An integrated model of Plasmodium falciparum dynamics

The blood-stage dynamics of malaria infection with immune response

This article is concerned with the dynamics of malaria infection model with diffusion and delay. The disease free threshold ℜ0 and the immune response threshold value ℜ1 of the malaria infection are obtained, which characterize the stability of the disease free equilibrium and infection equilibrium (with or without immune response). In addition, fluctuations occur when the model undergoes Hopf bifurcation as the delay passes through a certain critical value τ0. In this case, periodic oscillation appears near the positive steady state, which implies the recurrent attacks of disease. Finally, numerical simulations are provided to illustrate the theoretical results.

Declines in prevalence alter the optimal level of sexual investment for the malaria parasite Plasmodium falciparum

Like most human pathogens, the malaria parasite Plasmodium falciparum experiences strong selection pressure from public health interventions such as drug treatment. While most commonly studied in the context of drug targets and related pathways, parasite adaptation to control measures likely extends to phenotypes beyond drug resistance. Here, we use modeling to explore how control measures can reduce levels of within-host competition between P. falciparum genotypes and favor higher rates of sexual investment. We validate these predictions with longitudinally sampled genomic data from French Guiana during a period of malaria decline and find that the most strongly selected genes are enriched for transcription factors involved in commitment to and development of the parasite’s sexual gametocyte form.

Successful infectious disease interventions can result in large reductions in parasite prevalence. Such demographic change has fitness implications for individual parasites and may shift the parasite’s optimal life history strategy. Here, we explore whether declining infection rates can alter Plasmodium falciparum’s investment in sexual versus asexual growth. Using a multiscale mathematical model, we demonstrate how the proportion of polyclonal infections, which decreases as parasite prevalence declines, affects the optimal sexual development strategy: Within-host competition in multiclone infections favors a greater investment in asexual growth whereas single-clone infections benefit from higher conversion to sexual forms. At the same time, drug treatment also imposes selection pressure on sexual development by shortening infection length and reducing within-host competition. We assess these models using 148 P. falciparum parasite genomes sampled in French Guiana over an 18-y period of intensive intervention (1998 to 2015). During this time frame, multiple public health measures, including the introduction of new drugs and expanded rapid diagnostic testing, were implemented, reducing P. falciparum malaria cases by an order of magnitude. Consistent with this prevalence decline, we see an increase in the relatedness among parasites, but no single clonal background grew to dominate the population. Analyzing individual allele frequency trajectories, we identify genes that likely experienced selective sweeps. Supporting our model predictions, genes showing the strongest signatures of selection include transcription factors involved in the development of P. falciparum’s sexual gametocyte form. These results highlight how public health interventions impose wide-ranging selection pressures that affect basic parasite life history traits.

Understanding dynamics of Plasmodium falciparum gametocytes production: Insights from an age-structured model

Many models of within-host malaria infection dynamics have been formulated since the pioneering work of Anderson et al. in 1989. Biologically, the goal of these models is to understand what governs the severity of infections, the patterns of infectiousness, and the variation thereof across individual hosts. Mathematically, these models are based on dynamical systems, with standard approaches ranging from K-compartments ordinary differential equations (ODEs) to delay differential equations (DDEs), to capture the relatively constant duration of replication and bursting once a parasite infects a host red blood cell. Using malariatherapy data, which offers fine-scale resolution on the dynamics of infection across a number of individual hosts, we compare the fit and robustness of one of these standard approaches (K-compartments ODE) with a partial differential equations (PDEs) model, which explicitly tracks the "age" of an infected cell. While both models perform quite similarly in terms of goodness-of-fit for suitably chosen K, the K-compartments ODE model particularly overestimates parasite densities early on in infections when the number of repeated compartments is not large enough. Finally, the K-compartments ODE model (for suitably chosen K) and the PDE model highlight a strong qualitative connection between the density of transmissible parasite stages (i.e., gametocytes) and the density of host-damaging (and asexually-replicating) parasite stages. This finding provides a simple tool for predicting which hosts are most infectious to mosquitoes -vectors of Plasmodium parasites- which is a crucial component of global efforts to control and eliminate malaria.

Mechanistic within-host models of the asexual Plasmodium falciparum infection: a review and analytical assessment

Background

Malaria blood-stage infection length and intensity are important drivers of disease and transmission; however, the underlying mechanisms of parasite growth and the host’s immune response during infection remain largely unknown. Over the last 30 years, several mechanistic mathematical models of malaria parasite within-host dynamics have been published and used in malaria transmission models.

Methods

Mechanistic within-host models of parasite dynamics were identified through a review of published literature. For a subset of these, model code was reproduced and descriptive statistics compared between the models using fitted data. Through simulation and model analysis, key features of the models were compared, including assumptions on growth, immune response components, variant switching mechanisms, and inter-individual variability.

Results

The assessed within-host malaria models generally replicate infection dynamics in malaria-naïve individuals. However, there are substantial differences between the model dynamics after disease onset, and models do not always reproduce late infection parasitaemia data used for calibration of the within host infections. Models have attempted to capture the considerable variability in parasite dynamics between individuals by including stochastic parasite multiplication rates; variant switching dynamics leading to immune escape; variable effects of the host immune responses; or via probabilistic events. For models that capture realistic length of infections, model representations of innate immunity explain early peaks in infection density that cause clinical symptoms, and model representations of antibody immune responses control the length of infection. Models differed in their assumptions concerning variant switching dynamics, reflecting uncertainty in the underlying mechanisms of variant switching revealed by recent clinical data during early infection. Overall, given the scarce availability of the biological evidence there is limited support for complex models.

