On the Exact Measure of Disease Spread in Stochastic Epidemic Models

The basic reproduction number, R ₀, is probably the most important quantity in epidemiology. It is used to measure the transmission potential during the initial phase of an epidemic. In this paper, we are specifically concerned with the quantification of the spread of a disease modeled by a Markov chain. Due to the occurrence of repeated contacts taking place between a typical infective individual and other individuals already infected before, R ₀ overestimates the average number of secondary infections. We present two alternative measures, namely, the exact reproduction number, R _e0, and the population transmission number, R _p , that overcome this difficulty and provide valuable insight. The applicability of R _e0 and R _p to control of disease spread is also examined.     

Impact of group mixing on disease dynamics

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀<1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀>1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.     

Self-sustained oscillations in epidemic models with infective immigrants

A relevant issue related to eco-epidemiological studies concerns the demographic mechanisms that can lead to self-sustained oscillations in the composition of a host population subject to infection. In particular, why does the prevalence of some contagious diseases oscillate over time? Here, we address this question by using susceptible-infective-recovered-empty models including migration of infective foreigners and variable population size. These models are described in terms of ordinary differential equations (ODE) and also in terms of probabilistic cellular automaton (PCA), in which each cell is connected to others either by a regular lattice or by a random graph favoring local contacts. Each cell in the PCA model can be either empty or occupied by a single individual. The amount of neighbors per cell affects the value of the basic reproduction number R₀, which is, in fact, a bifurcation parameter. We show that, by varying the amount of neighbors per cell (and consequently R₀), the number of infective individuals can start to exhibit periodic behavior, which corresponds to a Hopf bifurcation in the ODE model. This bifurcation gives rise to a self-sustained oscillation and it can only occur if the immigration rate of infective individuals is above a critical value. We also investigate how the sum of new infections, within the considered time window, depends on the number of neighbors per cell.     

The control of vector-borne disease epidemics

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, R₀, the measure of long term endemicity. Despite its widespread application, R₀ does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, E₀, which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on E₀ and R₀ may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector-host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index E₀ to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.     

A model of bi-mode transmission dynamics of hepatitis C with optimal control

In this paper, we present a rigorous mathematical analysis of a deterministic model for the transmission dynamics of hepatitis C. The model is suitable for populations where two frequent modes of transmission of hepatitis C virus, namely unsafe blood transfusions and intravenous drug use, are dominant. The susceptible population is divided into two distinct compartments, the intravenous drug users and individuals undergoing unsafe blood transfusions. Individuals belonging to each compartment may develop acute and then possibly chronic infections. Chronically infected individuals may be quarantined. The analysis indicates that the eradication and persistence of the disease is completely determined by the magnitude of basic reproduction number R _c. It is shown that for the basic reproduction number R _c < 1, the disease-free equilibrium is locally and globally asymptotically stable. For R _c > 1, an endemic equilibrium exists and the disease is uniformly persistent. In addition, we present the uncertainty and sensitivity analyses to investigate the influence of different important model parameters on the disease prevalence. When the infected population persists, we have designed a time-dependent optimal quarantine strategy to minimize it. The Pontryagin’s Maximum Principle is used to characterize the optimal control in terms of an optimality system which is solved numerically. Numerical results for the optimal control are compared against the constant controls and their efficiency is discussed.     

An epidemic model with post-contact prophylaxis of distributed length I. Thresholds for disease persistence and extinction

A possible control strategy against the spread of an infectious disease is the treatment with antimicrobials that are given prophylactically to those who had a contact with an infective person. The treatment continues until recovery or until it becomes obvious that there was no infection in the first place. The model considers susceptible, treated uninfected exposed, treated infected, (untreated) infectious, and recovered individuals. Since treatment lengths have an arbitrary distribution, the model system consists of ordinary differential and integral equations. A sharp threshold condition is established in terms of a basic replacement ratio (disease reproduction number) that separates disease extinction from uniform disease persistence. We use results from dynamical systems persistence theory proving the existence of a global compact attractor along the way. Existence and multiplicity of endemic equilibria are also studied.     

