Analysis of a heroin epidemic model with saturated treatment function

A mathematical model is developed that examines how heroin addiction spreads in society. The model is formulated to take into account the treatment of heroin users by incorporating a realistic functional form that "saturates" representing the limited availability of treatment. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the saturation parameter is larger than a fixed threshold. We are particularly interested in studying the model's global stability. In the absence of backward bifurcations, Lyapunov functions can often be found and used to prove global stability. However, in the presence of backward bifurcations, such Lyapunov functions may not exist or may be difficult to construct. We make use of the geometric approach to global stability to derive a condition that ensures that the system is globally asymptotically stable. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users are mechanisms fuelling heroin epidemic proliferation

How Hepatitis D Virus Can Hinder the Control of Hepatitis B Virus

BACKGROUND: Hepatitis D (or hepatitis delta) virus is a defective virus that relies on hepatitis B virus (HBV) for transmission; infection with hepatitis D can occur only as coinfection with HBV or superinfection of an existing HBV infection. Because of the bond between the two viruses, control measures for HBV may have also affected the spread of hepatitis D, as evidenced by the decline of hepatitis D in recent years. Since the presence of hepatitis D is associated with suppressed HBV replication and possibly infectivity, it is reasonable to speculate that hepatitis D may facilitate the control of HBV. METHODOLOGY AND PRINCIPAL FINDINGS: We introduced a mathematical model for the transmission of HBV and hepatitis D, where individuals with dual HBV and hepatitis D infection transmit both viruses. We calculated the reproduction numbers of single HBV infections and dual HBV and hepatitis D infections and examined the endemic prevalences of the two viruses. The results show that hepatitis D virus modulates not only the severity of the HBV epidemic, but also the impact of interventions for HBV. Surprisingly we find that the presence of hepatitis D virus may hamper the eradication of HBV. Interventions that aim to reduce the basic reproduction number of HBV below one may not be sufficient to eradicate the virus, as control of HBV depends also on the reproduction numbers of dual infections. CONCLUSIONS AND SIGNIFICANCE: For populations where hepatitis D is endemic, plans for control programs ignoring the presence of hepatitis D may underestimate the HBV epidemic and produce overoptimistic results. The current HBV surveillance should be augmented with monitoring of hepatitis D, in order to improve accuracy of the monitoring and the efficacy of control measures

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

Modelling the transmission dynamics of RSV and the impact of routine vaccination

Introduction: Respiratory Syncytial Virus is the major viral cause of lower respiratory tract disease in young children worldwide, with the greatest burden of disease in infants aged 1-3 months. Consequently, vaccine development has centered on a vaccine to directly protect the infants in this age group. The fundamental problem is that these young infants are poor responders to candidate RSV vaccines. This thesis focuses on the use of mathematical models to explore the merits of vaccination. Methods: Following development and analysis of a simple non-age-structured ODE model, we elaborate this to a Realistic Age Structured model (RAS) capturing the key epidemiological characteristics of RSV and incorporating age-specific vaccination options. The compartmental ODE model was calibrated using agespecific and time series hospitalization data from a rural coastal Kenyan population. The determination of Who Acquires Infection From Whom (WAIFW) matrix was done using social contact data from 1) a synthetic mixing matrix generated from primarily household occupancy data and 2) a diary study that we conducted in the Kilifi Health and Demographic Surveillance System (KHDSS). The vaccine was assumed to elicit partial immunity equivalent to wild type infection and its impact was measured by the ratio of hospitalized RSV cases after to before introduction. of vaccination. Uncertainty and sensitivity analysis were undertaken using Latin Hypercube Sampling (LHS) and partial rank correlation respectively. Given the importance of households in the transmission of respiratory infections, an exploratory household model was developed to capture the transmission dynamics of RSV A and B in a population of households. Results: From the analytical work of the simple ODE model, we have demonstrated that the model has the potential to exhibit a backward bifurcation curve within realistic parameter ranges. Both the diary and the synthetic mixing matrices had similar characteristics i.e. strong assortative mixing in individuals less than 30 years old and strong mixing between children less than 5 years and adults between 20 and 50 years old. When the two matrices were jointly linearly regressed, their elements were well correlated with an R2 ~ 0.6. The RAS model was capable of capturing the age-specific disease and the temporal epidemic nature of RSV in the specified location. Introduction of routine universal vaccination at ages varying from the first month of life to the 10th year of life resulted in optimal long-term benefit at 7 months (for the diary contact model) and 5 months (for the synthetic contact model). The greatest benefit arose under the assumption of age-related mixing with the contact diary data with no great deal of effectiveness lost when the vaccine is delayed between 5 and 12 months of age from birth. Vaccination was also shown to change the temporal dynamics of RSV hospitalizations and also to increase the average age at primary infection. From the sensitivity analysis, we identified the duration of RSV specific maternal antibodies, duration of primary and tertiary infections as the most important parameters in explaining the imprecision observed in predicting both the age specific hospitalizations and the optimal month at vaccination. Results from the household model have demonstrated that the household epidemic profile may be different from the general population with strong interaction of the viruses in the household that do not necessarily reflect at the population level. Conclusion: The synthetic matrix method would be a preferable alternative route in estimating mixing patterns in populations with the required socio-demographic data. Retrospectively, the synthetic mixing data can be used to reconstruct contact patterns in the past and therefore beneficial in assessing the effect of demographic transition in disease transmission. Universal infant vaccination has the potential to significantly reduce the burden of RSV associated disease, even with delayed vaccination between 5 and 12 months. This age class represents the group that is being targeted by vaccines that are currently under development. More accurate data measuring the duration of RSV specific maternal antibodies and the duration of infections are required to reduce the uncertainty in the model predictions

Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay

AbstractIn this study, we propose a new SVEIR epidemic disease model with time delay, and analyze the dynamic behavior of the model under pulse vaccination. Pulse vaccination is an effective strategy for the elimination of infectious disease. Using the discrete dynamical system determined by the stroboscopic map, we obtain an ‘infection-free’ periodic solution. We also show that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model under appropriate conditions. The permanence of the model is investigated analytically. Our results indicate that a large vaccination rate or a short pulse of vaccination or a long latent period is a sufficient condition for the extinction of the disease

Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers

In this paper, an epidemic model is investigated for infectious diseases that can be transmitted through both the infectious individuals and the asymptomatic carriers (i.e., infected individuals who are contagious but do not show any disease symptoms). We propose a dose-structured vaccination model with multiple transmission pathways. Based on the range of the explic- itly computed basic reproduction number, we prove the global stability of the disease-free when this threshold number is less or equal to the unity. Moreover, whenever it is greater than one, the existence of the unique endemic equilibrium is shown and its global stability is established for the case where the changes of displaying the disease symptoms are independent of the vulnerable classes. Further, the model is shown to exhibit a transcritical bifurcation with the unit basic reproduction number being the bifurcation parameter. The impacts of the asymptomatic carriers and the e ectiveness of vaccination on the disease transmission are discussed through through the local and the global sensitivity analyses of the basic reproduction number. Finally, a case study of hepatitis B virus disease (HBV) is considered, with the numerical simulations presented to support the analytical results. They further suggest that, in high HBV prevalence countries, the combination of e ective vaccination (i.e. 3 doses of HepB vaccine), the diagnosis of asymptomatic carriers and the treatment of symptomatic carriers may have a much greater positive impact on the disease control.South African Research Chairs Initiatives (SARChI Chair), in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.elsevier.com/locate/msec2017-08-31hb2016Mathematics and Applied Mathematic

Bayesian inference of sampled ancestor trees for epidemiology and fossil calibration

Phylogenetic analyses which include fossils or molecular sequences that are sampled through time require models that allow one sample to be a direct ancestor of another sample. As previously available phylogenetic inference tools assume that all samples are tips, they do not allow for this possibility. We have developed and implemented a Bayesian Markov Chain Monte Carlo (MCMC) algorithm to infer what we call sampled ancestor trees, that is, trees in which sampled individuals can be direct ancestors of other sampled individuals. We use a family of birth-death models where individuals may remain in the tree process after the sampling, in particular we extend the birth-death skyline model [Stadler et al, 2013] to sampled ancestor trees. This method allows the detection of sampled ancestors as well as estimation of the probability that an individual will be removed from the process when it is sampled. We show that sampled ancestor birth-death models where all samples come from different time points are non-identifiable and thus require one parameter to be known in order to infer other parameters. We apply this method to epidemiological data, where the possibility of sampled ancestors enables us to identify individuals that infected other individuals after being sampled and to infer fundamental epidemiological parameters. We also apply the method to infer divergence times and diversification rates when fossils are included among the species samples, so that fossilisation events are modelled as a part of the tree branching process. Such modelling has many advantages as argued in literature. The sampler is available as an open-source BEAST2 package (https://github.com/gavryushkina/sampled-ancestors).Comment: 34 pages (including Supporting Information), 8 figures, 1 table. Part of the work presented at Epidemics 2013 and The 18th Annual New Zealand Phylogenomics Meeting, 201

Global stability of an age-structured infection model in vivo with two compartments and two routes

In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio R-0 gives the threshold of the stability. If R-0 > 1, the interior equilibrium is unique and globally stable, and if R-0 <= 1, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model