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

Investigating the dose-dependency of the midgut escape barrier using a mechanistic model of within-mosquito dengue virus population dynamics

Arboviruses can emerge rapidly and cause explosive epidemics of severe disease. Some of the most epidemiologically important arboviruses, including dengue virus (DENV), Zika virus (ZIKV), Chikungunya (CHIKV) and yellow fever virus (YFV), are transmitted by Aedes mosquitoes, most notably Aedes aegypti and Aedes albopictus. After a mosquito blood feeds on an infected host, virus enters the midgut and infects the midgut epithelium. The virus must then overcome a series of barriers before reaching the mosquito saliva and being transmitted to a new host. The virus must escape from the midgut (known as the midgut escape barrier; MEB), which is thought to be mediated by transient changes in the permeability of the midgut-surrounding basal lamina layer (BL) following blood feeding. Here, we present a mathematical model of the within-mosquito population dynamics of DENV (as a model system for mosquito-borne viruses more generally) that includes the interaction of the midgut and BL which can account for the MEB. Our results indicate a dose-dependency of midgut establishment of infection as well as rate of escape from the midgut: collectively, these suggest that the extrinsic incubation period (EIP)-the time taken for DENV virus to be transmissible after infection-is shortened when mosquitoes imbibe more virus. Additionally, our experimental data indicate that multiple blood feeding events, which more closely mimic mosquito-feeding behavior in the wild, can hasten the course of infections, and our model predicts that this effect is sensitive to the amount of virus imbibed. Our model indicates that mutations to the virus which impact its replication rate in the midgut could lead to even shorter EIPs when double-feeding occurs. Mechanistic models of within-vector viral infection dynamics provide a quantitative understanding of infection dynamics and could be used to evaluate novel interventions that target the mosquito stages of the infection.

Investigating the dose-dependency of the midgut escape barrier using a mechanistic model of within-mosquito dengue virus population dynamics

Flaviviruses are arthropod-borne (arbo)viruses which can emerge rapidly and cause explosive epidemics of severe disease. Some of the most epidemiologically important flaviviruses, including dengue virus (DENV), Zika virus (ZIKV) and yellow fever virus (YFV), are transmitted by Aedes mosquitoes, most notably Aedes aegypti and Aedes albopictus. After a mosquito blood feeds on an infected host, virus enters the midgut and infects the midgut epithelium. The virus must then overcome a series of barriers before reaching the mosquito saliva and being transmitted to a new host. The virus must escape from the midgut (known as the midgut escape barrier; MEB), which is thought to be mediated by transient changes in the permeability of the midgut-surrounding basal lamina layer (BL) following blood feeding. Here, we present a mathematical model of the within-mosquito population dynamics of flaviviruses that includes the interaction of the midgut and BL which can account for the MEB. Our results indicate a dose-dependency of midgut establishment of infection as well as rate of escape from the midgut: collectively, these suggest that the extrinsic incubation period (EIP) - the time taken for DENV virus to be transmissible after infection - is shortened when mosquitoes imbibe more virus. Additionally, our experimental data indicates that multiple blood feeding events, which more closely mimic mosquito-feeding behavior in the wild, can hasten the course of infections, and our model predicts that this effect is sensitive to the amount of virus imbibed. Our model indicates that mutations to the virus which impact its replication rate in the midgut could lead to even shorter EIPs when double-feeding occurs. Mechanistic models of within-vector viral infection dynamics provide a quantitative understanding of infection dynamics and could be used to evaluate novel interventions that target the mosquito stages of the infection.

Author summary

Aedes mosquitoes are the main vectors of dengue virus (DENV), Zika virus (ZIKV) and yellow fever virus (YFV), all of which can cause severe disease in humans with dengue alone infecting an estimated 100-400 million people each year. Understanding the processes that affect whether, and at which rate, mosquitoes may transmit such viruses is, hence, paramount. Here, we present a mathematical model of virus dynamics within infected mosquitoes. By combining the model with novel experimental data, we show that the course of infection is sensitive to the initial dose of virus ingested by the mosquito. The data also indicates that mosquitoes which blood feed subsequent to becoming infected may be able to transmit infection earlier, which is reproduced in the model. This is important as many mosquito species feed multiple times during their lifespan and, any reduction in time to dissemination will increase the number of days that a mosquito is infectious and so enhance the risk of transmission. Our study highlights the key and complementary roles played by mathematical models and experimental data for understanding within-mosquito virus dynamics.

