Transmission dynamics of an influenza model with vaccination and antiviral treatment

Generation of "OP7 chimera" defective interfering influenza A particle preparations free of infectious virus that show antiviral efficacy in mice

Influenza A virus (IAV) defective interfering particles (DIPs) are considered as new promising antiviral agents. Conventional DIPs (cDIPs) contain a deletion in the genome and can only replicate upon co-infection with infectious standard virus (STV), during which they suppress STV replication. We previously discovered a new type of IAV DIP "OP7" that entails genomic point mutations and displays higher antiviral efficacy than cDIPs. To avoid safety concerns for the medical use of OP7 preparations, we developed a production system that does not depend on infectious IAV. We reconstituted a mixture of DIPs consisting of cDIPs and OP7 chimera DIPs, in which both harbor a deletion in their genome. To complement the defect, the deleted viral protein is expressed by the suspension cell line used for production in shake flasks. Here, DIP preparations harvested are not contaminated with infectious virions, and the fraction of OP7 chimera DIPs depended on the multiplicity of infection. Intranasal administration of OP7 chimera DIP material was well tolerated in mice. A rescue from an otherwise lethal IAV infection and no signs of disease upon OP7 chimera DIP co-infection demonstrated the remarkable antiviral efficacy. The clinical development of this new class of broad-spectrum antiviral may contribute to pandemic preparedness.

Stability switches, periodic oscillations and global stability in an infectious disease model with multiple time delays

A delay differential equation model of an infectious disease is considered and analyzed. In this model, the impact of information due to the presence of infection is considered explicitly. As information propagation is dependent on the prevalence of the disease, the delay in reporting the prevalence is an important factor. Further, the time lag in waning immunity related to protective measures (such as vaccination, self-protection, responsive behaviour etc.) is also accounted. Qualitative analysis of the equilibrium points of the model is executed and it is observed that when the basic reproduction number is less unity, the local stability of the disease free equilibrium (DFE) depends on the rate of immunity loss as well as on the time delay for the waning of immunity. If the delay in immunity loss is less than a threshold quantity, the DFE is stable, whereas, it loses its stability when the delay parameter crosses the threshold value. When, the basic reproduction number is greater than unity, the unique endemic equilibrium point is found locally stable irrespective of the delay effect under certain parametric conditions. Further, we have analyzed the model system for different scenarios of both delays (i.e., no delay, only one delay, and both delay present). Due to these delays, oscillatory nature of the population is obtained with the help of Hopf bifurcation analysis in each scenario. Moreover, at two different time delays (delay in information's propagation), the emergence of multiple stability switches is investigated for the model system which is termed as Hopf-Hopf (double) bifurcation. Also, the global stability of the endemic equilibrium point is established under some parametric conditions by constructing a suitable Lyapunov function irrespective of time lags. In order to support and explore qualitative results, exhaustive numerical experimentations are carried out which lead to important biological insights and also, these results are compared with existing results.

A Spatial Kinetic Model of Crowd Evacuation Dynamics with Infectious Disease Contagion

This paper proposes a kinetic theory approach coupling together the modeling of crowd evacuation from a bounded domain with exit doors and infectious disease contagion. The spatial movement of individuals in the crowd is modeled by a proper description of the interactions with people in the crowd and the environment, including walls and exits. At the same time, interactions among healthy and infectious individuals may generate disease spreading if exposure time is long enough. Immunization of the population and individual awareness to contagion is considered as well. Interactions are modeled by tools of game theory, that let us propose the so-called tables of games that are introduced in the general kinetic equations. The proposed model is qualitatively studied and, through a series of case studies, we explore different scenarios related to crowding and gathering formation within indoor venues under the spread of a respiratory infectious disease, obtaining insights on specific policies to reduce contagion that may be implemented.

Modelling the impact of maternal pneumococcal vaccination on infant pneumococcal disease in low-income settings

Pneumococcal disease is a leading cause of mortality in young children. The largest burden of pneumococcal disease is in the first six months of life before protection from a complete schedule of direct immunisation is possible. Maternal pneumococcal vaccination has been proposed as a strategy for protection in this period of early childhood; however, limited clinical trial data exists. In this study, we developed an age-structured compartmental mathematical model to estimate the impact of maternal pneumococcal vaccination. Our model demonstrates how maternal pneumococcal vaccination could prevent 73% (range 49-88%) of cases in those aged <1 month and 55% (range 36-66%) in those 1-2 months old. This translates to an estimated 17% reduction in deaths due to invasive pneumococcal disease in children under five. Overall, this study demonstrates the potential for maternal pneumococcal vaccination to meaningfully reduce the burden of infant pneumococcal disease, supporting the case for appropriate field-based clinical studies.

