Investigating the impact of multiple feeding attempts on mosquito dynamics via mathematical models.

A deterministic differential equation model for the dynamics of terrestrial forms of mosquito populations is studied. The model assesses the impact of multiple probing attempts by mosquitoes that quest for blood within human populations by including a waiting class for mosquitoes that failed a blood feeding attempt. The equations are derived based on the idea that the reproductive cycle of the mosquito can be viewed as a set of alternating egg laying and blood feeding outcomes realised on a directed path called the gonotrophic cycle pathway. There exists a threshold parameter, the basic offspring number for mosquitoes, whose nature is affected by the way we interpret the transitions involving the different classes on the gonotrophic cycle path. The trivial steady state for the system, which always exists, can be globally asymptomatically stable whenever the threshold parameter is less than unity. The non-trivial steady state, when it exists, is stable for a range of values of the threshold parameter but can also be driven to instability via a Hopf bifurcation. The model's output reveals that the waiting class mosquitoes do contribute positively to sustain mosquito populations as well as increase their interactions with humans via increased frequency and initial amplitude of oscillations. We conclude that to understand human-mosquito interactions, it is informative to consider multiple probing attempts; known to occur when mosquitoes quest for blood meals within human populations.

Mathematical assessment of the impact of human-antibodies on sporogony during the within-mosquito dynamics of Plasmodium falciparum parasites.

We develop and analyze a deterministic ordinary differential equation mathematical model for the within-mosquito dynamics of the Plasmodium falciparum malaria parasite. Our model takes into account the action and effect of blood resident human-antibodies, ingested by the mosquito during a blood meal from humans, in inhibiting gamete fertilization. The model also captures subsequent developmental processes that lead to the different forms of the parasite within the mosquito. Continuous functions are used to model the switching transition from oocyst to sporozoites as well as human antibody density variations within the mosquito gut are proposed and used. In sum, our model integrates the developmental stages of the parasite within the mosquito such as gametogenesis, fertilization and sporogenesis culminating in the formation of sporozoites. Quantitative and qualitative analyses including a sensitivity analysis for influential parameters are performed. We quantify the average sporozoite load produced at the end of the within-mosquito malaria parasite's developmental stages. Our analysis shows that an increase in the efficiency of the ingested human antibodies in inhibiting fertilization within the mosquito's gut results in lowering the density of oocysts and hence sporozoites that are eventually produced by each mosquito vector. So, it is possible to control and limit oocysts development and hence sporozoites development within a mosquito by boosting the efficiency of antibodies as a pathway to the development of transmission-blocking vaccines which could potentially reduce oocysts prevalence among mosquitoes and hence reduce the transmission potential from mosquitoes to human.

On a three-stage structured model for the dynamics of malaria transmission with human treatment, adult vector demographics and one aquatic stage.

A modelling framework that describes the dynamics of populations of the female Anopheles sp mosquitoes is used to develop and analyse a deterministic ordinary differential equation model for dynamics and transmission of malaria amongst humans and varying mosquito populations. The framework includes a characterization of the gonotrophic cycle of the female mosquito. The epidemiological model also captures a novel feature whereby treated human's blood can become mosquitocidal to the questing mosquitoes upon the successful ingestion of the treated human's blood. Analysis of the disease free system, that is the model in the absence of infection in the human and mosquito populations, reveals the presence of a basic offspring number, N, whose size determines the existence and stability of a thriving mosquito population in the sense that when N≤1 we have only the mosquito extinction steady state which is globally asymptotically stable, while for N > 1 we have the persistent mosquito population steady state which is also globally asymptotically stable for these range of values of N. In the presence of disease, N still strongly affects the properties of the epidemiological model in the sense that for N≤1 the only steady state for the system is the mosquito extinction steady state, which is globally and asymptotically stable. As N increases beyond unity in the epidemiological model, we obtained the epidemiological basic reproduction number, R0. For R0 < 1, the disease free equilibrium, with both healthy thriving susceptible human and mosquito populations, is globally asymptotically stable. Both N and R0 are studied for control purposes and our study highlights that multiple control schemes would have a stronger impact on reducing both N and R0 to values small enough for a possible disease vector control and disease eradication. Our model further illustrates that newly emerged mosquitoes that are infected with the malaria parasite during their first blood meal play an important and strong role in the malaria disease dynamics. Additionally, mosquitoes at later gonotrophic cycle stages also impact the dynamics but their contributions to the total mosquito population size decreases with increasing number of gonotrophic cycles. The size of the contribution into the young mosquito population is also dependent on the length of the gonotrophic cycles, an important bionomic parameter, as well as on how the mosquitoes at the final gonotrophic cycles are incorporated into the modelling scheme.