Targeting malaria parasites inside mosquitoes: ecoevolutionary consequences

Proof-of-concept studies demonstrate that antimalarial drugs designed for human treatment can also be applied to mosquitoes to interrupt malaria transmission. Deploying a new control tool is ideally undertaken within a stewardship programme that maximises a drug's lifespan by minimising the risk of resistance evolution and slowing its spread once emerged. We ask: what are the epidemiological and evolutionary consequences of targeting parasites within mosquitoes? Our synthesis argues that targeting parasites inside mosquitoes (i) can be modelled by readily expanding existing epidemiological frameworks; (ii) provides a functionally novel control method that has potential to be more robust to resistance evolution than targeting parasites in humans; and (iii) could extend the lifespan and clinical benefit of antimalarials used exclusively to treat humans.

Modeling cellular co-infection and reassortment of bluetongue virus in Culicoides midges

When related segmented RNA viruses co-infect a single cell, viral reassortment can occur, potentially leading to new strains with pandemic potential. One virus capable of reassortment is bluetongue virus (BTV), which causes substantial health impacts in ruminants and is transmitted via Culicoides midges. Because midges can become co-infected by feeding on multiple different host species and remain infected for their entire life span, there is a high potential for reassortment to occur. Once a midge is co-infected, additional barriers must be crossed for a reassortant virus to emerge, such as cellular co-infection and dissemination of reassortant viruses to the salivary glands. We developed three mathematical models of within-midge BTV dynamics of increasing complexity, allowing us to explore the conditions leading to the emergence of reassortant viruses. In confronting the simplest model with published data, we estimate that the average life span of a bluetongue virion in the midge midgut is about 6 h, a key determinant of establishing a successful infection. Examination of the full model, which permits cellular co-infection and reassortment, shows that small differences in fitness of the two infecting strains can have a large impact on the frequency with which reassortant virions are observed. This is consistent with experimental co-infection studies with BTV strains with different relative fitnesses that did not produce reassortant progeny. Our models also highlight several gaps in existing data that would allow us to elucidate these dynamics in more detail, in particular the times it takes the virus to disseminate to different tissues, and measurements of viral load and reassortant frequency at different temperatures.

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 realized 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.

Bluetongue Research at a Crossroads: Modern Genomics Tools Can Pave the Way to New Insights

Bluetongue virus (BTV) is an arthropod-borne, segmented double-stranded RNA virus that can cause severe disease in both wild and domestic ruminants. BTV evolves via several key mechanisms, including the accumulation of mutations over time and the reassortment of genome segments.Additionally, BTV must maintain fitness in two disparate hosts, the insect vector and the ruminant. The specific features of viral adaptation in each host that permit host-switching are poorly characterized. Limited field studies and experimental work have alluded to the presence of these phenomena at work, but our understanding of the factors that drive or constrain BTV's genetic diversification remains incomplete. Current research leveraging novel approaches and whole genome sequencing applications promises to improve our understanding of BTV's evolution, ultimately contributing to the development of better predictive models and management strategies to reduce future impacts of bluetongue epizootics.

Estimating the extrinsic incubation period of malaria using a mechanistic model of sporogony

During sporogony, malaria-causing parasites infect a mosquito, reproduce and migrate to the mosquito salivary glands where they can be transmitted the next time blood feeding occurs. The time required for sporogony, known as the extrinsic incubation period (EIP), is an important determinant of malaria transmission intensity. The EIP is typically estimated as the time for a given percentile, x, of infected mosquitoes to develop salivary gland sporozoites (the infectious parasite life stage), which is denoted by EIPx. Many mechanisms, however, affect the observed sporozoite prevalence including the human-to-mosquito transmission probability and possibly differences in mosquito mortality according to infection status. To account for these various mechanisms, we present a mechanistic mathematical model, which explicitly models key processes at the parasite, mosquito and observational scales. Fitting this model to experimental data, we find greater variation in the EIP than previously thought: we estimated the range between EIP10 and EIP90 (at 27°C) as 4.5 days compared to 0.9 days using existing statistical methods. This pattern holds over the range of study temperatures included in the dataset. Increasing temperature from 21°C to 34°C decreased the EIP50 from 16.1 to 8.8 days. Our work highlights the importance of mechanistic modelling of sporogony to (1) improve estimates of malaria transmission under different environmental conditions or disease control programs and (2) evaluate novel interventions that target the mosquito life stages of the parasite.

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.

The impact of within-vector parasite development on the extrinsic incubation period

Mosquito-borne diseases, in particular malaria, have a significant burden worldwide leading to nearly half a million deaths each year. The malaria parasite requires a vertebrate host, such as a human, and a vector host, the Anopheles mosquito, to complete its full life cycle. Here, we focus on the parasite dynamics within the vector to examine the first appearance of sporozoites in the salivary glands, which indicates a first time of infectiousness of mosquitoes. The timing of this period of pathogen development in the mosquito until transmissibility, known as the extrinsic incubation period, remains poorly understood. We develop compartmental models of within-mosquito parasite dynamics fitted with experimental data on oocyst and sporozoite counts. We find that only a fraction of oocysts burst to release sporozoites and bursting must be delayed either via a time-dependent function or a gamma-distributed set of compartments. We use Bayesian inference to estimate distributions of parameters and determine that bursting rate is a key epidemiological parameter. A better understanding of the factors impacting the extrinsic incubation period will aid in the development of interventions to slow or stop the spread of malaria.

