Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.     

Modeling transmission dynamics of stage-specific malaria vaccines

Population effects of malaria vaccination programs will depend on a complex interaction of the stage specificity of the vaccine, its duration of effectiveness, whether it is responsive to natural boosting, the strategy implemented, the proportion vaccinated and the pre-existing endemic conditions. In this article, Elizabeth Halloran and Claudio Struchiner review models of malaria transmission that incorporate aspects of immunity relevant to studying the effects of stage-specific malaria vaccination programs. They discuss the difference in the assumptions and applicability of the models and compare their predictions. Experience with malaria has demonstrated the difficulty in eliminating transmission, so emphasis needs to be on the new host-parasite balance that will be induced by the vaccination program. Although Halloran and Struchiner advise caution in interpreting the results of such models, they conclude that quantitative and theoretical analysis will be important in planning and evaluating interventions with malaria vaccines.     

A weather-driven model of malaria transmission

The classic formulae in malaria epidemiology are reviewed that relate entomological parameters to malaria transmission, including mosquito survivorship and age-at-infection, the stability index (S), the human blood index (HBI), proportion of infected mosquitoes, the sporozoite rate, the entomological inoculation rate (EIR), vectorial capacity (C) and the basic reproductive number (R ₀). The synthesis emphasizes the relationships among classic formulae and reformulates a simple dynamic model for the proportion of infected humans. The classic formulae are related to formulae from cyclical feeding models, and some inconsistencies are noted. The classic formulae are used to to illustrate how malaria control reduces malaria transmission and show that increased mosquito mortality has an effect even larger than was proposed by Macdonald in the 1950's.     

Mathematics of a model for Zika transmission dynamics

Zika is a flavivirus transmitted to humans through either the bite of infected Aedes mosquitoes or sexual intercourse with infected individuals. In this paper, we present a mathematical model based on these two modes of transmission. Using the next-generation matrix method, a threshold parameter called basic reproduction number is determined. Sensitivity analysis of the basic reproduction number in terms of parameters involved in its formulation is discussed. A dynamically consistent nonstandard finite difference scheme is designed to replicate the properties of the continuous model. Numerical simulations of the nonstandard finite difference scheme that we have constructed show the number of infected humans due to sexual intercourse with Zika virus infectious individuals. The numerical simulations also support the theoretical analysis of the model.     

Inference and dynamic simulation of malaria using a simple climate-driven entomological model of malaria transmission

Given the crucial role of climate in malaria transmission, many mechanistic models of malaria represent vector biology and the parasite lifecycle as functions of climate variables in order to accurately capture malaria transmission dynamics. Lower dimension mechanistic models that utilize implicit vector dynamics have relied on indirect climate modulation of transmission processes, which compromises investigation of the ecological role played by climate in malaria transmission. In this study, we develop an implicit process-based malaria model with direct climate-mediated modulation of transmission pressure borne through the Entomological Inoculation Rate (EIR). The EIR, a measure of the number of infectious bites per person per unit time, includes the effects of vector dynamics, resulting from mosquito development, survivorship, feeding activity and parasite development, all of which are moderated by climate. We combine this EIR-model framework, which is driven by rainfall and temperature, with Bayesian inference methods, and evaluate the model’s ability to simulate local transmission across 42 regions in Rwanda over four years. Our findings indicate that the biologically-motivated, EIR-model framework is capable of accurately simulating seasonal malaria dynamics and capturing of some of the inter-annual variation in malaria incidence. However, the model unsurprisingly failed to reproduce large declines in malaria transmission during 2018 and 2019 due to elevated anti-malaria measures, which were not accounted for in the model structure. The climate-driven transmission model also captured regional variation in malaria incidence across Rwanda’s diverse climate, while identifying key entomological and epidemiological parameters important to seasonal malaria dynamics. In general, this new model construct advances the capabilities of implicitly-forced lower dimension dynamical malaria models by leveraging climate drivers of malaria ecology and transmission.     

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.     

Dynamics of malaria transmission model with sterile mosquitoes

In this paper, a malaria transmission model with sterile mosquitoes is considered. We first formulate a simple SEIR malaria transmission model as our baseline model. Then sterile mosquitoes are introduced into the baseline model. We consider the case that the release rate of sterile mosquitoes is proportional to the wild mosquito population size. To investigate the impact of releasing sterile mosquitoes on the malaria transmission, the dynamics of the baseline model and the models with the sterile mosquitoes are discussed. We derive formulas of the reproductive numbers and explore the existence of endemic equilibrium as the reproductive number is more than unity for these models. It is shown that both the baseline model and the models with the sterile mosquitoes undergo backward bifurcations. Based on theoretical analysis and numerical simulation, we investigate the impact of releasing sterile mosquitoes on malaria transmission.     