Modeling Relapsing Disease Dynamics in a Host-Vector Community

Vector-borne diseases represent a threat to human and wildlife populations and mathematical models provide a means to understand and control epidemics involved in complex host-vector systems. The disease model studied here is a host-vector system with a relapsing class of host individuals, used to investigate tick-borne relapsing fever (TBRF). Equilibrium analysis is performed for models with increasing numbers of relapses and multiple hosts and the disease reproduction number, R₀, is generalized to establish relationships with parameters that would result in the elimination of the disease. We show that host relapses in a single competent host-vector system is needed to maintain an endemic state. We show that the addition of an incompetent second host with no relapses increases the number of relapses needed for maintaining the pathogen in the first competent host system. Further, coupling of the system with hosts of differing competencies will always reduce R₀, making it more difficult for the system to reach an endemic state.     

Contact rate calculation for a basic epidemic model

Mass-action epidemic models are the foundation of the majority of studies of disease dynamics in human and animal populations. Here, a kinetic model of mobile susceptible and infective individuals in a two-dimensional domain is introduced, and an examination of the contact process results in a mass-action-like term for the generation of new infectives. The conditions under which density dependent and frequency dependent transmission terms emerge are clarified. Moreover, this model suggests that epidemics in large mobile spatially distributed populations can be well described by homogeneously mixing mass-action models. The analysis generates an analytic formula for the contact rate (β) and the basic reproductive ratio (R₀) of an infectious pathogen, which contains a mixture of demographic and epidemiological parameters. The analytic results are compared with a simulation and are shown to give good agreement. The simulation permits the exploration of more realistic movement strategies and their consequent effect on epidemic dynamics.     

Predicting and Evaluating the Epidemic Trend of Ebola Virus Disease in the 2014-2015 Outbreak and the Effects of Intervention Measures

We constructed dynamic Ebola virus disease (EVD) transmission models to predict epidemic trends and evaluate intervention measure efficacy following the 2014 EVD epidemic in West Africa. We estimated the effective vaccination rate for the population, with basic reproduction number (R₀) as the intermediate variable. Periodic EVD fluctuation was analyzed by solving a Jacobian matrix of differential equations based on a SIR (susceptible, infective, and removed) model. A comprehensive compartment model was constructed to fit and predict EVD transmission patterns, and to evaluate the effects of control and prevention measures. Effective EVD vaccination rates were estimated to be 42% (31–50%), 45% (42–48%), and 51% (44–56%) among susceptible individuals in Guinea, Liberia and Sierra Leone, respectively. In the absence of control measures, there would be rapid mortality in these three countries, and an EVD epidemic would be likely recur in 2035, and then again 8~9 years later. Oscillation intervals would shorten and outbreak severity would decrease until the periodicity reached ~5.3 years. Measures that reduced the spread of EVD included: early diagnosis, treatment in isolation, isolating/monitoring close contacts, timely corpse removal, post-recovery condom use, and preventing or quarantining imported cases. EVD may re-emerge within two decades without control and prevention measures. Mass vaccination campaigns and control and prevention measures should be instituted to prevent future EVD epidemics.     

Malaria's Missing Number: Calculating the Human Component of R0 by a Within-Host Mechanistic Model of Plasmodium falciparum Infection and Transmission