A MATHEMATICAL STUDY OF THE IMPLICIT ROLE OF INNATE AND ADAPTIVE IMMUNE RESPONSES ON WITHIN-HUMAN PLASMODIUM FALCIPARUM PARASITE LEVELS

A within-human-host malaria parasite model, integrating key variables that influence parasite evolution-progression-advancement, under innate and adaptive immune responses, is analyzed. The implicit role of immunity on the steady state parasite loads and parasitemia reproduction number ([Formula: see text]), a threshold parameter measuring the parasite’s annexing ability of healthy red blood cells (HRBCs), eventually rendering a human infectious to mosquitoes, is investigated. The impact of the type of recruitment function used to model HRBC growth is also investigated. The model steady states and [Formula: see text], both obtained as functions of immune system variables, are analyzed at snapshots of immune sizes. Model results indicate that the more the immune cells, innate and adaptive, the more efficient they are at inhibiting parasite development and progression; consequently, the less severe the malaria disease in a patient. Our analysis also illustrates the existence of a Hopf bifurcation leading to a limit cycle, observable only for the nonlinear recruitment functions, at reasonably large [Formula: see text].

The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria

This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates, and also from the random delays of the incubation and immunity periods. Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state, and also for the disease to remain permanently in the system over time. Moreover, the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined, and the statistical characteristics of the distribution are given numerically. The results of this study show that the disease will persist and become permanent in the system, regardless of (1) whether the noises are from the disease transmission rate and/or from the natural death rates or (2) whether the delays in the system are constant or random for individuals in the system. Furthermore, it is shown that “weak” noise is associated with the existence of an endemic stationary distribution for the disease, while “strong” noise is associated with extinction of the population over time. Numerical simulation examples for Plasmodium vivax malaria are given.

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.

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.

Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment

A family of deterministic SEIRS epidemic dynamic models for malaria is presented. The family type is determined by a general functional response for the nonlinear incidence rate of the disease. Furthermore, the malaria models exhibit three random delays — the incubation periods of the plasmodium inside the female mosquito and human hosts, and also the period of effective acquired natural immunity against the disease. Insights about the effects of the delays and the nonlinear incidence rate of the disease on (1) eradication and (2) persistence of malaria in the human population are obtained via analyzing and interpreting the global asymptotic stability results of the disease-free and endemic equilibrium of the system. The basic reproduction numbers and other threshold values for malaria are calculated, and superior threshold conditions for the stability of the equilibria are found. Numerical simulation results are presented.

Simulating within-vector generation of the malaria parasite diversity

Plasmodium falciparum, the most virulent human malaria parasite, undergoes asexual reproduction within the human host, but reproduces sexually within its vector host, the Anopheles mosquito. Consequently, the mosquito stage of the parasite life cycle provides an opportunity to create genetically novel parasites in multiply-infected mosquitoes, potentially increasing parasite population diversity. Despite the important implications for disease transmission and malaria control, a quantitative mapping of how parasite diversity entering a mosquito relates to diversity of the parasite exiting, has not been undertaken. To examine the role that vector biology plays in modulating parasite diversity, we develop a two-part model framework that estimates the diversity as a consequence of different bottlenecks and expansion events occurring during the vector-stage of the parasite life cycle. For the underlying framework, we develop the first stochastic model of within-vector P. falciparum parasite dynamics and go on to simulate the dynamics of two parasite subpopulations, emulating multiply infected mosquitoes. We show that incorporating stochasticity is essential to capture the extensive variation in parasite dynamics, particularly in the presence of multiple parasites. In particular, unlike deterministic models, which always predict the most fit parasites to produce the most sporozoites, we find that occasionally only parasites with lower fitness survive to the sporozoite stage. This has important implications for onward transmission. The second part of our framework includes a model of sequence diversity generation resulting from recombination and reassortment between parasites within a mosquito. Our two-part model framework shows that bottlenecks entering the oocyst stage decrease parasite diversity from what is present in the initial gametocyte population in a mosquito’s blood meal. However, diversity increases with the possibility for recombination and proliferation in the formation of sporozoites. Furthermore, when we begin with two parasite subpopulations in the initial gametocyte population, the probability of transmitting more than two unique parasites from mosquito to human is over 50% for a wide range of initial gametocyte densities.

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.

Electronic supplementary material

The online version of this article (doi:10.1186/1475-2875-13-428) contains supplementary material, which is available to authorized users.

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.

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.