affects the basic reproduction number. We, therefore, conclude that the local stability of the disease-free equilibrium relies more on the variance than on the mean.     

An SIR pairwise epidemic model with infection age and demography

The demography and infection age play an important role in the spread of slowly progressive diseases. To investigate their effects on the disease spreading, we propose a pairwise epidemic model with infection age and demography on dynamic networks. The basic reproduction number of this model is derived. It is proved that there is a disease-free equilibrium which is globally asymptotically stable if the basic reproduction number is less that unity. Besides, sensitivity analysis is performed and shows that increasing the variance in recovery time and decreasing the variance in infection time can effectively control the diseases. The complex interaction between the death rate and equilibrium prevalence suggests that it is imperative to correctly estimate the parameters of demography in order to assess the disease transmission dynamics accurately. Moreover, numerical simulations show that the endemic equilibrium is globally asymptotically stable.     

Pleiotropic Stabilizing Selection Limits the Number of Polymorphic Loci to at Most the Number of Characters

We demonstrate that, in a model incorporating weak Gaussian stabilizing selection on n additively determined characters, at most n loci are polymorphic at a stable equilibrium. The number of characters is defined to be the number of independent components in the Gaussian selection scheme. We also assume linkage equilibrium, and that either the number of loci is large enough that the phenotypic distribution in the population can be approximated as multivariate Gaussian or that selection is weak enough that the mean fitness of the population can be approximated using only the mean and the variance of the characters in the population. Our results appear to rule out antagonistic pleiotropy without epistasis as a major force in maintaining additive genetic variation in a uniform environment. However, they are consistent with the maintenance of variability by genotype-environment interaction if a trait in different environments corresponds to different characters and the number of different environments exceeds the number of polymorphic loci that affect the trait.     

Global dynamics of an epidemiological model with age of infection and disease relapse

In this paper, an epidemiological model with age of infection and disease relapse is investigated. The basic reproduction number for the model is identified, and it is shown to be a sharp threshold to completely determine the global dynamics of the model. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is verified that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable, and hence the disease dies out; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease becomes endemic.     

Mathematical analysis of an age-structured population model with space-limited recruitment

In this paper, we investigate structured population model of marine invertebrate whose life stage is composed of sessile adults and pelagic larvae, such as barnacles contained in a local habitat. First we formulate the basic model as an Cauchy problem on a Banach space to discuss the existence and uniqueness of non-negative solution. Next we define the basic reproduction number R0 to formulate the invasion condition under which the larvae can successfully settle down in the completely vacant habitat. Subsequently we examine existence and stability of steady states. We show that the trivial steady state is globally asymptotically stable if R0 < or = 1, whereas it is unstable if R0 > 1. Furthermore, we show that a positive (non-trivial) steady state uniquely exists if R0 > 1 and it is locally asymptotically stable as far as absolute value of R0 - 1 is small enough.     

Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.     

A mathematical model for Chagas disease with infection-age-dependent infectivity

In this paper we develop a mathematical model for Chagas disease with infection-age-dependent infectivity. The effects of vector and blood transfusion transmission are considered, and the infected population is structured by the infection age (the time elapsed from infection). The authors identify the basic reproduction ratio R0 and show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We show that depending on parameters, backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. We also discuss local and global stability of steady states.     

A two-age-classes dengue transmission model

In this paper, we discuss a two-age-classes dengue transmission model with vaccination. The reason to divide the human population into two age classes is for practical purpose, as vaccination is usually concentrated in one age class. We assume that a constant rate of individuals in the child-class is vaccinated. We analyze a threshold number which is equivalent to the basic reproduction number. If there is an undeliberate vaccination to infectious children, which worsens their condition as the time span of being infectious increases, then paradoxically, vaccination can be counter productive. The paradox, stating that vaccination makes the basic reproduction number even bigger, can occur if the worsening effect is greater than a certain threshold, a function of the human demographic and epidemiological parameters, which is independent of the level of vaccination. However, if the worsening effect is to increase virulence so that one will develop symptoms, then the vaccination is always productive. In both situations, screening should take place before vaccination. In general, the presence of class division has obscured the known rule of thumb for vaccination.     