Human infection by malarial parasites of the genus Plasmodium begins with the bite of an infected Anopheles mosquito. Current estimates place malaria mortality at over 650,000 individuals each year, mostly in African children. Efforts to reduce disease burden can benefit from the development of mathematical models of disease transmission. To date, however, comprehensive modeling of the parameters defining human infectivity to mosquitoes has remained elusive. Here, we describe a mechanistic within-host model of Plasmodium falciparum infection in humans and pathogen transmission to the mosquito vector. Our model incorporates the entire parasite lifecycle, including the intra-erythrocytic asexual forms responsible for disease, the onset of symptoms, the development and maturation of intra-erythrocytic gametocytes that are transmissible to Anopheles mosquitoes, and human-to-mosquito infectivity. These model components were parameterized from malaria therapy data and other studies to simulate individual infections, and the ensemble of outputs was found to reproduce the full range of patient responses to infection. Using this model, we assessed human infectivity over the course of untreated infections and examined the effects in relation to transmission intensity, expressed by the basic reproduction number R₀ (defined as the number of secondary cases produced by a single typical infection in a completely susceptible population). Our studies predict that net human-to-mosquito infectivity from a single non-immune individual is on average equal to 32 fully infectious days. This estimate of mean infectivity is equivalent to calculating the human component of malarial R₀. We also predict that mean daily infectivity exceeds five percent for approximately 138 days. The mechanistic framework described herein, made available as stand-alone software, will enable investigators to conduct detailed studies into theories of malaria control, including the effects of drug treatment and drug resistance on transmission.     

The Effect of Myofilament Compliance on Kinetics of Force Generation by Myosin Motors in Muscle

We use the inhibitor of isometric force of skeletal muscle N-benzyl-p-toluene sulfonamide (BTS) to decrease, in a dose dependent way, the number of myosin motors attached to actin during the steady isometric contraction of single fibers from frog skeletal muscle (4°C, 2.1 μm sarcomere length). In this way we can reduce the strain in the myofilament compliance during the isometric tetanus (T₀) from 3.54 nm in the control solution (T_0,NR) to ∼0.5 nm in 1 μM BTS, where T₀ is reduced to ∼0.15 T_0,NR. The quick force recovery after a step release (1-3 nm per half-sarcomere) becomes faster with the increase of BTS concentration and the decrease of T₀. The simulation of quick force recovery with a multistate model of force generation, that adapts Huxley and Simmons model to account for both the high stiffness of the myosin motor (∼3 pN/nm) and the myofilament compliance, shows that the increase in the rate of quick force recovery by BTS is explained by the reduced strain in the myofilaments, consequent to the decrease in half-sarcomere force. The model estimates that i), for the same half-sarcomere release the state transition kinetics in the myosin motor are five times faster in the absence of filament compliance than in the control; and ii), the rate of force recovery from zero to T₀ is ∼6000/s in the absence of filament compliance.     

Dynamics of plant mosaic disease propagation and the usefulness of roguing as an alternative biological control

In this research article, an epidemiological model is formulated for mosaic disease considering plant and vector populations. Plant host population has been divided into three compartments namely healthy, latently infected and infected ones, and vector population is divided into two compartments: non-infective and infective vectors. The system possesses three equilibria: plant-only, disease-free and endemic equilibrium. Plant-only equilibrium is always unstable; disease-free equilibrium is stable when the basic reproduction number, R₀, is less than unity and unstable for when it crosses unity, and ensure existence of an endemic equilibrium which may be stable or can undergo a Hopf bifurcation. Finally, impulse periodic roguing with varied rate and time interval is adopted for cost effective and eco-friendly disease control and future direction of agriculture management. The dynamics of the impulsive system has also been analysed. Detailed numerical simulations are employed to support the analytical results. We found that roguing is most cost effective and useful management for mosaic disease eradication of plants if applied at proper rate and interval.     