Conclusions

This study suggests that much of the inter-individual variability observed in clinical malaria infections has traditionally been attributed in models to random variability, rather than mechanistic disease dynamics. Thus, it is proposed that newly developed models should assume simple immune dynamics that minimally capture mechanistic understandings and avoid over-parameterization and large stochasticity which inaccurately represent unknown disease mechanisms.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12936-021-03813-z.

A general method for multiscale modelling of vector-borne disease systems

The inability to develop multiscale models which can describe vector-borne disease systems in terms of the complete pathogen life cycle which represents multiple targets for control has hindered progress in our efforts to control, eliminate and even eradicate these multi-host infections. This is because it is currently not easy to determine precisely where and how in the life cycles of vector-borne disease systems the key constrains which are regarded as crucial in regulating pathogen population dynamics in both the vertebrate host and vector host operate. In this article, we present a general method for development of multiscale models of vector-borne disease systems which integrate the within-host and between-host scales for the two hosts (a vertebrate host and a vector host) that are implicated in vector-borne disease dynamics. The general multiscale modelling method is an extension of our previous work on multiscale models of infectious disease systems which established a basic science and accompanying theory of how pathogen population dynamics at within-host scale scales up to between-host scale and in turn how it scales down from between-host scale to within-host scale. Further, the general method is applied to multiscale modelling of human onchocerciasis-a vector-borne disease system which is sometimes called river blindness as a case study.

Transition from endemic behavior to eradication of malaria due to combined drug therapies: An agent-model approach

We introduce an agent-based model describing a susceptible-infectious-susceptible (SIS) system of humans and mosquitoes to predict malaria epidemiological scenarios in realistic biological conditions. Emphasis is given to the transition from endemic behavior to eradication of malaria transmission induced by combined drug therapies acting on both the gametocytemia reduction and on the selective mosquito mortality during parasite development in the mosquito. Our mathematical framework enables to uncover the critical values of the parameters characterizing the effect of each drug therapy. Moreover, our results provide quantitative evidence of what was up to now only partially assumed with empirical support: interventions combining gametocytemia reduction through the use of gametocidal drugs, with the selective action of ivermectin during parasite development in the mosquito, may actively promote disease eradication in the long run. In the agent model, the main properties of human-mosquito interactions are implemented as parameters and the model is validated by comparing simulations with real data of malaria incidence collected in the endemic malaria region of Chimoio in Mozambique. Finally, we discuss our findings in light of current drug administration strategies for malaria prevention, which may interfere with human-to-mosquito transmission process.

Within-host modeling of blood-stage malaria

Malaria infection continues to be a major health problem worldwide and drug resistance in the major human parasite species, Plasmodium falciparum, is increasing in South East Asia. Control measures including novel drugs and vaccines are in development, and contributions to the rational design and optimal usage of these interventions are urgently needed. Infection involves the complex interaction of parasite dynamics, host immunity, and drug effects. The long life cycle (48 hours in the common human species) and synchronized replication cycle of the parasite population present significant challenges to modeling the dynamics of Plasmodium infection. Coupled with these, variation in immune recognition and drug action at different life cycle stages leads to further complexity. We review the development and progress of "within-host" models of Plasmodium infection, and how these have been applied to understanding and interpreting human infection and animal models of infection.

Linked within-host and between-host models and data for infectious diseases: a systematic review

The observed dynamics of infectious diseases are driven by processes across multiple scales. Here we focus on two: within-host, that is, how an infection progresses inside a single individual (for instance viral and immune dynamics), and between-host, that is, how the infection is transmitted between multiple individuals of a host population. The dynamics of each of these may be influenced by the other, particularly across evolutionary time. Thus understanding each of these scales, and the links between them, is necessary for a holistic understanding of the spread of infectious diseases. One approach to combining these scales is through mathematical modeling. We conducted a systematic review of the published literature on multi-scale mathematical models of disease transmission (as defined by combining within-host and between-host scales) to determine the extent to which mathematical models are being used to understand across-scale transmission, and the extent to which these models are being confronted with data. Following the PRISMA guidelines for systematic reviews, we identified 24 of 197 qualifying papers across 30 years that include both linked models at the within and between host scales and that used data to parameterize/calibrate models. We find that the approach that incorporates both modeling with data is under-utilized, if increasing. This highlights the need for better communication and collaboration between modelers and empiricists to build well-calibrated models that both improve understanding and may be used for prediction.