The Impact of Recruitment on the Dynamics of an Immune-Suppressed Within-Human-Host Model of the Plasmodium falciparum Parasite.

A model is developed and used to study within-human malaria parasite dynamics. The model integrates actors involved in the development-progression of parasitemia, gametocytogenesis and mechanisms for immune response activation. Model analyses under immune suppression reveal different dynamical behaviours for different healthy red blood cell (HRBC) generation functions. Existence of a threshold parameter determines conditions for HRBCs depletion. Oscillatory dynamics reminiscent of malaria parasitemia are obtained. A dependence exists on the type of recruitment function used to generate HRBCs, with complexities observed for a more nonlinear function. An upper bound that delimits the size of feasible parasitized steady-state solution exists for a logistic function but not a constant function. The upper bound is completely characterized and is affected by parameters associated with HRBCs recruitment, parasitized red blood cells generation and the release and time-to-release of free merozoites. A stable density size for mature gametocytes, the bridge to invertebrate hosts, is derived.

Intermittent Preventive Treatment (IPT): Its Role in Averting Disease-Induced Mortality in Children and in Promoting the Spread of Antimalarial Drug Resistance.

We develop an age-structured ODE model to investigate the role of intermittent preventive treatment (IPT) in averting malaria-induced mortality in children, and its related cost in promoting the spread of antimalarial drug resistance. IPT, a malaria control strategy in which a full curative dose of an antimalarial medication is administered to vulnerable asymptomatic individuals at specified intervals, has been shown to reduce malaria transmission and deaths in children and pregnant women. However, it can also promote drug resistance spread. Our mathematical model is used to explore IPT effects on drug resistance and deaths averted in holoendemic malaria regions. The model includes drug-sensitive and drug-resistant strains as well as human hosts and mosquitoes. The basic reproduction, and invasion reproduction numbers for both strains are derived. Numerical simulations show the individual and combined effects of IPT and treatment of symptomatic infections on the prevalence of both strains and the number of lives saved. Our results suggest that while IPT can indeed save lives, particularly in high transmission regions, certain combinations of drugs used for IPT and to treat symptomatic infection may result in more deaths when resistant parasite strains are circulating. Moreover, the half-lives of the treatment and IPT drugs used play an important role in the extent to which IPT may influence spread of the resistant strain. A sensitivity analysis indicates the model outcomes are most sensitive to the reduction factor of transmission for the resistant strain, rate of immunity loss, and the natural clearance rate of sensitive infections.

A Mathematical Model with Quarantine States for the Dynamics of Ebola Virus Disease in Human Populations.

A deterministic ordinary differential equation model for the dynamics and spread of Ebola Virus Disease is derived and studied. The model contains quarantine and nonquarantine states and can be used to evaluate transmission both in treatment centres and in the community. Possible sources of exposure to infection, including cadavers of Ebola Virus victims, are included in the model derivation and analysis. Our model's results show that there exists a threshold parameter, R 0, with the property that when its value is above unity, an endemic equilibrium exists whose value and size are determined by the size of this threshold parameter, and when its value is less than unity, the infection does not spread into the community. The equilibrium state, when it exists, is locally and asymptotically stable with oscillatory returns to the equilibrium point. The basic reproduction number, R 0, is shown to be strongly dependent on the initial response of the emergency services to suspected cases of Ebola infection. When intervention measures such as quarantining are instituted fully at the beginning, the value of the reproduction number reduces and any further infections can only occur at the treatment centres. Effective control measures, to reduce R 0 to values below unity, are discussed.