Negative selection on BRCA1 susceptibility alleles sheds light on the population genetics of late-onset diseases and aging theory

The magnitude of negative selection on alleles involved in age-specific mortality decreases with age. This is the foundation of the evolutionary theory of senescence. Because of this decrease in negative selection with age, and because of the absence of reproduction after menopause, alleles involved in women's late-onset diseases are generally considered evolutionarily neutral. Recently, genetic and epidemiological data on alleles involved in late onset-diseases have become available. It is therefore timely to estimate selection on these alleles. Here, we estimate selection on BRCA1 alleles leading to susceptibility to late-onset breast and ovarian cancer. For this, we integrate estimates of the risk of developing a cancer for BRCA1-carriers into population genetics frameworks, and calculate selection coefficients on BRCA1 alleles for different demographic scenarios varying across the extent of human demography. We then explore the magnitude of negative selection on alleles leading to a diverse range of risk patterns, to capture a variety of late-onset diseases. We show that BRCA1 alleles may have been under significant negative selection during human history. Although the mean age of onset of the disease is long after menopause, variance in age of onset means that there are always enough cases occurring before the end of reproductive life to compromise the selective value of women carrying a susceptibility allele in BRCA1. This seems to be the case for an extended range of risk of onset functions varying both in mean and variance. This finding may explain the distribution of BRCA1 alleles' frequency, and also why alleles for many late-onset diseases, like certain familial forms of cancer, coronary artery diseases and Alzheimer dementia, are typically recent and rare. Finally, we discuss why the two most popular evolutionary theories of aging, mutation accumulation and antagonistic pleiotropy, may underestimate the effect of selection on survival at old ages.     

Stochastic model of an influenza epidemic with drug resistance

A continuous-time Markov chain (CTMC) model is formulated for an influenza epidemic with drug resistance. This stochastic model is based on an influenza epidemic model, expressed in terms of a system of ordinary differential equations (ODE), developed by Stilianakis, N.I., Perelson, A.S., Hayden, F.G., [1998. Emergence of drug resistance during an influenza epidemic: insights from a mathematical model. J. Inf. Dis. 177, 863-873]. Three different treatments-chemoprophylaxis, treatment after exposure but before symptoms, and treatment after symptoms appear, are considered. The basic reproduction number, R(0), is calculated for the deterministic-model under different treatment strategies. It is shown that chemoprophylaxis always reduces the basic reproduction number. In addition, numerical simulations illustrate that the basic reproduction number is generally reduced with realistic treatment rates. Comparisons are made among the different models and the different treatment strategies with respect to the number of infected individuals during an outbreak. The final size distribution is computed for the CTMC model and, in some cases, it is shown to have a bimodal distribution corresponding to two situations: when there is no outbreak and when an outbreak occurs. Given an outbreak occurs, the total number of cases for the CTMC model is in good agreement with the ODE model. The greatest number of drug resistant cases occurs if treatment is delayed or if only symptomatic individuals are treated.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.     

The joint evolution of seed dormancy and flowering time in annual plants living in variable environments

In annual plants, both seed germination (as opposed to dormancy) and delayed flowering are assumed to confer both greater and more variable reproduction. With this assumption, a simple model of these two characters, together with a reparameterization of survival and reproductive rates and a general approximation, leads to several predictions based on maximizing geometric growth rates. Each character optimum is a ratio of mean to variance of future reproduction, and the two optimums are interdependent, with compensation by one favored if the other changes. Optimal character variances, possible with prediction of future reproduction, are roughly squared correlations of characters with future reproduction, divided by variances of future reproduction. The optimal mean of each character is more sensitive to the other character's variance than to its own variance, and the optimal variance of each character is more sensitive to the other character's mean than to its own mean. Optimal germination is increased with better correlation between flowering time and future reproduction. A diffusion approximation for the single locus, single character genetic analog of the phenotypic model showed fluctuating natural selection to favor, in expectation and under suitable conditions, the allele frequency maximizing the geometric population growth rate. Conditions for a protected polymorphism at a second locus, controlling the other character, show genetic variation at the first locus favors alleles at the second locus adapted to greater environmental variability. Computer simulations check some results, and the general predictions should be relevant for studies of correlated life history characters.     

