Theoretical analysis of mixed Plasmodium malariae and Plasmodium falciparum infections with partial cross-immunity

A deterministic model for assessing the dynamics of mixed species malaria infections in a human population is presented to investigate the effects of dual infection with Plasmodium malariae and Plasmodium falciparum. Qualitative analysis of the model including positivity and boundedness is performed. In addition to the disease free equilibrium, we show that there exists a boundary equilibrium corresponding to each species. The isolation reproductive number of each species is computed as well as the reproductive number of the full model. Conditions for global stability of the disease free equilibrium as well as local stability of the boundary equilibria are derived. The model has an interior equilibrium which exists if at least one of the isolation reproductive numbers is greater than unity. Among the interesting dynamical behaviours of the model, the phenomenon of backward bifurcation where a stable boundary equilibrium coexists with a stable interior equilibrium, for a certain range of the associated invasion reproductive number less than unity is observed. Results from analysis of the model show that, when cross-immunity between the two species is weak, there is a high probability of coexistence of the two species and when cross-immunity is strong, competitive exclusion is high. Further, an increase in the reproductive number of species i increases the stability of its boundary equilibrium and its ability to invade an equilibrium of species j. Numerical simulations support our analytical conclusions and illustrate possible behaviour scenarios of the model.     

A deterministic analysis of self-incompatibility alleles

A number of plant species have a self-incompatibility locus that prevents self-fertilization. We 'analyse a deterministic model with an arbitrary number of alleles. We prove that the only polymorphic equilibrium is the one for which all (heterozygous) genotypes are equally frequent, and we prove that all (initially) polymorphic populations converge to this equilibrium.     

Mathematical model for Zika virus dynamics with sexual transmission route

Zika virus is a flavivirus transmitted to humans primarily through the bite of infected Aedes mosquitoes. In addition to vector-borne spread, however, the virus can also be transmitted through sexual contact. In this paper, we formulate and analyze a new system of ordinary differential equations which incorporates both vector and sexual transmission routes. Theoretical analysis of this model when there is no disease induced mortality shows that the disease-free equilibrium is locally and globally asymptotically stable whenever the associated reproduction number is less than unity and unstable otherwise. However, when we extend this same model to include Zika induced mortality, which have been documented in Latin America, we find that the model exhibits a backward bifurcation. Specifically, a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. To further explore model predictions, we use numerical simulations to assess the importance of sexual transmission to disease dynamics. This analysis shows that risky behavior involving multiple sexual partners, particularly among male populations, substantially increases the number of infected individuals in the population, contributing significantly to the disease burden in the community.     

一类带有非线性感染率传染病模型的动力学性质（英文）

主要介绍了一类带有非线性感染率的传染病模型.并且证明了当基本再生数Ro≤1时,无病平衡点是全局稳定的,当基本再生数R_0〉1时,疾病持续.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

一个带有ATL过程的HTL V-I感染的时滞模型

本文提出并分析了两个关于人体T-细胞淋巴回归Ⅰ型病毒（HTL V-I）感染并带有坏死白血病细胞（ATL）进程的数学模型,一个常微分方程模型,一个离散时滞模型.首先对常微分方程模型进行了分析,运用相应的特征方程得到一个阈值Ro（CD4⁺ T-细胞的基本再生数）.当R₀≤1时,仅有未染病平衡态存在,并且给出了其稳定性;当R₀>1时,有一个染病稳定态存在,并且此时它是稳定的.然后,我们在常微分方程模型中引入了一个离散时滞,通过对时滞模型的超越特征方程的分析,导出了与常微分方程模型中同样的稳定性条件,即时滞模型平衡态的稳定性与时滞的具体值无关.     

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.     

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.     