Evolutionary implications for the determination of gametocyte sex ratios under fecundity variation for the malaria parasite.

We investigate sex ratio determination strategies for the Malaria parasite based on putative changes in its male fecundity over the lifetime of an infection, and how such strategies might have evolved. We model fitness using the incomplete fertilization limit developed in Teboh-Ewungkem and Yuster (2010). We divide the infection lifetime of a strain into two periods, assume each human is infected by two different strains, and assume that there are two different strategies present among the many strains in the general malaria parasite population. A unique parameter dependent ESS exists for all parameter values in both of our main models, with many such strategies unbeatable. These strategies produce both male and female biased population sex ratios with female bias predominating over most of the parameter space. The first model (SKM) suggests that strains without the ability to detect characteristics of other strains present could still have evolved strategies to vary sex ratio over their lifetimes, and the second model (DKM) suggests strains with detection abilities might have evolved after that. Our analysis suggests that once the ability to detect the population sizes and fecundities of other strains has developed, detection of their sex ratio choices confers no additional selective advantage in that a DKM ESS is still an ESS among sex ratio detecting strategies. The sex ratio choices for each DKM ESS are given by the equilibrium values of the parameter equivalent sex ratio detecting strategy described in Teboh-Ewungkem and Wang (2012), in the case where two strains employing that strategy encounter each other.

Mathematical assessment of the effect of traditional beliefs and customs on the transmission dynamics of the 2014 Ebola outbreaks.

BACKGROUND: Ebola is one of the most virulent human viral diseases, with a case fatality ratio between 25% to 90%. The 2014 West African outbreaks are the largest and worst in history. There is no specific treatment or effective/safe vaccine against the disease. Hence, control efforts are restricted to basic public health preventive (non-pharmaceutical) measures. Such efforts are undermined by traditional/cultural belief systems and customs, characterized by general mistrust and skepticism against government efforts to combat the disease. This study assesses the roles of traditional customs and public healthcare systems on the disease spread. METHODS: A mathematical model is designed and used to assess population-level impact of basic non-pharmaceutical control measures on the 2014 Ebola outbreaks. The model incorporates the effects of traditional belief systems and customs, along with disease transmission within health-care settings and by Ebola-deceased individuals. A sensitivity analysis is performed to determine model parameters that most affect disease transmission. The model is parameterized using data from Guinea, one of the three Ebola-stricken countries. Numerical simulations are performed and the parameters that drive disease transmission, with or without basic public health control measures, determined. Three effectiveness levels of such basic measures are considered. RESULTS: The distribution of the basic reproduction number ([Formula: see text]) for Guinea (in the absence of basic control measures) is such that [Formula: see text], for the case when the belief systems do not result in more unreported Ebola cases. When such systems inhibit control efforts, the distribution increases to [Formula: see text]. The total Ebola cases are contributed by Ebola-deceased individuals (22%), symptomatic individuals in the early (33%) and latter (45%) infection stages. A significant reduction of new Ebola cases can be achieved by increasing health-care workers' daily shifts from 8 to 24 hours, limiting hospital visitation to 1 hour and educating the populace to abandon detrimental traditional/cultural belief systems. CONCLUSIONS: The 2014 outbreaks are controllable using a moderately-effective basic public health intervention strategy alone. A much higher (>50%) disease burden would have been recorded in the absence of such intervention. 2000 Mathematics Subject Classifications 92B05, 93A30, 93C15.

Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito populations: implications for control.