Epiplastome variation of the number of chloroplasts in stomata guard cells of sugar beet (Beta vulgaris L.)

In stomata guard cells of sugar beet, variation in the number of chloroplasts was studied in successive generations: (1) hybrid generation; (2) generation yielded by uniparental apozygotic seed reproduction; (3) generation obtained after seed treatment with a colchicine solution; (4) generation obtained after seed treatment with 5-azacytidine. As compared to hybrid generation, uniparental seed reproduction increases the average number of chloroplasts in stomata guard cells (from 13.5 to 15.0) and decreases distribution variance of this trait by a factor of 3 to 4. Colchicine increases both average number of chloroplasts in stomata guard cells (from 13.5 to 18.2) and distribution variance (about twice). 5-Azacytindine reduces the number of chloroplasts in cells (from 15.0 to 12.9) but enhances distribution variance (about 1.5 times). Variation in the number of chromosomes in stomata cells is related to myxoploidy in meristem tissue, on the one hand, and to the rate of cell division, on the other. Uniparental seed reproduction is suggested to enhance the number of organelles per cell due to high myxoploidy in cell populations, which is typical of inbred plants. Colchicine blocks spindle division and sharply increases the level of myxoploidy in cell populations and the number of organelles per cell. 5-Azacytidine hypomethylates chromosome DNA, increases the rate of cell divisions, and reduces the number of organelles per cell. The described changes in the number of chloroplasts are inherited in cell lineage ("cell hereditary memory") and successive sporophyte generations.     

An advantage of sexual reproduction in a rapidly changing environment.

When an environmental change imposes strong directional selection, there are two advantages of sexual reproduction. First, an asexual population is limited to the most extreme individual in the population, and progress under directional selection can go no farther without mutation; no such limitation applies to a sexual population. Second, more quantitatively, directional selection in an asexual population monotonically decreases the variance, whereas the variance of a sexual population quickly reaches a steady value; this difference remains even if the direction of selection occasionally changes. With realistic environmental changes small alterations in any particular measurement or trait are usually sufficient to keep up with the changes, but fitness, since it depends on a large number of traits, will be selected with greater intensity, which may be enough to confer a distinct advantage on sexual reproduction. This applies particularly to a large or rapid environmental change. Eventually mutation will enhance the variance, but by then it may be too late to prevent extinction of asexual strains.     

A discrete model of avian influenza with seasonal reproduction and transmission

In this paper, we formulate a discrete-time model with the reproductive and overwintering periods to assess the impact of avian influenza transmission in poultry. It is shown that the disease is extinct if the basic reproduction number is less than one and is persistent if the basic reproductive number is greater than one. Furthermore, the model admits a closed invariant cycle, which means that avian influenza fluctuates in poultry.     

Modeling the Geographic Spread of Rabies in China

In order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and vaccinated subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The existence of the disease-free equilibrium is discussed, the basic reproduction number is calculated, and the effect of moving rates of dogs between patches on the basic reproduction number is studied. To investigate the rabies virus clades lineages, the two-patch submodel is used to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. It is found that the basic reproduction number of the two-patch model could be larger than one even if the isolated basic reproduction number of each patch is less than one. This indicates that the immigration of dogs may make the disease endemic even if the disease dies out in each isolated patch when there is no immigration. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog markets and trades needs to be regulated, and transportation of dogs has to be better monitored and under constant surveillance.     

Epiplastome Variation of the Number of Chloroplasts in Stomata Guard Cells of Sugar Beet (Beta vulgaris L.)