The effects of intraspecific competition and stabilizing selection on a polygenic trait

The equilibrium properties of an additive multilocus model of a quantitative trait under frequency- and density-dependent selection are investigated. Two opposing evolutionary forces are assumed to act: (i) stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and (ii) intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the equilibrium structure, in particular, number, degree of polymorphism, and genetic variance of stable equilibria, is affected by the strength of frequency dependence, and what role the number of loci, the amount of recombination, and the demographic parameters play. To this end, we employ a statistical and numerical approach, complemented by analytical results, and explore how the equilibrium properties averaged over a large number of genetic systems with a given number of loci and average amount of recombination depend on the ecological and demographic parameters. We identify two parameter regions with a transitory region in between, in which the equilibrium properties of genetic systems are distinctively different. These regions depend on the strength of frequency dependence relative to pure stabilizing selection and on the demographic parameters, but not on the number of loci or the amount of recombination. We further study the shape of the fitness function observed at equilibrium and the extent to which the dynamics in this model are adaptive, and we present examples of equilibrium distributions of genotypic values under strong frequency dependence. Consequences for the maintenance of genetic variation, the detection of disruptive selection, and models of sympatric speciation are discussed.     

On the Evolution of the Timing of Reproduction with Non-equilibrium Resident Dynamics

We study the evolution of an individual’s reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.     

Impact of Allee effect on an eco-epidemiological system

In this paper, we propose a general ratio-dependent prey-predator model with disease in predator subject to the strong Allee effect in prey. We obtain the complete dynamics of both models: (a) full model with Allee effect; (b) full model without Allee effect. Model (a) may have more than one interior equilibrium point, but model (b) has only one interior equilibrium point. Numerical results reveal that the coexistence of all the populations at the endemic state is possible for both the models. But for the model with Allee effect, the coexistence can be destroyed by an increased supply of alternative food for the predators. It can also be proved that for the full model with Allee effect, the disease can be suppressed under certain parametric conditions. Also by comparing models (a) and (b), we conclude that Allee effect can create or destroy the interior attractor. Finally, we have studied the disease free-submodel (prey and susceptible predator model) with and without Allee effect. The comparative study between these two submodels leads to the following conclusions: 1) In the presence of Allee effect, the number of interior equilibrium points can change from zero to two whereas the submodel without Allee effect has unique interior equilibrium point; 2) Both with and without Allee effect, initial conditions play an important role on the survival and extinction of prey as well as its corresponding predator; 3) In the presence of Allee effect, bi-stability occurs with stable or periodic coexistence of prey and susceptible predator and the extinction of prey and susceptible predator; 4) Allee effect can generate or destroy the interior equilibrium points.     

Studies of effects of macromolecular crowding and confinement on protein folding and protein stability

In cells, proteins execute specific tasks in crowded environments; these environments influence their stability and dynamics. Similarly, for an enzyme molecule encapsulated in an inorganic cavity as in biosensors or biocatalysts, confinement or excluded volume plays an important role in its stability and dynamics. In this article we present results of our experimental and theoretical investigations of the confinement and macromolecular crowding effects on protein. On the experimental side we study the stability of encapsulated cytochrome c against unfolding induced by the presence of denaturants, such as urea. Results show that, as the pore size in which protein is trapped is reduced, protein shows higher stability against denaturant-induced unfolding. On the theoretical side, after reviewing our previous study of the confinement effects on the equilibrium and dynamic properties of protein using a minimalist (two-dimensional lattice, Monte Carlo, Brownian dynamics) model, we have extended the model so that the effects of macromolecular crowding on such properties can be studied. Our simulations show that both folding and unfolding times increase with the number of crowders in solution, however, the equilibrium constant is affected such that the equilibrium is shifted towards the folded state. Furthermore, our results show that, for a fixed number of crowders as the size of crowder (or excluded volume) increases, the average size of protein at equilibrium decreases.     

On the behavior of solutions in viral dynamical models

We consider simple mathematical models for the early population dynamics of the human immunodefficiency type 1 virus (HIV-1). Although these systems of differential equations may be solved by numerical methods, few general theoretical results are available due to nonlinearities. We analyze a model whose components are plasma densities of uninfected CD4+ T-cells and infected cells (assumed in this model to be proportional to virion density). In addition to analyzing the nature of the equilibrium points, we show that there are no periodic or limit-cycle solutions. Depending on the values of the parameters, solutions either tend without oscillation to an equilibrium point with zero virion density or to an equilibrium point in which there are a nonzero number of virions. In the latter case the approach to equilibrium may be through damped oscillations or without oscillation.     