Time-varying sensitivity analysis of an influenza model with interventions

We formulate an influenza model with treatment and vaccination. Both time invariant and time-dependent uncertainty analyses and sensitivity analysis of the model parameter values are carried out to understand the dependence of the reproduction numbers and model state variables on their components. Results show that the relationship between treatment and epidemic size is nonlinear and that there exists a critical threshold treatment rate under which treatment is beneficial. Sensitivity analysis suggests that the most significant parameters are those related to infection transmission, infectiousness, duration of infectiousness and waning immunity. Further, there are important instances when the relationship between some parameters and model outputs changes behavior from negatively to positively correlated or vice versa because all sensitivity indices, except [Formula: see text] are functions of other parameters and thus will change with the change in parameter values. For example, treatment helps to lower the epidemic size, but may then become a “source” of infection likely due to resistance de novo. This knowledge is critical for proper public health planning and guidance of control strategies.

Optimizing antiviral treatment for seasonal influenza in the USA: a mathematical modeling analysis

BACKGROUND

Seasonal influenza remains a major cause of morbidity and mortality in the USA. Despite the US Centers for Disease Control and Prevention recommendation promoting the early antiviral treatment of high-risk patients, treatment coverage remains low.

METHODS

To evaluate the population-level impact of increasing antiviral treatment timeliness and coverage among high-risk patients in the USA, we developed an influenza transmission model that incorporates data on infectious viral load, social contact, and healthcare-seeking behavior. We modeled the reduction in transmissibility in treated individuals based on their reduced daily viral load. The reduction in hospitalizations following treatment was based on estimates from clinical trials. We calibrated the model to weekly influenza data from Texas, California, Connecticut, and Virginia between 2014 and 2019. We considered in the baseline scenario that 2.7-4.8% are treated within 48 h of symptom onset while an additional 7.3-12.8% are treated after 48 h of symptom onset. We evaluated the impact of improving the timeliness and uptake of antiviral treatment on influenza cases and hospitalizations.

RESULTS

Model projections suggest that treating high-risk individuals as early as 48 h after symptom onset while maintaining the current treatment coverage level would avert 2.9-4.5% of all symptomatic cases and 5.5-7.1% of all hospitalizations. Geographic variability in the effectiveness of earlier treatment arises primarily from variabilities in vaccination coverage and population demographics. Regardless of these variabilities, we found that when 20% of the high-risk individuals were treated within 48 h, the reduction in hospitalizations doubled. We found that treatment of the elderly population (> 65 years old) had the highest impact on reducing hospitalizations, whereas treating high-risk individuals aged 5-19 years old had the highest impact on reducing transmission. Furthermore, the population-level benefit per treated individual is enhanced under conditions of high vaccination coverage and a low attack rate during an influenza season.

CONCLUSIONS

Increased timeliness and coverage of antiviral treatment among high-risk patients have the potential to substantially reduce the burden of seasonal influenza in the USA, regardless of influenza vaccination coverage and the severity of the influenza season.

Global dynamics for a Filippov epidemic system with imperfect vaccination

Given imperfect vaccination we extend the existing non-smooth models by considering susceptible and vaccinated individuals enhance the protection and control strategies once the number of infected individuals exceeds a certain level. On the basis of global dynamics of two subsystems, for the formulated Filippov system, we examine the sliding mode dynamics, the boundary equilibrium bifurcations, and the global dynamics. Our main results show that it is possible that the pseudo-equilibrium exists and is globally stable, or the pseudo-equilibrium, the disease-free equilibrium and the real equilibrium are tri-stable, or the pseudo-equilibrium and the real equilibrium are bi-stable, or the pseudo-equilibrium and disease-free equilibrium are bi-stable, which depend on the threshold value and other parameter values. The global stability of the disease-free equilibrium or pseudo-equilibrium reveals that we may eradicate the disease or maintain the number of infected individuals at a previously given value. Further, the bi-stability and tri-stability imply that whether the number of infected individuals tends to zero or a previously given value or other positive values depends on the parameter values and the initial states of the system. This emphasizes the importance of threshold policy and challenges in the control of infectious diseases if without perfect vaccines.