We derive and study a deterministic compartmental model for malaria transmission with varying human and mosquito populations. Our model considers disease-related deaths, asymptomatic immune humans who are also infectious, as well as mosquito demography, reproduction and feeding habits. Analysis of the model reveals the existence of a backward bifurcation and persistent limit cycles whose period and size is determined by two threshold parameters: the vectorial basic reproduction number Rm, and the disease basic reproduction number R0, whose size can be reduced by reducing Rm. We conclude that malaria dynamics are indeed oscillatory when the methodology of explicitly incorporating the mosquito's demography, feeding and reproductive patterns is considered in modeling the mosquito population dynamics. A sensitivity analysis reveals important control parameters that can affect the magnitudes of Rm and R0, threshold quantities to be taken into consideration when designing control strategies. Both Rm and the intrinsic period of oscillation are shown to be highly sensitive to the mosquito's birth constant λm and the mosquito's feeding success probability pw. Control of λm can be achieved by spraying, eliminating breeding sites or moving them away from human habitats, while pw can be controlled via the use of mosquito repellant and insecticide-treated bed-nets. The disease threshold parameter R0 is shown to be highly sensitive to pw, and the intrinsic period of oscillation is also sensitive to the rate at which reproducing mosquitoes return to breeding sites. A global sensitivity and uncertainty analysis reveals that the ability of the mosquito to reproduce and uncertainties in the estimations of the rates at which exposed humans become infectious and infectious humans recover from malaria are critical in generating uncertainties in the disease classes.

The effect of intermittent preventive treatment on anti-malarial drug resistance spread in areas with population movement.

BACKGROUND: The use of intermittent preventive treatment in pregnant women (IPTp), children (IPTc) and infant (IPTi) is an increasingly popular preventive strategy aimed at reducing malaria risk in these vulnerable groups. Studies to understand how this preventive intervention can affect the spread of anti-malarial drug resistance are important especially when there is human movement between neighbouring low and high transmission areas. Because the same drug is sometimes utilized for IPTi and for symptomatic malaria treatment, distinguishing their individual roles on accelerating the spread of drug resistant malaria, with or without human movement, may be difficult to isolate experimentally or by analysing data. A theoretical framework, as presented here, is thus relevant as the role of IPTi on accelerating the spread of drug resistance can be isolated in individual populations and when the populations are interconnected and interact. METHODS: A previously published model is expanded to include human movement between neighbouring high and low transmission areas, with focus placed on the malaria parasites. Parasite fitness functions, determined by how many humans the parasites can infect, are used to investigate how fast resistance can spread within the neighbouring communities linked by movement, when the populations are at endemic equilibrium. RESULTS: Model simulations indicate that population movement results in resistance spreading fastest in high transmission areas, and the more complete the anti-malarial resistance the faster the resistant parasite will tend to spread through a population. Moreover, the demography of infection in low transmission areas tends to change to reflect the demography of high transmission areas. Additionally, when regions are strongly connected the rate of spread of partially resistant parasites (R1) relative to drug sensitive parasites (RS), and fully resistant parasites (R2) relative to partially resistant parasites (R1) tend to behave the same in both populations, as should be expected. CONCLUSIONS: In fighting anti-malarial drug resistance, different drug resistance monitoring and management policies are needed when the area in question is an isolated high or low transmission area, or when it is close and interacting with a neighbouring high or low transmission area, with human movement between them.

On a reproductive stage-structured model for the population dynamics of the malaria vector.