A coupled multiscale model to guide malaria control and elimination

In this paper, we share with the biomathematics community a new coupled multiscale model which has the potential to inform policy and guide malaria control and elimination. The formulation of this multiscale model is based on integrating four submodels which are: (i) a sub-model for the mosquito-to-human transmission of malaria parasite, (ii) a sub-model for the human-to-mosquito transmission of malaria parasite, (iii) a within-mosquito malaria parasite population dynamics sub-model and (iv) a within-human malaria parasite population dynamics sub-model. The integration of the four submodels is achieved by assuming that the transmission parameters of the sub-model for the mosquito-to-human transmission of malaria at the epidemiological scale are functions of the dependent variables of the within-mosquito sporozoite population dynamics while the transmission parameters of the sub-model for the human-to-mosquito transmission of malaria are functions of the dependent variables of the within-human gametocyte population dynamics. This establishes a unidirectionally coupled multiscale model where the within-human and within-mosquito submodels are unidirectionally coupled to the human-to-mosquito and mosquito-to-human submodels. A fast and slow time scale analysis is performed on this system. The result is a simple multiscale model which describes the mechanics of malaria transmission in terms of the major components of the complete malaria parasite life-cycle. This multiscale modelling approach may be found useful in guiding malaria control and elimination.

Agent-based models of malaria transmission: a systematic review

BACKGROUND

Much of the extensive research regarding transmission of malaria is underpinned by mathematical modelling. Compartmental models, which focus on interactions and transitions between population strata, have been a mainstay of such modelling for more than a century. However, modellers are increasingly adopting agent-based approaches, which model hosts, vectors and/or their interactions on an individual level. One reason for the increasing popularity of such models is their potential to provide enhanced realism by allowing system-level behaviours to emerge as a consequence of accumulated individual-level interactions, as occurs in real populations.

METHODS

A systematic review of 90 articles published between 1998 and May 2018 was performed, characterizing agent-based models (ABMs) relevant to malaria transmission. The review provides an overview of approaches used to date, determines the advantages of these approaches, and proposes ideas for progressing the field.

RESULTS

The rationale for ABM use over other modelling approaches centres around three points: the need to accurately represent increased stochasticity in low-transmission settings; the benefits of high-resolution spatial simulations; and heterogeneities in drug and vaccine efficacies due to individual patient characteristics. The success of these approaches provides avenues for further exploration of agent-based techniques for modelling malaria transmission. Potential extensions include varying elimination strategies across spatial landscapes, extending the size of spatial models, incorporating human movement dynamics, and developing increasingly comprehensive parameter estimation and optimization techniques.

CONCLUSION

Collectively, the literature covers an extensive array of topics, including the full spectrum of transmission and intervention regimes. Bringing these elements together under a common framework may enhance knowledge of, and guide policies towards, malaria elimination. However, because of the diversity of available models, endorsing a standardized approach to ABM implementation may not be possible. Instead it is recommended that model frameworks be contextually appropriate and sufficiently described. One key recommendation is to develop enhanced parameter estimation and optimization techniques. Extensions of current techniques will provide the robust results required to enhance current elimination efforts.

A complete categorization of multiscale models of infectious disease systems

Modelling of infectious disease systems has entered a new era in which disease modellers are increasingly turning to multiscale modelling to extend traditional modelling frameworks into new application areas and to achieve higher levels of detail and accuracy in characterizing infectious disease systems. In this paper we present a categorization framework for categorizing multiscale models of infectious disease systems. The categorization framework consists of five integration frameworks and five criteria. We use the categorization framework to give a complete categorization of host-level immuno-epidemiological models (HL-IEMs). This categorization framework is also shown to be applicable in categorizing other types of multiscale models of infectious diseases beyond HL-IEMs through modifying the initial categorization framework presented in this study. Categorization of multiscale models of infectious disease systems in this way is useful in bringing some order to the discussion on the structure of these multiscale models.

Model-Informed Drug Development for Malaria Therapeutics

Malaria is a critical public health problem resulting in substantial morbidity and mortality, particularly in developing countries. Owing to the development of resistance toward current therapies, novel approaches to accelerate the development efforts of new malaria therapeutics are urgently needed. There have been significant advancements in the development of in vitro and in vivo experiments that generate data used to inform decisions about the potential merit of new compounds. A comprehensive disease-drug model capable of integrating discrete data from different preclinical and clinical components would be a valuable tool across all stages of drug development. This could have an enormous impact on the otherwise slow and resource-intensive process of traditional clinical drug development.

Dynamics of Plasmodium vivax sporogony in wild Anopheles stephensi in a malaria-endemic region of Western India

Background

In global efforts to track mosquito infectivity and parasite elimination, controlled mosquito-feeding experiments can help in understanding the dynamics of parasite development in vectors. Anopheles stephensi is often accepted as the major urban malaria vector that transmits Plasmodium in Goa and elsewhere in South Asia. However, much needs to be learned about the interactions of Plasmodium vivax with An. stephensi. As a component of the US NIH International Center of Excellence for Malaria Research (ICEMR) for Malaria Evolution in South Asia (MESA), a series of membrane-feeding experiments with wild An. stephensi and P. vivax were carried out to better understand this vector-parasite interaction.

Methods

Wild An. stephensi larvae and pupae were collected from curing water in construction sites in the city of Ponda, Goa, India. The larvae and pupae were reared at the MESA ICEMR insectary within the National Institute of Malaria Research (NIMR) field unit in Goa until they emerged into adult mosquitoes. Blood for membrane-feeding experiments was obtained from malaria patients at the local Goa Medical College and Hospital who volunteered for the study. Parasites were counted by Miller reticule technique and correlation between gametocytaemia/parasitaemia and successful mosquito infection was studied.