In stomata guard cells of sugar beet, variation in the number of chloroplasts was studied in successive generations: (1) hybrid generation; (2) generation yielded by uniparental apozygotic seed reproduction; (3) generation obtained after seed treatment with a colchicine solution; (4) generation obtained after seed treatment with 5-azacytidine. As compared to hybrid generation, uniparental seed reproduction increases the average number of chloroplasts in stomata guard cells (from 13.5 to 15.0) and decreases distribution variance of this trait by a factor of 3 to 4. Colchicine increases both average number of chloroplasts in stomata guard cells (from 13.5 to 18.2) and distribution variance (about twice). 5-Azacytindine reduces the number of chloroplasts in cells (from 15.0 to 12.9) but enhances distribution variance (about 1.5 times). Variation in the number of chromosomes in stomata cells is related to myxoploidy in meristem tissue, on the one hand, and to the rate of cell division, on the other. Uniparental seed reproduction is suggested to enhance the number of organelles per cell due to high myxoploidy in cell populations. Colchicine blocks spindle division and sharply increases the level of myxoploidy in cell populations and the number of organelles per cell. 5-Azacytidine hypomethylates chromosome DNA, increases the rate of cell divisions, and reduces the number of organelles per cell. The described changes in the number of chloroplasts are inherited in cell lineage (cell hereditary memory) and successive sporophyte generations.     

Chromosome Copy Number Variation and Control in the Ciliate Chilodonella uncinata

Copy number variations are widespread in eukaryotes. The unusual genome architecture of ciliates, in particular, with its process of amitosis in macronuclear division, provides a valuable model in which to study copy number variation. The current model of amitosis envisions stochastic distribution of macronuclear chromosomes during asexual reproduction. This suggests that amitosis is likely to result in high levels of copy number variation in ciliates, as dividing daughter cells can have variable copy numbers of chromosomes if chromosomal distribution during amitosis is a stochastic process. We examined chromosomal distribution during amitosis in Chilodonella uncinata, a ciliate with gene-size macronuclear chromosomes. We quantified 4 chromosomes in evolving populations of C. uncinata and modeled the amitotic distribution process. We found that macronuclear chromosomes differ in copy number from one another but that copy number does not change as expected under a stochastic process. The chromosome carrying SSU increased in copy number, which is consistent with selection to increase abundance; however, two other studied chromosomes displayed much lower than expected among-line variance. Our models suggest that balancing selection is sufficient to explain the observed maintenance of chromosome copy during asexual reproduction.     

The basic reproduction number and the probability of extinction for a dynamic epidemic model

We consider the spread of an epidemic through a population divided into n sub-populations, in which individuals move between populations according to a Markov transition matrix Σ and infectives can only make infectious contacts with members of their current population. Expressions for the basic reproduction number, R₀, and the probability of extinction of the epidemic are derived. It is shown that in contrast to contact distribution models, the distribution of the infectious period effects both the basic reproduction number and the probability of extinction of the epidemic in the limit as the total population size N → ∞. The interactions between the infectious period distribution and the transition matrix Σ mean that it is not possible to draw general conclusions about the effects on R₀ and the probability of extinction. However, it is shown that for n = 2, the basic reproduction number, R₀, is maximised by a constant length infectious period and is decreasing in ?, the speed of movement between the two populations.     

Evolution and maintenance of the environmental component of the phenotypic variance: benefit of plastic traits under changing environments

How environmental variances in quantitative traits are influenced by variable environments is an important problem in evolutionary biology. In this study, the evolution and maintenance of phenotypic variance in a plastic trait under stabilizing selection are investigated. The mapping from genotypic value to phenotypic value of the quantitative trait is approximated by a linear reaction norm, with genotypic effects on its phenotypic mean and sensitivity to environment. The environmental deviation is assumed to be decomposed into environmental quality, which interacts with genotypic value, and residual developmental noise, which is independent of genotype. Environmental quality and the optimal phenotype of stabilizing selection are allowed to randomly fluctuate in both space and time, and individuals migrate equally before development and reproduction among different niches. Analyses show that phenotypic plasticity is adaptive within variable environments if correlations have become established between the optimal phenotype and environmental quality in space and/or time. The evolved plasticity increases with variances in optimal phenotypes and correlations between optimal phenotype and environmental quality; this further induces increases in mean fitness and the environmental variance in the trait. Under certain circumstances, however, the environmental variance may decrease with increase in variation in environmental quality.