A mathematical model for the dynamics of malaria in mosquitoes feeding on a heterogeneous host population

We describe and develop a difference equation model for the dynamics of malaria in a mosquito population feeding on, infecting and getting infected from a heterogeneous population of hosts. Using the force of infection from different classes of humans to mosquitoes as parameters, we evaluate a number of entomological parameters, indicating malaria transmission levels, which can be compared to field data. By assigning different types of vector control interventions to different classes of humans and by evaluating the corresponding levels of malaria transmission, we can compare the effectiveness of these interventions. We show a numerical example of the effects of increasing coverage of insecticide-treated bed nets in a human population where the predominant malaria vector is Anopheles gambiae.     

Within-host population dynamics and the evolution of microparasites in a heterogeneous host population

Abstract Why do parasites harm their hosts? The general understanding is that if the transmission rate and virulence of a parasite are linked, then the parasite must harm its host to maximize its transmission. The exact nature of such trade‐offs remains largely unclear, but for vertebrate hosts it probably involves interactions between a microparasite and the host immune system. Previous results have suggested that in a homogeneous host population in the absence of super‐ or coinfection, within‐host dynamics lead to selection of the parasite with an intermediate growth rate that is just being controlled by the immune system before it kills the host (Antia et al. 1994). In this paper, we examine how this result changes when heterogeneity is introduced to the host population. We incorporate the simplest form of heterogeneity–random heterogeneity in the parameters describing the size of the initial parasite inoculum, the immune response of the host, and the lethal density at which the parasite kills the host. We find that the general conclusion of the previous model holds: parasites evolve some intermediate growth rate. However, in contrast with the generally accepted view, we find that virulence (measured by the case mortality or the rate of parasite‐induced host mortality) increases with heterogeneity. Finally, we link the within‐host and between‐host dynamics of parasites. We show how the parameters for epidemiological spread of the disease can be estimated from the within‐host dynamics, and in doing so examine the way in which trade‐offs between these epidemiological parameters arise as a consequence of the interaction of the parasite and the immune response of the host.     

A mathematical model for the population dynamics of army ants

A stochastic cellular automata model for the population dynamics of the army antEciton burchelli on Barro Colorado Island in Panama is set up. It is simulated on the computer and shown to give good agreement with biological data. It is analysed using two approximations akin to the mean field approximation in statistical mechanics, and good agreement with the simulations is obtained. Finally, the role of distance between successive statary phase bivouacs is discussed with regard to the rate of colony growth. There are two aspects of the biological system studied here that make it of general importance. First, the population is structured, since the size of each colony of army ants is crucial. Second, the spatial behaviour of the population, as in many others, is not diffusion-like, although it is random. This has implications for the kind of model that is chosen.     

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

Evolution of rotifer population dynamics in a heterogeneous habitat: mathematical modeling

We present the results of mathematical modeling of a rotifer species inhabiting two coupled habitats with different environmental conditions. We use the modified Consensus model and show that the exchange between the habitats can lead to chaotization of originally regular plankton dynamics and synchronization of plankton biomass oscillations. As a result, the invasion of a chaotic regime takes place.     

A negative binomial model for sampling mosquitoes in a malaria survey

Sampling models are investigated for counts of mosquitoes from a malaria field survey conducted by the World Health Organization in Nigeria. The data can be described by a negative binomial model for two-way classified counted data, where the cell means are constrained to satisfy row-by-column independence and the parameter k is constant across rows. An algorithm, based on iterative proportional fitting, is devised for finding maximum likelihood estimates. Sampling properties of the estimates and likelihood-ratio statistics for the small sample sizes of the data are investigated by Monte Carlo experiments. The WHO reported an observation that the relative efficiencies of four trapping methods vary over time. Out of eight villages in the survey area, this observation is found to be true in only the one village that is near a swamp.     

Dynamical consequences of optimal host feeding on host-parasitoid population dynamics

This study examines the influence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka-Volterra continuous-time model. This study shows that when parasitoids behave optimally, i.e. they maximize their fitness measured by the instantaneous per capita growth rate, the non-destructive type of host feeding stabilizes host-parasitoid dynamics. Other types of host feeding, i.e. destructive, concurrent, or non-concurrent, do not qualitatively change the neutral stability of the Lotka-Volterra model. Moreover, it is shown that the pattern of host feeding which maximizes parasitoid fitness is either non-concurrent and destructive, or concurrent and non-destructive host feeding, depending on the host abundance and parameters of the model. The effects of the adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.     

Kinetic study of a heterogeneous tumor cell population using a mathematical model

A mathematical model of a heterogenous tumor as a system of interrelating cell populations is described, including a pool of quiescent cells, cell-to-cell variability in maturation rates, and cell migration from growth area to necrotic one. Computer simulation results are given, model labeled mitoses and labeled index curves for the Lewis carcinoma are compared with experimental data.     

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.     

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.     

A quantitative model of population dynamics in malaria with drug treatment

In this paper we build a population dynamics of malaria including drug treatment. By taking into account both sensitive and resistant parasites, we want to see the effect of treatments on resistance phenomenon and prevent it from overspreading. Our main results include a new dynamics model, its mathematical properties which are found through analysis, the determination of unknown parameters with help of a data set for malaria from Burkina Faso and the numerical simulations of the fitted model. Based on these results, treatment strategies to reduce drug resistance can be elaborated.     