Results

A weak but significant correlation was found between patient blood gametocytaemia/parasitaemia and mosquito oocyst load. No correlation was observed between gametocytaemia/parasitaemia and oocyst infection rates, and between gametocyte sex ratio and oocyst load. When it came to development of the parasite in the mosquito, a strong positive correlation was observed between oocyst midgut levels and sporozoite infection rates, and between oocyst levels and salivary gland sporozoite loads. Kinetic studies showed that sporozoites appeared in the salivary gland as early as day 7, post-infection.

Conclusions

This is the first study in India to carry out membrane-feeding experiments with wild An. stephensi and P. vivax. A wide range of mosquito infection loads and infection rates were observed, pointing to a strong interplay between parasite, vector and human factors. Most of the present observations are in agreement with feeding experiments conducted with P. vivax elsewhere in the world.

Electronic supplementary material

The online version of this article (doi:10.1186/s12936-017-1931-8) contains supplementary material, which is available to authorized users.

Dissecting the determinants of malaria chronicity: why within-host models struggle to reproduce infection dynamics

The duration of infection is fundamental to the epidemiological behaviour of any infectious disease, but remains one of the most poorly understood aspects of malaria. In endemic areas, the malaria parasite Plasmodium falciparum can cause both acute, severe infections and asymptomatic, chronic infections through its interaction with the host immune system. Frequent superinfection and massive parasite genetic diversity make it extremely difficult to accurately measure the distribution of infection lengths, complicating the estimation of basic epidemiological parameters and the prediction of the impact of interventions. Mathematical models have qualitatively reproduced parasite dynamics early during infection, but reproducing long-lived chronic infections remains much more challenging. Here, we construct a model of infection dynamics to examine the consequences of common biological assumptions for the generation of chronicity and the impact of co-infection. We find that although a combination of host and parasite heterogeneities are capable of generating chronic infections, they do so only under restricted parameter choices. Furthermore, under biologically plausible assumptions, co-infection of parasite genotypes can alter the course of infection of both the resident and co-infecting strain in complex non-intuitive ways. We outline the most important puzzles for within-host models of malaria arising from our analysis, and their implications for malaria epidemiology and control.

Comparison of mathematical frameworks for modeling erythropoiesis in the context of malaria infection

Malaria is an infectious disease present all around the globe and responsible for half a million deaths per year. A within-host model of this infection requires a framework capable of properly approximating not only the blood stage of the infection but also the erythropoietic process that is in charge of overcoming the malaria induced anemia. Within this context, we compare ordinary differential equations (ODEs) with and without age classes, delayed differential equations (DDEs), and discrete recursive equations (DREs) with age classes. Results show that ODEs without age classes are fair approximations that do not provide a crisp temporal representation of the processes involved, and inclusion of age classes only mitigates the problem to some degree. DDEs perform well with respect to generating the essentially fixed delay between cell production and cell removal due to age, but the inclusion of any other processes, such as sudden blood loss, becomes cumbersome. The framework that was found to perform best in representing the dynamics of red blood cells during malaria infection is a DRE with age classes. In this model structure, the amount of time a cell remains alive is easily controlled, and the addition of age dependent or independent processes is straightforward. All events that populations of cells face during their lifespan, like growth or adaptation in differentiation or maturation rate, are properly represented in this framework.

Experimental study of the relationship between Plasmodium gametocyte density and infection success in mosquitoes; implications for the evaluation of malaria transmission-reducing interventions

The evaluation of transmission reducing interventions (TRI) to control malaria widely uses membrane feeding assays. In such assays, the intensity of Plasmodium infection in the vector might affect the measured efficacy of the candidates to block transmission. Gametocyte density in the host blood is a determinant of the infection success in the mosquito, however, uncertain estimates of parasite densities and intrinsic characteristics of the infected blood can induce variability. To reduce this variation, a feasible method is to dilute infectious blood samples. We describe the effect of diluting samples of Plasmodium-containing blood samples to allow accurate relative measures of gametocyte densities and their impact on mosquito infectivity and TRI efficacy. Natural Plasmodium falciparum samples were diluted to generate a wide range of parasite densities, and fed to Anopheles coluzzii mosquitoes. This was compared with parallel dilutions conducted on Plasmodium berghei infections. We examined how blood dilution influences the observed blocking activity of anti-Pbs28 monoclonal antibody using the P. berghei/Anopheles stephensi system. In the natural species combination P. falciparum/An. coluzzii, blood dilution using heat-inactivated, infected blood as diluents, revealed positive near linear relationships, between gametocyte densities and oocyst loads in the range tested. A similar relationship was observed in the P. berghei/An. stephensi system when using a similar dilution method. In contrast, diluting infected mice blood with fresh uninfected blood dramatically increases the infectiousness. This suggests that highly infected mice blood contains inhibitory factors or reduced blood moieties, which impede infection and may in turn, lead to misinterpretation when comparing individual TRI evaluation assays. In the lab system, the transmission blocking activity of an antibody specific for Pbs28 was confirmed to be density-dependent. This highlights the need to carefully interpret evaluations of TRI candidates, regarding gametocyte densities in the P. berghei/An. stephensi system.