Optimal control strategies and cost-effectiveness analysis of a malaria model

The aim of this paper is to investigate the effectiveness and cost-effectiveness of three malaria preventive measures (use of treated bednets, spray of insecticides and a possible treatment of infective humans that blocks transmission to mosquitoes). For this, we consider a mathematical model for the transmission dynamics of the disease that includes these measures. We first consider the constant control parameters’ case, we calculate the basic reproduction number and investigate the existence and stability of equilibria; the model is found to exhibit backward bifurcation. We then assess the relative impact of each of the constant control parameters measures by calculating the sensitivity index of the basic reproductive number to the model's parameters. In the time-dependent constant control case, we use Pontryagin's Maximum Principle to derive necessary conditions for the optimal control of the disease. We also calculate the Infection Averted Ratio (IAR) and the Incremental Cost-Effectiveness Ratio (ICER) to investigate the cost-effectiveness of all possible combinations of the three control measures. One of our findings is that the most cost-effective strategy for malaria control, is the combination of the spray of insecticides and treatment of infective individuals. This strategy requires a 100% effort in both treatment (for 20 days) and spray of insecticides (for 57 days). In practice, this will be extremely difficult, if not impossible to achieve. The second most cost-effective strategy which consists of a 100% use of treated bednets and 87% treatment of infective individuals for 42 and 100 days, respectively, is sustainable and therefore preferable.     

Genetic management of infectious diseases: a heterogeneous epidemio-genetic model illustrated with S. aureus mastitis

Given that individuals are genetically heterogeneous in their degree of resistance to infection, a model is proposed to formulate appropriate choices that will limit the spread of an infectious disease. The model is illustrated with data on S. aureus mastitis and is based on parameters characterizing the spread of the disease (contact rate, probability of infection after contact, and rate of recovery after infection), the demography (replacement and culling rates) and the genetic composition (degree of relationship and heritability of the disease trait) of the animal population. To decrease infection pressure, it is possible to apply non-genetic procedures that increase the culling (e.g., culling of chronically infected cows) and recovery (e.g., antibiotic therapy) rates of infected cows. But the contribution of the paper is to show that genetic management of infectious disease is also theoretically possible as a control measure complementary to non-genetic actions. Indeed, the probability for an uninfected individual to become infected after contact with an infected one is partially related to their degree of kinship: the more closely they are related, the more likely they are to share identical genes like those associated to the non-resistance to infection. Different prospective genetic management procedures are proposed to decrease the contact rate between infected and uninfected relatives and keep the number of secondary cases generated by one infected animal below 1.     

Potential negative effects on biological control by Sancassania polyphyllae (Acari: Acaridae) on an entomopathogenic nematode species

Sancassania polyphyllae (Acari: Acaridae) is associated with larvae of the white grub, Polyphylla fullo (Coleoptera: Scarabaeidae), and will feed on the infective juveniles of entomopathogenic nematodes in the families Steinernematidae and Heterorhabditidae which are important biological control agents of soil insect pests. We conducted laboratory studies to determine the potential negative effects this mite species might have on biological control of soil insect pests. Our objectives in this study were to (1) determine the response of S. polyphyllae adult mites to a nematode-killed insects on agar, (2) evaluate the predation by mites on Steinernema feltiae infective juveniles from nematode-killed insects on agar and in soil, and (3) assess predation efficiency of the mite on the infective juveniles in the soil. On agar, we found (1) significantly more adult female mites near or on a nematode-killed Ceratitis capitata (Diptera: Tephritidae) larva than near or on the freeze-killed larva or a bamboo mimic suggesting that a chemical or an odor from the nematode-killed larva attracted the mites, and (2) 10 mites consumed 96% of infective juveniles that emerged from an insect cadaver. In soil with a nematode-killed insect, the average number of infective juveniles recovered was <30 when mites were present, whereas the average number of infective juveniles recovered was >375 when the mites were absent. When the infective juveniles alone were placed in different depths in relation to the mites in the soil column for 4 and 10 days, S. polyphyllae was not as efficient at finding the infective juveniles when they were separated from each other in the soil lending support to the idea that the mites were cueing in on the cadaver as a food resource. Our data suggest that emerging infective juveniles from an insect cadaver in the soil in the presence of S. polyphyllae can adversely affect biological control because of nematode consumption by the mites.     