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.     

Efficacy model for mosquito stage transmission blocking vaccines for malaria

Vaccines that target antigens found on the mosquito stages of Plasmodium falciparum and Plasmodium vivax parasites are under development as transmission blocking vaccines. Antisera from vaccinated animals and humans are able to block oocyst development in artificially fed mosquitoes but it is not clear from these data what level of antibody response would be required for a useful vaccine in a field setting. This paper describes a mathematical model that takes into account the relationship between antibody levels and blocking of oocyst levels in artificial feeds, the distribution of antibody responses seen in human populations and the distribution of oocyst densities in infected mosquitoes in the field to calculate the levels of antibody in the host population that would be required to achieve a level of herd immunity in a vaccinated human population that would give an operationally useful level of transmission blocking. The model predicts that current formulations of Pfs25 are likely to achieve useful reductions in transmission when tested in human field trials.     

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.     

Controlling malaria transmission with genetically-engineered, Plasmodium-resistant mosquitoes: milestones in a model system

We are developing transgenic mosquitoes resistant to malaria parasites to test the hypothesis that genetically-engineered mosquitoes can be used to block the transmission of the parasites. We are developing and testing many of the necessary methodologies with the avian malaria parasite, Plasmodium gallinaceum, and its laboratory vector, Aedes aegypti, in anticipation of engaging the technical challenges presented by the malaria parasite, P. falciparum, and its major African vector, Anopheles gambiae. Transformation technology will be used to insert into the mosquito a synthetic gene for resistance to P. gallinaceum. The resistance gene will consist of a promoter of a mosquito gene controlling the expression of an effector protein that interferes with parasite development and/or infectivity. Mosquito genes whose promoter sequences are capable of sex- and tissue-specific expression of exogenous coding sequences have been identified, and stable transformation of the mosquito has been developed. We now are developing the expressed effector portion of the synthetic gene that will interfere with the transmission of the parasites. Mouse monoclonal antibodies that recognize the circumsporozoite protein of P. gallinaceum block sporozoite invasion of mosquito salivary glands, as well as abrogate the infectivity of sporozoites to a vertebrate host, the chicken, Gallus gallus, and block sporozoite invasion and development in susceptible cell lines in vitro. Using the genes encoding these antibodies, we propose to clone and express single-chain antibody constructs (scFv) that will serve as the effector portion of the gene that interferes with transmission of P. gallinaceum sporozoites.     

Spatial simulation of malaria transmission and its control by malaria transmission blocking vaccination

A simple, visual representation of spatial aspects of malaria transmission in successive snap-shots in time, is presented. The spatial components of the simulation involve (i) the identification of mosquito vector breeding sites of defined shape and area, (ii) the identification of a zone of malaria transmission determined by the shapes and areas of the vector breeding sites and the distance from these sites that the mosquitoes disperse, (iii) a human population dispersed in relation to the malaria transmission zone, (iv) perimeters around each individual human within which his or her infection can be transmitted by the local vector mosquitoes. The intensity of transmission within a malaria transmission zone is given by a number which is the number of new cases of malaria that each existing case will distribute through the human population within the duration of an infection. The simulation has been used here to examine the effects of vaccination against malaria transmission. Different levels of vaccine coverage are represented under endemic and epidemic malaria. The consequences of full or partial coverage of a zone of malaria transmission are also examined. The results are numerically compatible with the predictions of previous simple mathematical simulations of malaria transmission and interventions. The present simulation allows the nature of malaria transmission and the effects of interventions to be communicated easily and directly to an audience. It could have practical value in discussions of malaria control strategies with health planners.     

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.     

Global dynamics of a cholera model with age structures and multiple transmission modes

In this paper,we propose and analyze a cholera model.The model incorporates both direct transmission (person-to-person transmission) and indirect transmission (contaminated environment-to-person transmission: hyper-infectivity and lower-infcctivity).Moreover,we employ general nonlinear incidences and introduce infection age of infectious individuals and biological ages of pathogens in the environment.After considering the well-posedness of the system,we study the existence and local stability of steady states,which is determined by the basic reproduction number.To establish the attractivity of the infection steady state,we also get the uniform persistence and existence of compact global attractors.The main result is a threshold dynamics obtained by applying the Fluctuation Lemma and the approach of Lyapunov functionals.When the basic reproduction number is less than one,the infection-free steady state is globally asymptotically stable while when the basic reproduction number is larger than one,the infection steady state attracts each solution with nonzero infection force at some time point.The effect of multiple transmission modes on the disease dynamics is also discussed.     

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.     

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.