Cross-reactive immune responses as primary drivers of malaria chronicity

The within-host dynamics of an infection with the malaria parasite Plasmodium falciparum are the result of a complex interplay between the host immune system and parasite. Continual variation of the P. falciparum erythrocyte membrane protein (PfEMP1) antigens displayed on the surface of infected red blood cells enables the parasite to evade the immune system and prolong infection. Despite the importance of antigenic variation in generating the dynamics of infection, our understanding of the mechanisms by which antigenic variation generates long-term chronic infections is still limited. We developed a model to examine the role of cross-reactivity in generating infection dynamics that are comparable to those of experimental infections. The hybrid computational model we developed is attuned to the biology of malaria by mixing discrete replication events, which mimics the synchrony of parasite replication and invasion, with continuous interaction with the immune system. Using simulations, we evaluated the dynamics of a single malaria infection over time. We then examined three major mechanisms by which the dynamics of a malaria infection can be structured: cross-reactivity of the immune response to PfEMP1, differences in parasite clearance rates, and heterogeneity in the rate at which antigens switch. The results of our simulations demonstrate that cross-reactive immune responses play a primary role in generating the dynamics observed in experimentally untreated infections and in lengthening the period of infection. Importantly, we also find that it is the primary response to the initially expressed PfEMP1, or small subset thereof, that structures the cascading cross-immune dynamics and allows for elongation of the infection.

Modeling the human infectious reservoir for malaria control: does heterogeneity matter?

The complex biological relationships underlying malaria transmission make it difficult to predict the impact of interventions. Mathematical models simplify these relationships and capture essential components of malaria transmission and epidemiology. Models designed to predict the impact of control programs generally infer a relationship between transmission intensity and human infectiousness to the mosquito, requiring assumptions about how infectiousness varies between individuals. A lack of understanding of human infectiousness precludes a standard approach to this inference, however, and field data reveal no obvious correlation between transmission intensity and human population infectiousness. We argue that model assumptions will have important consequences for predicting the impact of control programs.

Linking the antigen archive structure to pathogen fitness in African trypanosomes

Systems that generate antigenic variation enable pathogens to evade host immune responses and are intricately interwoven with major pathogen traits, such as host choice, growth, virulence and transmission. Although much is understood about antigen switching at the molecular level, little is known about the cross-scale links between these molecular processes and the larger-scale within and between host population dynamics that they must ultimately drive. Inspired by the antigenic variation system of African trypanosomes, we apply modelling approaches to our expanding understanding of the organization and expression of antigen repertoires, and explore links across these scales. We predict how pathogen population processes are determined by underlying molecular genetics and infer resulting selective pressures on important emergent repertoire traits.

Pierre TG, Davis TME. Mathematical modeling of malaria infection with innate and adaptive immunity in individuals and agent-based communities

Background

Agent-based modeling of Plasmodium falciparum infection offers an attractive alternative to the conventional Ross-Macdonald methodology, as it allows simulation of heterogeneous communities subjected to realistic transmission (inoculation patterns).

Methodology/Principal Findings

We developed a new, agent based model that accounts for the essential in-host processes: parasite replication and its regulation by innate and adaptive immunity. The model also incorporates a simplified version of antigenic variation by Plasmodium falciparum. We calibrated the model using data from malaria-therapy (MT) studies, and developed a novel calibration procedure that accounts for a deterministic and a pseudo-random component in the observed parasite density patterns. Using the parasite density patterns of 122 MT patients, we generated a large number of calibrated parameters. The resulting data set served as a basis for constructing and simulating heterogeneous agent-based (AB) communities of MT-like hosts. We conducted several numerical experiments subjecting AB communities to realistic inoculation patterns reported from previous field studies, and compared the model output to the observed malaria prevalence in the field. There was overall consistency, supporting the potential of this agent-based methodology to represent transmission in realistic communities.

Conclusions/Significance

Our approach represents a novel, convenient and versatile method to model Plasmodium falciparum infection.

The within-host dynamics of malaria infection with immune response

Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value R0, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if R0 < 1. If R0 > 1, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.

Mathematical models of malaria--a review

Mathematical models have been used to provide an explicit framework for understanding malaria transmission dynamics in human population for over 100 years. With the disease still thriving and threatening to be a major source of death and disability due to changed environmental and socio-economic conditions, it is necessary to make a critical assessment of the existing models, and study their evolution and efficacy in describing the host-parasite biology. In this article, starting from the basic Ross model, the key mathematical models and their underlying features, based on their specific contributions in the understanding of spread and transmission of malaria have been discussed. The first aim of this article is to develop, starting from the basic models, a hierarchical structure of a range of deterministic models of different levels of complexity. The second objective is to elaborate, using some of the representative mathematical models, the evolution of modelling strategies to describe malaria incidence by including the critical features of host-vector-parasite interactions. Emphasis is more on the evolution of the deterministic differential equation based epidemiological compartment models with a brief discussion on data based statistical models. In this comprehensive survey, the approach has been to summarize the modelling activity in this area so that it helps reach a wider range of researchers working on epidemiology, transmission, and other aspects of malaria. This may facilitate the mathematicians to further develop suitable models in this direction relevant to the present scenario, and help the biologists and public health personnel to adopt better understanding of the modelling strategies to control the disease.