A reproductive stage-structured deterministic differential equation model for the population dynamics of the human malaria vector is derived and analysed. The model captures the gonotrophic and behavioural life characteristics of the female Anopheles sp. mosquito and takes into consideration the fact that for the purposes of reproduction, the female Anopheles sp. mosquito must visit and bite humans (or animals) to harvest necessary proteins from blood that it needs for the development of its eggs. Focusing on mosquitoes that feed exclusively on humans, our results indicate the existence of a threshold parameter, the vectorial reproduction number, whose size increases with increasing number of gonotrophic cycles, and is also affected by the female mosquito's birth rate, its attraction and visitation rate to human residences, and its contact rate with humans. A stability analysis of the model indicates that the mosquito can establish itself in the environment if and only if the value of the vectorial reproduction number exceeds unity and that mosquito eradication is possible if the vectorial reproduction number is less than unity, since, then, the trivial steady state which always exist is unique and is globally and asymptotically stable. When a persistent vector population steady state exists, it is locally and asymptotically stable for a range of reproduction numbers, but can also be driven to instability via a Hopf bifurcation as the reproduction number increases further away from unity. The model derivation identifies and characterizes control parameters relating to activities such as human-mosquito contact and the mosquito's survival chances between blood meals and egg laying. Our results show that the total mosquito population size increases with increasing number of gonotrophic cycles. Therefore understanding the fundamental aspects of the mosquito's behaviour provides a pathway for the study of human-mosquito contact and mosquito population control. Control of the mosquito population densities would ultimately lead to malaria control.

Male fecundity and optimal gametocyte sex ratios for Plasmodium falciparum during incomplete fertilization.

A mathematical model of the within-vector dynamics of Plasmodium falciparum in an Anopheles mosquito with unbiased random mating and incomplete fertilization is used to investigate the effects of varying fecundity and population size on the gametocyte sex ratio when strains maximize their individual fitnesses. Previous studies considered either the effects of variable fecundity or the effects of incomplete fertilization. Here we investigate the simultaneous effects of variable fecundity, incomplete fertilization, and variable number of ingested gametocytes per strain on the optimal gametocyte sex ratio. Our model results agree with others in the case of two identically sized populations in that large differences in fecundities lead to female-biased total population sex ratios. When the assumption of identically sized populations is relaxed in our model, population sex ratios vary from highly female-biased to slightly male-biased, depending on relative strain fecundities and population sizes. Our results provide a plausible explanation for the high variation in gametocyte sex ratios of P. falciparum observed in nature.

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.

A within-vector mathematical model of Plasmodium falciparum and implications of incomplete fertilization on optimal gametocyte sex ratio.

A mathematical model that simulates the within-vector dynamics of Plasmodium falciparum in an Anopheles mosquito is developed, based on experimental data. The model takes a mosquito's blood meal as input and computes the salivary gland sporozoite load as the final output, a probable measure of mosquito infectivity. Computational model results are consistent with observed results in nature. Sensitivity analysis of the model parameters suggests that reducing the gametocyte density in the blood meal most significantly lowers sporozoite load in the salivary glands and hence mosquito infectivity, and is thus an attractive target for malaria control. The model is used to investigate the implication of incomplete fertilization on optimal gametocyte sex ratio. For a single strain, the transition from complete fertilization to increasingly incomplete fertilization shifts that ratio from 1 to N, where N is the number of viable male gametes produced by a single male gametocyte, towards 1 to 1, which is demonstrated to be the limiting ratio analytically. This ratio is then shown to be an evolutionarily stable strategy as well in the limiting case.

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.

The role of counter-current exchange in preventing hypoxia in skeletal muscle.

Mathematical models that describe oxygen transport from a single capillary into a region of surrounding tissue often predict that the tissue is hypoxic, whereas in reality diffusion from more richly perfused nearby capillaries prevents hypoxia from forming in the tissue. In this manuscript, a mathematical model of oxygen transport is presented that is applicable to vascular beds consisting of a large number of non-uniformly perfused parallel capillaries arranged in a manner characteristic of skeletal muscle. The model is used to examine conditions under which counter-current flow and myoglobin-facilitated diffusion provides sufficient oxygen to poorly perfused regions to prevent the occurrence of hypoxia. The method developed here leads to a coupled system of nonlinear ordinary differential equations for the oxygen concentration in the capillaries, and is easy to apply even for vascular beds containing a large number of capillaries.