Population Growth in Planaria Dugesia tigrina (Gerard) : Regulation by the absolute number in the population

Planaria reproduce by transverse fission. Isolated worms increase in number exponentially, while social animals at the same density are inhibited in terms of numerical increase, but over a 25 day period undergo a larger increase in mass. Isolated posterior fission products reproduce faster than isolated anterior fission products. Regulation of population growth is independent of density over a 16-fold range and regulatory factors cannot be demonstrated in the medium. Regulation of population growth depends on direct contact between animals. Fission period varies from individual to individual and from period to period for a given individual. Doubling time is related to the absolute number of individuals comprising the population as follows: P_N = (P_M · N)/(K + N), where P_N is the doubling period of a population of N individuals, P_M is the doubling time of an infinitely large population, N is the number of individuals in the population, and K is the number of individuals in a population the period of which is one-half P_M. At 22°–24°C P_M is estimated to be 43.3 days and K is 1.87 individuals. A model system assumes that inhibitor flows through the population from animal to animal from the slowest to the fastest animal in the population thus acting to synchronize population increase as well as to determine the rate of population growth. A possible source of the inhibitor is discussed.     

Global analysis of an SAIS model

This paper is concerned with global analysis of an SAIS epidemiological model in a population of varying size introduced by Busenberg and van den Driessche. In this model the population is divided into three subgroups of susceptible, asymptomatic and infective individuals. It has been shown that this system has no periodic solutions and all its trajectories tend to the equilibria of the system. We use the Poincaré Index theorem to determine the number of the equilibria and their stability properties. We have shown that bistability occurs for suitable values of parameters and found a set of examples of all possible dynamics of the system.     

Emergence and Dispersal Patterns of Two Isolates of the Entomopathogenic Nematode Steinernema feltiae

Few studies have addressed the role of the sexes in the emergence and dispersal of entomopathogenic nematodes from host insects. Individuals of two isolates of Steinernema feltiae, UK76 and SBIl, emerging from Galleria mellonella cadavers were classed as Non-Dispersed (remaining on the cadaver for up to nine days) and Dispersed (actively moving away from the cadaver). Sex ratios within both classes were examined in infective (individuals that successfully invaded bait G. mellonella) and entire (infective and noninfective individuals that matured in hanging drops of G. mellonella haemolymph) populations. Sex ratios differed significantly from 1:1 only in the SBIl Non-Dispersed entire population (female bias) and SBIl Non-Dispersed infective population (male bias). For each isolate, Dispersed individuals were significantly more infective than Non-Dispersed. However, only 11% of SBIl and 22% of UK76 Non-Dispersed individuals were found to be mature infective juveniles (IJ) compared with 78% of SBIl and 82% of UK76 Dispersed individuals (based on survival in SDS). Infective juveniles dispersing towards distant radial bait G. mellonella tended to migrate from the head region of the natal cadaver. For each isolate, a higher proportion of males than females arrived early at distant baits. SBIl males survived alone in G. mellonella cadavers for longer periods than did females, which supports the “male colonization” hypothesis.     

An efficient method for viable cryopreservation and recovery of hookworms and other gastrointestinal nematodes in the laboratory

Hookworms (genera Ancylostoma and Necator) are amongst the most prevalent and important parasites of humans globally. These intestinal parasites ingest blood, resulting in anemia, growth stunting, malnutrition, and adverse pregnancy outcomes. They are also critical parasites of dogs and other animals. In addition, hookworms and hookworm products are being explored for their use in treatment of autoimmune and inflammatory diseases. There is thus a significant and growing interest in these mammalian host-obligate parasites. Laboratory research is hampered by the lack of good means of cryopreservation and recovery of parasites. Here, we describe a robust method for long-term (≥3 year) cryopreservation and recovery of both Ancylostoma and Necator hookworms that is also applicable to two other intestinal parasites that passage through the infective L3 stage, Strongyloides ratti and Heligmosomoides polygyrus bakeri. The key is a revised recovery method, in which cryopreserved L1s are thawed and raised to the infective L3 stage using activated charcoal mixed with uninfected feces from a permissive host. This technique will greatly facilitate research on and availability of gastrointestinal parasitic nematodes with great importance to global health, companion animal health, and autoimmune/inflammatory disease therapies.