Synchronous versus asynchronous oscillations for antigenically varying Plasmodium falciparum with host immune response

We consider a deterministic intra-host model for Plasmodium falciparum (Pf) malaria infection, which accounts for antigenic variation between n clonal variants of PfEMP1 and the corresponding host immune response (IR). Specifically, the model separates the IR into two components, specific and cross-reactive, respectively, in order to demonstrate that the latter can be a mechanism for the sequential appearance of variants observed in actual Pf infections. We show that a strong variant-specific IR relative to the cross-reactive IR favours the asynchronous oscillations (sequential dominance) over the synchronous oscillations in a number of ways. The decay rate of asynchronous oscillations is smaller than that for the synchronous oscillations, allowing for the parasite to survive longer. With the introduction of a delay in the stimulation of the IR, we show that only a small delay is necessary to cause persistent asynchronous oscillations and that a strong variant-specific IR increases the amplitude of the asynchronous oscillations.

A research agenda to underpin malaria eradication

The interruption of malaria transmission worldwide is one of the greatest challenges for international health and development communities. The current expert view suggests that, by aggressively scaling up control with currently available tools and strategies, much greater gains could be achieved against malaria, including elimination from a number of countries and regions; however, even with maximal effort we will fall short of global eradication. The Malaria Eradication Research Agenda (malERA) complements the current research agenda--primarily directed towards reducing morbidity and mortality--with one that aims to identify key knowledge gaps and define the strategies and tools that will result in reducing the basic reproduction rate to less than 1, with the ultimate aim of eradication of the parasite from the human population. Sustained commitment from local communities, civil society, policy leaders, and the scientific community, together with a massive effort to build a strong base of researchers from the endemic areas will be critical factors in the success of this new agenda.

Modelling malaria population structure and its implications for control

Mathematical models of malaria transmission have been used to inform the design of malaria control programs since the mid 20th century, and many of these models have provided useful insights into the complexity of the disease. Among developing countries, however and particularly in sub-Saharan Africa, malaria remains a major cause of morbidity and mortality. One of the main difficulties in controlling the most virulent human malaria parasite, Plasmodium falciparum, is its genetic diversity, which confounds attempts to design an effective vaccine. The population structure of P. falciparum remains poorly understood but plays a key role in determining epidemiological patterns of disease and the development of immunity. We discuss the seminal model of malaria transmission developed by Ross and MacDonald, and the modifications that have been made since to include more realism. We show that age profiles of disease and serological data support a theoretical model in which the parasite population is diverse and structured into several antigenic types and highlight the implications of this structure for controlling malaria. Lastly, we discuss the current sequence data on parasite antigen genes that are important for the aquisition of immunity, and the results of a new analysis of P. falciparum population structure at the genomic level.

Uncertainty in mapping malaria epidemiology: implications for control

Malaria is a location-specific, dynamic infectious disease transmitted by mosquitoes to humans and is influenced by environmental, vector, parasite, and host factors. The principal purposes of malarial epidemiology are 1) to describe the malarial distribution in space and time along with the physical, biologic, and social etiologic factors and 2) to guide control objectives for either modeling impact or measuring progress of control tactics. Mapping malaria and many of its causative factors has been achieved on many different levels from global distribution to biologic quantitative trait localization in humans, parasites, and mosquitoes. Despite these important achievements, a large degree of uncertainty still exists on the annual burden of malarial cases. Accurate, sensitive detection and treatment of asymptomatic reservoirs important to infectious transmission are additional components necessary for future control measures. Presently spurred by the leadership and funding of Bill and Melinda Gates, the malarial community is developing and implementing plans for elimination of malaria. The challenge for malariologists is to digitally integrate and map epidemiologic factors and intervention measures in space and time to target effective, sustainable control alongside research efforts.

An intra-host mathematical model on interaction between HIV and malaria

In this paper a mathematical model is proposed for the interaction of the immune system with HIV viruses and malaria parasites in an individual host. It consists of a system of three coupled ordinary differential equations, which represents the rate of change in the concentration of malaria parasites, HIV viruses and immunity effector within a host, respectively. The theoretical model gives insight into the biological balance between pathogen replication and the immune response to the pathogen: persistence versus elimination of the pathogen, which determines the outcome of infection. Dynamical analysis shows that the outcomes of the interactions between the immune system of the host with either malaria parasites or HIV viruses are dramatic such as malaria infection promoting proliferation of HIV virus, HIV infection increasing the risk from malaria and the immune system of the host failing to keep the diseases under control, etc. The results provide a new perspective for understanding of the complexity mechanisms of the co-infection (or dual infection) with malaria and HIV in a host.

