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.     

Habitat suitability of Anopheles vector species and association with human malaria in the Atlantic Forest in south-eastern Brazil

Every year, autochthonous cases of Plasmodium vivax malaria occur in low-endemicity areas of Vale do Ribeira in the south-eastern part of the Atlantic Forest, state of S?o Paulo, where Anopheles cruzii and Anopheles bellator are considered the primary vectors. However, other species in the subgenus Nyssorhynchus of Anopheles (e.g., Anopheles marajoara) are abundant and may participate in the dynamics of malarial transmission in that region. The objectives of the present study were to assess the spatial distribution of An. cruzii, An. bellator and An. marajoara and to associate the presence of these species with malaria cases in the municipalities of the Vale do Ribeira. Potential habitat suitability modelling was applied to determine both the spatial distribution of An. cruzii, An. bellator and An. marajoara and to establish the density of each species. Poisson regression was utilized to associate malaria cases with estimated vector densities. As a result, An. cruzii was correlated with the forested slopes of the Serra do Mar, An. bellator with the coastal plain and An. marajoara with the deforested areas. Moreover, both An. marajoara and An. cruzii were positively associated with malaria cases. Considering that An. marajoara was demonstrated to be a primary vector of human Plasmodium in the rural areas of the state of Amapá, more attention should be given to the species in the deforested areas of the Atlantic Forest, where it might be a secondary vector.     

Determination of the processes driving the acquisition of immunity to malaria using a mathematical transmission model

Acquisition of partially protective immunity is a dominant feature of the epidemiology of malaria among exposed individuals. The processes that determine the acquisition of immunity to clinical disease and to asymptomatic carriage of malaria parasites are poorly understood, in part because of a lack of validated immunological markers of protection. Using mathematical models, we seek to better understand the processes that determine observed epidemiological patterns. We have developed an age-structured mathematical model of malaria transmission in which acquired immunity can act in three ways (“immunity functions”): reducing the probability of clinical disease, speeding the clearance of parasites, and increasing tolerance to subpatent infections. Each immunity function was allowed to vary in efficacy depending on both age and malaria transmission intensity. The results were compared to age patterns of parasite prevalence and clinical disease in endemic settings in northeastern Tanzania and The Gambia. Two types of immune function were required to reproduce the epidemiological age-prevalence curves seen in the empirical data; a form of clinical immunity that reduces susceptibility to clinical disease and develops with age and exposure (with half-life of the order of five years or more) and a form of anti-parasite immunity which results in more rapid clearance of parasitaemia, is acquired later in life and is longer lasting (half-life of >20 y). The development of anti-parasite immunity better reproduced observed epidemiological patterns if it was dominated by age-dependent physiological processes rather than by the magnitude of exposure (provided some exposure occurs). Tolerance to subpatent infections was not required to explain the empirical data. The model comprising immunity to clinical disease which develops early in life and is exposure-dependent, and anti-parasite immunity which develops later in life and is not dependent on the magnitude of exposure, appears to best reproduce the pattern of parasite prevalence and clinical disease by age in different malaria transmission settings. Understanding the effector mechanisms underlying these two immune functions will assist in the design of transmission-reducing interventions against malaria.     

Diversity and abundance of climbers from the Atlantic Forest, southeastern Brazil

Climbing plant community composition and abundance was compared between the Alto da Serra de Paranapiacaba Biological Reserve (ASPRB) and the Nascentes de Paranapiacaba Municipal Natural Park (NPMNP), two protected areas in the largest Atlantic Forest remnant, southeastern Brazil. Climbing plants ≥1 cm in diameter were sampled in 52 quadrats of 10 × 20 m (1.04 ha). From these data we calculated the Shannon–Wiener diversity index, calculated abundance and importance values for families, performed a principal coordinate analysis, and created a similarity dendrogram (TWINSPAN). The composition and abundance of species between ASPRB and NPMNP differed significantly due to the successional state of the ASPRB forest, which was affected by more recent disturbance. There we found 34 species, while in the NPMNP, species richness was 72, with only 24 species in common. The two areas formed two groups corresponding to two distinct floristic subsets. These montane forests were characterized by a predominance of species of the Asteraceae. The high species richness, floristic composition, and community composition reflect the greater degree of conservation in the area of NPMNP. The high density and community composition of climbers in the more conserved area of rainforest may be due to the low intensity of disturbance.     

A model for predicting the transmission rate of malaria from serological data

A model is developed to estimate the duration for which malaria antibody levels in the blood remain high in a closed population. This estimate can be used to calculate the transmission rate within a region, in conjunction with the serological information contained in the population. The model is used on data obtained from a study of malaria in the Philippines and shows excellent agreement. It is subsequently utilised for predictions and seems to be an appropriate vehicle for this purpose.     