Rational behavioral response and the transmission of STDs

The susceptible-infected (SI) model is extended by allowing for individual optimal choices of self-protective actions against infection, where agents differ with respect to preferences and costs of self-protection. It is shown that a unique endemic equilibrium prevalence exists when the basic reproductive number of a STD is strictly greater than unity, and that the disease-free equilibrium is the unique steady state equilibrium when the basic reproductive number is less than or equal to one. Unlike in models that take individual behavior as given and fixed, the endemic equilibrium prevalence need not vary monotonically with respect to the basic reproductive number. Specifically, with endogenously determined self-protective behavior, a reduction in the basic reproductive number may in fact increase the endemic equilibrium prevalence. The global stability of the endemic steady state is established for the case of a homogeneous population by showing that, for any non-zero initial disease prevalence, there exists an equilibrium path which converges to the endemic steady state.     

Global dynamics of a staged-progression model with amelioration for infectious diseases

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R(0). If R(0)≤1, then the disease-free equilibrium P(0) is globally asymptotically stable, and the disease always dies out. If R(0)>1, P(0) is unstable, a unique endemic equilibrium P* is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated.     

Combining Perturbations and Parameter Variation to Influence Mean First Passage Times

Perturbations are relatively large shocks to state variables that can drive transitions between stable states, while drift in parameter values gradually alters equilibrium magnitudes. This latter effect can lead to equilibrium bifurcation, the generation, or annihilation of equilibria. Equilibrium annihilations reduce the number of equilibria and so are associated with catastrophic population collapse. We study the combination of perturbations and parameter drift, using a two-species intraguild predation (IGP) model. For example, we use bifurcation analysis to understand how parameter drift affects equilibrium number, showing that both competition and predation rates in this model are bifurcating parameters. We then introduce a stochastic process to model the effects of population perturbations. We demonstrate how to evaluate the joint effects of perturbations and drift using the common currency of mean first passage time to transitions between stable states. Our methods and results are quite general, and for example, can relate to issues in both pest control and sustainable harvest. Our results show that parameter drift (1) does not importantly change the expected time to reach target points within a basin of attraction, but (2) can dramatically change the expected time to shift between basins of attraction, through its effects on equilibrium resilience.     

The content and availability of information affects the evolution of social-information gathering strategies

Social animals can gather information by observing the other members of their groups. Strategies for gathering this type of social information have many components. In particular, an animal can vary the number of other animals it observes. European starlings (Sturnus vulgaris) in flight pay attention to a number of neighbors that allows the flock to reach consensus quickly and robustly. The birds may do this because being in such a flock confers benefits on its members, or the birds may use the strategy that is individually beneficial without regard for the flock’s structure. To understand when individual-level optimization results in a group-level optimum, we develop a model of animals gathering social information about environmental cues, where the cue can be about either predators or resources, and we analyze two processes through which the number of neighbors changes over time. We then identify the number of neighbors the birds use when the two dynamics reach equilibrium. First, we find that the equilibrium number of neighbors is much lower when the birds are learning about the presence of resources rather than predators. Second, when the information is about the presence of predators, we find that the equilibrium number of neighbors increases as the information becomes more widespread. Third, we find that an optimization process converges on strategies that allow the flock to reach consensus when the information is about the presence of abundant resources, but not when it is about the presence of scarce resources or predators.     

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible-infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.     

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.     

The general mixing of addicts and needles in a variable-infectivity needle-sharing environment

 In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. The model we discuss focuses on the transmission of HIV through the sharing of contaminated drug injection equipment and in particular we examine the mixing of addicts and needles when the AIDS incubation period is divided into three distinct infectious stages. The impact of this assumption is to greatly increase the complexity of the HIV transmission mechanism. We begin the paper with a brief literature review followed by the derivation of a model which incorporates three classes of infectious addicts and three classes of infectious needles and where a general probability structure is used to represent the interaction of addicts and needles of varying levels of infectivity. We find that if the basic reproductive number is less than or equal to unity then there exists a globally stable disease free equilibrium. The model possesses an endemic equilibrium solution if the basic reproductive number exceeds unity. We then conduct a brief simulation study of our model. We find that the spread of disease is heavily influenced by the way addicts and needles of different levels of infectivity interact. Received: 20 September 2001 / Revised version: 21 December 2001 / Published online: 17 May 2002