A new mathematical model of the dynamics of an age cohort population

A new generalized conception of an organism is given. Based on this conception, a new mathematical model of ontogenesis of an individual and the survival of the age cohort of population was proposed. By using real data on the dynamics of the survival of the age cohort of population, the model enables one to determine the parameters characterizing the relationship man-environment in the context of survival and calculate the dynamics (from birth to death) of the model variables of the state of the organism.     

Effect of resource subsidies on predator-prey population dynamics: a mathematical model

The influence of a resource subsidy on predator-prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator-prey-subsidy interactions also increases.     

A mathematical model for the transmission of Plasmodium vivax malaria

We have proposed a mathematical model for the transmission of Plasmodium vivax malaria quantitatively, which is adjusted to the infected region, Guadalcanal, in the Solomon Islands. The simulation of a transmission model will be instrumental in planning the malaria control strategy. A characteristic of the life cycle of P. vivax is that a sporozoite injected into the blood stream by a mosquito bite may sometimes stay in a hepatocyte as a hypnozoite. Therefore, we have incorporated a phenomenon of renewed infections caused by a relapse into the transmission model. Also through the simulations we have attempted to evaluate the decline in prevalence caused by the programs of selective mass drug administration (MDA) and vector control such as the distribution of permethrin-treated bednets. The simulations have indicated that the concentrated repetition of MDA at 1-week intervals would reduce the prevalence of vivax malaria swiftly in the beginning and would keep the parasite rate below 1% for a few years but the prevalence would increase thereafter. In contrast, the parasite rate would remain below 1% for a long time if a trial of 1 or 2 times MDA is accompanied with some reduction of the vectorial capacity by the enforcement of vector control. In any case, it is important to beware of relapse cases because even after the execution of MDA it takes a long time to decrease the proportion of hypnozoite carriers.     

A mathematical model for Crimean-Congo haemorrhagic fever: tick-borne dynamics with conferred host immunity

Crimean-Congo haemorrhagic fever (CCHF) is a highly contagious tick-borne disease that impacts many countries in parts of Africa, Europe, Asia, and the Middle East. Outbreaks are episodic, but deadly. Due to the highly contagious nature of this disease, suspected cases are taken extremely serious, with very strong control measures implemented almost immediately. It is primarily those living on farms, livestock workers, and medical workers who are at risk. The virus responsible for CCHF is transmitted asymptomatically and transiently to livestock, and symptomatically to humans. The fatality rate in human cases can be very high. The number of methods and directions of viral transmission is large, including tick-to-tick, tick-to-livestock, tick-to-human, livestock-to-tick, livestock-to-human, and human-to-human. We model CCHF using a deterministic system of nonlinear differential equations. This compartment model allows us to analyse threshold parameters and equilibria describing the magnitude and progression of cases of the disease in a hypothetical outbreak.     

Evolution of virulence in a heterogeneous host population

Abstract.— There is a large body of theoretical studies that investigate factors that affect the evolution of virulence, that is parasite-induced host mortality. In these studies the host population is assumed to be genetically homogeneous. However, many parasites have a broad range of host types they infect, and trade-offs between the parasite virulence in different host types may exist. The aim of this paper is to study the effect of host heterogeneity on the evolution of parasite virulence. By analyzing a simple model that describes the replication of different parasite strains in a population of two different host types, we determine the optimal level of virulence in both host types and find the conditions under which strains that specialize in one host type dominate the parasite population. Furthermore, we show that intrahost evolution of the parasite during an infection may lead to stable polymorphisms and could introduce evolutionary branching in the parasite population.     

Arm-specific dynamics of chromosome evolution in malaria mosquitoes

Background  

The malaria mosquito species of subgenus Cellia have rich inversion polymorphisms that correlate with environmental variables. Polymorphic inversions tend to cluster on the chromosomal arms 2R and 2L but not on X, 3R and 3L in Anopheles gambiae and homologous arms in other species. However, it is unknown whether polymorphic inversions on homologous chromosomal arms of distantly related species from subgenus Cellia nonrandomly share similar sets of genes. It is also unclear if the evolutionary breakage of inversion-poor chromosomal arms is under constraints.     

Implications of a simple mathematical model to cancer cell population dynamics

Recent research in cancer progression and treatment indicates that many forms of cancer arise from the development of a small subpopulation of abnormal cancer stem cells (CSCs) that promote cancer growth and spread. Many potential treatments preferentially interact with cells at certain stages of the cell cycle by either selective killing or halting the cell cycle, such as intense, nanosecond-duration pulsed electric fields (nsPEFs). Simple mathematical models of unfed cancer cell populations at the plateau of their growth characteristics may estimate the long-term consequences of these treatments on proliferating and quiescent cell populations. Applying such a model with no transition from the quiescent to proliferating state shows that it is possible for the proliferating cell population to fall below 1 if the quiescent cell population obtains a sufficient competitive advantage with respect to nutrient consumption and/or survival rate. Introducing small, realistic transition rates did not appreciably alter short-term or long-term population behaviour, indicating that the predicted small cell population behaviour (< 1 cell) is not an artefact of the simpler model. Experimental observations of nsPEF-induced effects on the cell cycle suggest that such a model may serve as a first step in assessing the viability of a given cancer treatment in vitro prior to clinical application.