Scrapie transmission in Britain: a recipe for a mathematical model

Responses to an anonymous postal survey concerning scrapie are analysed. Risk factors associated with farms that have had scrapie are identified as size, geographical region, lambing practices and holding of certain breeds. Further analysis of farms that have scrapie only in bought-in animals reveals that such farms tend to breed a smaller proportion of their replacement animals than farms without scrapie. Farms that have had scrapie in home-bred animals have attributes associated with breeding many animals: large numbers of rams bought, few ewes bought, and many animals that are home-bred. The demography of British sheep farms as described by size, breeds, purchasing behaviour, age structure and proportion of animals that are home-bred is summarized. British farms with scrapie reveal certain special features: they have more sheep that are found dead, more elderly ewes and more cases of scab.     

The role of vector control in stopping the transmission of malaria: threats and opportunities

Malaria control, and that of other insect borne diseases such as dengue, is heavily dependent on our ability to control the mosquito populations that transmit these diseases. The major push over the last decade to reduce the global burden of malaria has been driven by the distribution of pyrethroid insecticide-treated bednets and an increase in coverage of indoor residual spraying (IRS). This has reduced malaria deaths by a third. Progress towards the goal of reducing this further is threatened by lack of funding and the selection of drug and insecticide resistance. When malaria control was initially scaled up, there was little pyrethroid resistance in the major vectors, today there is no country in Africa where the vectors remain fully susceptible to pyrethroids. The first pyrethroid resistance mechanisms to be selected produced low-level resistance which had little or no operational significance. More recently, metabolically based resistance has been selected, primarily in West Africa, which in some mosquito populations produces more than 1000-fold resistance. As this spreads the effectiveness of pyrethroid-based bednets and IRS will be compromised. New public health insecticides are not readily available. The pipeline of agrochemical insecticides that can be re-purposed for public health dried up 30 years ago when the target product profile for agricultural insecticides shifted from broad spectrum, stable, contact-acting insecticides to narrow spectrum stomach poisons that could be delivered through the plant. A public–private partnership, the Innovative Vector Control Consortium, was established in 2005 to stimulate the development of new public health pesticides. Nine potential new classes of chemistry are in the pipeline, with the intention of developing three into new insecticides. While this has been successfully achieved, it will still take 6–9 years for new insecticides to reach the market. Careful management of the resistance situation in the interim will be needed if current gains in malaria control are not to be reversed.     

Exploring rarity using a general model for distribution and abundance

We describe a simple model for changes in the distribution and abundance of a metapopulation and use it to explore the conditions leading to different types of rarity. The model suggests that localized populations (those with low patch occupancy but high local abundance) arise from low dispersal, low heterogeneity in extant population size, and frequent local extinctions relative to the potential for recolonization. Scarce populations (with low distribution and abundance) arise when relative local extinction rate is low to moderate and heterogeneity is high or successful dispersal is relatively low. Sparse populations (widespread, but with low local abundance) arise when relative local extinction rate is very low and either spatial heterogeneity or mortality through unsuccessful dispersal is high. In sparse or common species, there may be unstable as well as stable equilibria, implying a threshold distribution and abundance for persistence. The model supports a general correlation between distribution and abundance and suggests that persistence may be threatened by dispersal rates being either too high or too low. The model provides a new perspective on rarity and suggests a simple theoretical foundation for understanding the population-dynamic mechanisms that determine distribution and abundance.     

Year to year and seasonal variations in vector bionomics and malaria transmission in a humid savannah village in west Burkina Faso

A longitudinal entomological study was carried out from 1999 to 2001 in Lena, a humid savannah village in the western region of Burkina Faso in order to establish malaria vector bionomics and the dynamics of malaria transmission. In the first year, malaria transmission was mainly due to An. gambiae s.s., but during the two later years was due to An. funestus, which were observed in high frequency towards the end of the rainy season. PCR identification of samples of An. gambiae s.1. showed 93% to be An. gambiae s.s. and 7% An. arabiensis. An. funestus constituting more than 60% of the vectors were identified in PCR as An. funestus s.s. The persistence of intense vectorial activity in this village was probably due to the road building in a swampy area creating a semi-permanent swamp that provided large sites for larval mosquitoes. These swampy sites seemed to be more favorable for An. funestus than for An. gambiae s.s. Thus, land development must be monitored and subjected to planning to minimize vector proliferation. Such a system of planning could lead to the restriction or even elimination of the swamp that is the source of larvae developing in the heart of the village.     

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.     

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.     