Random, top-down, or bottom-up coexistence of parasites: malaria population dynamics in multi-parasitic settings

Epidemiological models concerned with the control of malaria using interventions such as bed nets and vaccines increasingly incorporate realistic aspects of malaria biology. The increasing complexity of these models limits their ability to abstract ecological processes and to address questions on the regulation of population dynamics using time-series data, particularly in regards to interactions between different pathogens and the regulatory role of innate (bottom-up) and acquired (top-down) immunity. We use a theoretical framework to test hypotheses on the importance of population-level immunity and parasite abundance in regulating the population dynamics of malaria. We use qualitative loop analyses to examine the sign of the interaction between Plasmodium falciparum and P. vivax at the population level, and we discuss implications of this sign for the within-host regulation of parasites. Our analyses of monthly malaria time-series data from the island of Espirito Santo, Vanuatu (1983-1997), show that the dynamics of P. falciparum are not sensitive to P. vivax, whereas infections by the latter increase in response to those of the former. These results support a differential use of resources inside the hosts, a resource-consumer interaction between hosts and their immune system, and within-host regulation of parasites. Finally, our results emphasize the need to better understand factors regulating malaria dynamics before developing control strategies and call for the use of control strategies directed at the interruption of transmission, such as vector control and the use of bed nets.

Predicting changing malaria risk after expanded insecticide-treated net coverage in Africa

The Roll Back Malaria (RBM) partnership has established goals for protecting vulnerable populations with locally appropriate vector control. In many places, these goals will be achieved by the mass distribution of insecticide treated bednets (ITNs). Mathematical models can forecast an ITN-driven realignment of malaria endemicity, defined by the Plasmodium falciparum parasite rate (PfPR) in children, to predict PfPR endpoints and appropriate program timelines for this change in Africa. The relative ease of measuring PfPR and its widespread use make it particularly suitable for monitoring and evaluation. This theory provides a method for context-dependent evaluation of ITN programs and a basis for setting rational ITN coverage targets over the next decade.

Malaria model with periodic mosquito birth and death rates

In this paper, we introduce a model of malaria, a disease that involves a complex life cycle of parasites, requiring both human and mosquito hosts. The novelty of the model is the introduction of periodic coefficients into the system of one-dimensional equations, which account for the seasonal variations (wet and dry seasons) in the mosquito birth and death rates. We define a basic reproduction number R(0) that depends on the periodic coefficients and prove that if R(0)<1 then the disease becomes extinct, whereas if R(0)>1 then the disease is endemic and may even be periodic.

Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by R(0)), is less than unity. Further, it has a unique endemic equilibrium if R(0) > 1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by nabla) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by R(vac)) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if R(vac) is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.

Impact of antimicrobial usage on the transmission dynamics of antimicrobial resistant bacteria among pigs

There is increasing evidence showing that antimicrobial consumption provides a powerful selective force that promotes the emergence of resistance in pathogenic, commensal as well as zoonotic bacteria in animals. The main aim of this study was to develop a modeling framework that can be used to assess the impact of antimicrobial usage in pigs on the emergence and transmission of resistant bacteria within a finisher pig farm. The transmission dynamics of drug-sensitive and drug-resistant bacteria among pigs in the herd were characterized by studying the local and global stability properties of steady state solutions of the system. Numerical simulations demonstrating the influence of factors such as initial prevalence of infection, presence of pre-existing antimicrobial resistant mutants, and frequency of treatment on predicted prevalence were performed. Sensitivity analysis revealed that two parameters had a huge influence on the predicted proportion of pigs carrying resistant bacteria: (a) the transmission coefficient between uninfected pigs and those infected with drug-resistant bacteria during treatment (beta(2)) and after treatment stops (beta(3)), and (b) the spontaneous clear-out rate of drug-resistant bacteria during treatment (gamma(2)) and immediately after treatment stops (gamma(3)). Control measures should therefore be geared towards reducing the magnitudes of beta(2) and beta(3) or increasing those of gamma(2) and gamma(3).

Host control of malaria infections: constraints on immune and erythropoeitic response kinetics

The two main agents of human malaria, Plasmodium vivax and Plasmodium falciparum, can induce severe anemia and provoke strong, complex immune reactions. Which dynamical behaviors of host immune and erythropoietic responses would foster control of infection, and which would lead to runaway parasitemia and/or severe anemia? To answer these questions, we developed differential equation models of interacting parasite and red blood cell (RBC) populations modulated by host immune and erythropoietic responses. The model immune responses incorporate both a rapidly responding innate component and a slower-responding, long-term antibody component, with several parasite developmental stages considered as targets for each type of immune response. We found that simulated infections with the highest parasitemia tended to be those with ineffective innate immunity even if antibodies were present. We also compared infections with dyserythropoiesis (reduced RBC production during infection) to those with compensatory erythropoiesis (boosted RBC production) or a fixed basal RBC production rate. Dyserythropoiesis tended to reduce parasitemia slightly but at a cost to the host of aggravating anemia. On the other hand, compensatory erythropoiesis tended to reduce the severity of anemia but with enhanced parasitemia if the innate response was ineffective. For both parasite species, sharp transitions between the schizont and the merozoite stages of development (i.e., with standard deviation in intra-RBC development time <or=2.4 h) were associated with lower parasitemia and less severe anemia. Thus tight synchronization in asexual parasite development might help control parasitemia. Finally, our simulations suggest that P. vivax can induce severe anemia as readily as P. falciparum for the same type of immune response, though P. vivax attacks a much smaller subset of RBCs. Since most P. vivax infections are nonlethal (if debilitating) clinically, this suggests that P. falciparum adaptations for countering or evading immune responses are more effective than those of P. vivax.