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 mathematical model for the spread of Strepotococcus pneumoniae with transmission dependent on serotype

We examine a mathematical model for the transmission of Streptococcus Pneumoniae amongst young children when the carriage transmission coefficient depends on the serotype. Carriage means pneumococcal colonization. There are two sequence types (STs) spreading in a population each of which can be expressed as one of two serotypes. We derive the differential equation model for the carriage spread and perform an equilibrium and global stability analysis on it. A key parameter is the effective reproduction number R (e). For R (e) ≤ 1, there is only the carriage-free equilibrium (CFE) and the carriage will die out whatever be the starting values. For R (e) > 1, unless the effective reproduction numbers of the two STs are equal, in addition to the CFE there are two carriage equilibria, one for each ST. If the ST with the largest effective reproduction number is initially present, then in the long-term the carriage will tend to the corresponding equilibrium.     

A research agenda for malaria eradication: vector control

Different challenges are presented by the variety of malaria transmission environments present in the world today. In each setting, improved control for reduction of morbidity is a necessary first step towards the long-range goal of malaria eradication and a priority for regions where the disease burden is high. For many geographic areas where transmission rates are low to moderate, sustained and well-managed application of currently available tools may be sufficient to achieve local elimination. The research needs for these areas will be to sustain and perhaps improve the effectiveness of currently available tools. For other low-to-moderate transmission regions, notably areas where the vectors exhibit behaviours such as outdoor feeding and resting that are not well targeted by current strategies, new interventions that target predictable features of the biology/ecologies of the local vectors will be required. To achieve elimination in areas where high levels of transmission are sustained by very efficient vector species, radically new interventions that significantly reduce the vectorial capacity of wild populations will be needed. Ideally, such interventions should be implemented with a one-time application with a long-lasting impact, such as genetic modification of the vectorial capacity of the wild vector population.     

A climate-based distribution model of malaria transmission in sub-Saharan Africa.

Malaria remains the single largest threat to child survival in sub-Saharan Africa and warrants long-term investment for control. Previous malaria distribution maps have been vague and arbitrary. Marlies Craig, Bob Snow and David le Sueur here describe a simple numerical approach to defining distribution of malaria transmission, based upon biological constraints of climate on parasite and vector development. The model compared well with contemporary field data and historical 'expert opinion' maps, excepting small-scale ecological anomalies. The model provides a numerical basis for further refinement and prediction of the impact of climate change on transmission. Together with population, morbidity and mortality data, the model provides a fundamental tool for strategic control of malaria.     

A mathematical model for calculating the vector magnetic field of a single muscle fiber.

A mathematical model is described for calculating the volume-conducted magnetic field from active muscle fibers in an anisotropic bundle. With earlier models, the azimuthal magnetic field of a nerve bundle was calculated and the results were compared with the fields measured by toroidal pickup coils. The present model is capable of evaluating all three of the magnetic field components and is thus applicable for analyzing SQUID magnetometer recordings of fields from a muscle bundle. The component of the magnetic field parallel to the fiber axis is more than an order of magnitude smaller than either of the other two components. The amplitude of the magnetic signal is strongly dependent upon the anisotropy of the muscle bundle, the intracellular conductivity, the radius of the muscle fiber, the radius of the muscle bundle, and the location of the fiber in the muscle bundle. The peak-to-peak amplitude of the single-muscle-fiber action field increases linearly with increasing intracellular conductivity, as the square of the radius of the muscle fiber, and exponentially with the distance between the location of the fiber and the center of the bundle.     

A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission

Background

Malaria is one of the oldest and deadliest infectious diseases in humans. Many mathematical models of malaria have been developed during the past century, and applied to potential interventions. However, malaria remains uncontrolled and is increasing in many areas, as are vector and parasite resistance to insecticides and drugs.

Methods

This study presents a simulation model of African malaria vectors. This individual-based model incorporates current knowledge of the mechanisms underlying Anopheles population dynamics and their relations to the environment. One of its main strengths is that it is based on both biological and environmental variables.

Results

The model made it possible to structure existing knowledge, assembled in a comprehensive review of the literature, and also pointed out important aspects of basic Anopheles biology about which knowledge is lacking. One simulation showed several patterns similar to those seen in the field, and made it possible to examine different analyses and hypotheses for these patterns; sensitivity analyses on temperature, moisture, predation and preliminary investigations of nutrient competition were also conducted.

Conclusions

Although based on some mathematical formulae and parameters, this new tool has been developed in order to be as explicit as possible, transparent in use, close to reality and amenable to direct use by field workers. It allows a better understanding of the mechanisms underlying Anopheles population dynamics in general and also a better understanding of the dynamics in specific local geographic environments. It points out many important areas for new investigations that will be critical to effective, efficient, sustainable interventions.