Towards a comprehensive simulation model of malaria epidemiology and control

Planning of the control of Plasmodium falciparum malaria leads to a need for models of malaria epidemiology that provide realistic quantitative prediction of likely epidemiological outcomes of a wide range of control strategies. Predictions of the effects of control often ignore medium- and long-term dynamics. The complexities of the Plasmodium life-cycle, and of within-host dynamics, limit the applicability of conventional deterministic malaria models. We use individual-based stochastic simulations of malaria epidemiology to predict the impacts of interventions on infection, morbidity, mortality, health services use and costs. Individual infections are simulated by stochastic series of parasite densities, and naturally acquired immunity acts by reducing densities. Morbidity and mortality risks, and infectiousness to vectors, depend on parasite densities. The simulated infections are nested within simulations of individuals in human populations, and linked to models of interventions and health systems. We use numerous field datasets to optimise parameter estimates. By using a volunteer computing system we obtain the enormous computational power required for model fitting, sensitivity analysis, and exploration of many different intervention strategies. The project thus provides a general platform for comparing, fitting, and evaluating different model structures, and for quantitative prediction of effects of different interventions and integrated control programmes.

Modelling malaria pathogenesis

Almost 20 years after the development of models of malaria pathogenesis began, we are beyond the 'proof-of-concept' phase and these models are no longer abstract mathematical exercises. They have refined our knowledge of within-host processes, and have brought insights that could not easily have been obtained from experimentation alone. There is much potential that remains to be realized, however, both in terms of informing the design of interventions and health policy, and in terms of addressing lingering questions about the basic biology of malaria. Recent research has begun to iterate theory and data in a much more comprehensive way, and the use of statistical techniques for model fitting and comparison offers a promising approach for providing a quantitative understanding of the pathogenesis of such a complex disease.

Development of a two-dimensional agent-based model for chronic chagasic cardiomyopathy after stem cell transplantation

MOTIVATION

A significant issue in stem cell therapy is to understand the role of this type of cell in the tissue regeneration. To explain this mechanism, an experimental study has quantified that the bone marrow cell transplantation decreases the number of inflammatory cells and reduces the fibrosis area in chagasic mice. Using this experimental data, we have developed an agent-based computational model to investigate the regeneration of the chronic chagasic cardiomyopathy after bone marrow stem cell transplantation.

RESULTS

Our model includes six different types of agents: inflammatory cell, fibrosis area, cardiomyocyte, proinflammatory cytokine tumor necrosis factor-alpha, Trypanosoma cruzi parasite and bone marrow stem cell. This latter promotes apoptosis in inflammatory cells, reduction in the fibrosis area and can differentiate into cardiomyocyte. Proinflammatory cytokine tumor necrosis factor-alpha can increase the fibrosis area and T.cruzi can increase the number of inflammatory cells. Our results for both apoptosis of inflammatory cells and reduction in the fibrosis area were compared with experimental data. They suggest that the concentration pattern is the most important factor to characterize the kinetics of cardiac tissue regeneration after bone marrow stem cell transplantation.

AVAILABILITY

The source code of our software is available online at www.vivas.ufba.br/bone/bone.zip .br Supplementaty information: Supplementary data are available at Bioinformatics online.

On the delayed Ross-Macdonald model for malaria transmission

The feedback dynamics from mosquito to human and back to mosquito involve considerable time delays due to the incubation periods of the parasites. In this paper, taking explicit account of the incubation periods of parasites within the human and the mosquito, we first propose a delayed Ross-Macdonald model. Then we calculate the basic reproduction number R0 and carry out some sensitivity analysis of R0 on the incubation periods, that is, to study the effect of time delays on the basic reproduction number. It is shown that the basic reproduction number is a decreasing function of both time delays. Thus, prolonging the incubation periods in either humans or mosquitos (via medicine or control measures) could reduce the prevalence of infection.

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 stochastic model of immune-modulated malaria infection and disease in children

We develop a simple three-state stochastic description of individual malaria infections that relates dynamics of disease and immune status to age and previous exposure, under different intensities of transmission. We apply the resulting individual-based community models to examine the effects of drug treatment and vaccination on the frequency and severity of disease in ensembles of children. The several broad qualitative similarities between our results and field observations include potential rebound effects following intervals of drug treatment.

Mathematical model of the first wave of Plasmodium falciparum asexual parasitemia in non-immune and vaccinated individuals

We present a dynamic model of the highly pathogenic first wave of Plasmodium falciparum asexual parasitemia in non-immune persons. The model was successfully fitted to malaria therapy data. This required four case-specific parameters: the basic two-day multiplication factor, the time of onset of adaptive immunity, and the effective dose 50 densities for the innate and adaptive immune responses, respectively. All four parameters show large case-dependent variation that is mainly attributable to host factors. According to the model, the maximum value of the first wave is controlled mainly by the innate immune response. We used the model to explore the expected effects of vaccines targeting the parasite's asexual blood stages on the basis of what we consider to be the biologically most plausible assumptions concerning the parameter modifications induced by vaccination. According to our simulations, the benefit of antiparasitic vaccination is strongly host dependent and vaccine efficacy at low immunogenicity is much larger against severe disease than against fever. This has implications for the early testing of the protective efficacy of a vaccine in humans.