Traveling Waves and Estimation of Minimal Wave Speed for a Diffusive Influenza Model with Multiple Strains

Antiviral treatment remains one of the key pharmacological interventions against influenza pandemic. However, widespread use of antiviral drugs brings with it the danger of drug resistance evolution. To assess the risk of the emergence and diffusion of resistance, in this paper, we develop a diffusive influenza model where influenza infection involves both drug-sensitive and drug-resistant strains. We first analyze its corresponding reaction model, whose reproduction numbers and equilibria are derived. The results show that the sensitive strains can be eliminated by treatment. Then, we establish the existence of the three kinds of traveling waves starting from the disease-free equilibrium, i.e., semi-traveling waves, strong traveling waves and persistent traveling waves, from which we can get some useful information (such as whether influenza will spread, asymptotic speed of propagation, the final state of the wavefront). On the other hand, we discuss three situations in which semi-traveling waves do not exist. When the control reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{C}$$\end{document}RC is larger than 1, the conditions for the existence and nonexistence of traveling waves are determined completely by the reproduction numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{SC}$$\end{document}RSC, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{RC}$$\end{document}RRC and the wave speed c. Meanwhile, we give an interval estimation of minimal wave speed for influenza transmission, which has important guiding significance for the control of influenza in reality. Our findings demonstrate that the control of influenza depends not only on the rates of resistance emergence and transmission during treatment, but also on the diffusion rates of influenza strains, which have been overlooked in previous modeling studies. This suggests that antiviral treatment should be implemented appropriately, and infected individuals (especially with the resistant strain) should be tested and controlled effectively. Finally, we outline some future directions that deserve further investigation.

Global dynamics of a fractional-order SIR epidemic model with memory

In this paper, an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley–Martin type functional response and Holling type-II treatment rate are established along the memory. The existence and stability of the equilibrium points are investigated. The sufficient conditions for the persistence of the disease are provided. First, a threshold value, [Formula: see text], is obtained which determines the stability of equilibria, then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained. The outcome of the study reveals that the applied L1 scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics. The results show that order of the fractional derivative has a significant effect on the dynamic process. Also, from the results, it is obvious that the memory effect is zero for [Formula: see text]. When the fractional-order [Formula: see text] is decreased from [Formula: see text] the memory trace nonlinearly increases from [Formula: see text], and its dynamics strongly depends on time. The memory effect points out the difference between the derivatives of the fractional-order and integer order.

A two-thresholds policy for a Filippov model in combating influenza

This work designs a two-thresholds policy for a Filippov model in combating influenza, so as to estimate when and whether to take control strategies, including the media coverage, antiviral treatment of infected individuals and vaccination of susceptible population. By introducing two tolerance thresholds [Formula: see text] and [Formula: see text] of susceptible and infected individuals, the two-thresholds policy is designed as: a vaccination program is implemented when the number of susceptible individuals is above [Formula: see text]; an antiviral treatment strategy is taken and the mass media begins to report information about influenza when the infection number is larger than [Formula: see text]; no control strategies are required in other cases. Furthermore, the global dynamics of the model are analyzed by varying these two thresholds, including the existence and dynamics of sliding mode, and the existence and global stability of equilibrium. It is shown that the model solutions ultimately converge to a pseudoequilibrium or a pseudoattractor on the switching surface, or a real equilibrium. The obtained results indicate that, by choosing susceptible and infected thresholds properly, the infection number can be remained below or at an acceptable level.

Qualitative analysis of a stochastic SEITR epidemic model with multiple stages of infection and treatment

We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission, treatment and recovery rates. We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By defining RT,n and RT,n as the basic deterministic and stochastic reproduction numbers, respectively, in stage n of infection and treatment, we show mathematically that as the intensity of the noise in the transmission, treatment and recovery rates increases, the number of secondary cases of infection increases. The global stability of the disease-free and endemic equilibrium for the deterministic and stochastic SEITR models is also presented. The work presented is demonstrated using parameter values relevant to the transmission dynamics of Influenza in the United States from October 1, 2018 through May 4, 2019 influenza seasons.

Modelling and analysing the coexistence of dual dilemmas in the proactive vaccination game and retroactive treatment game in epidemic viral dynamics

The dynamics of a spreadable disease are largely governed by four factors: proactive vaccination, retroactive treatment, individual decisions, and the prescribing behaviour of physicians. Under the imposed vaccination policy and antiviral treatment in society, complex factors (costs and expected effects of the vaccines and treatments, and fear of being infected) trigger an emulous situation in which individuals avoid infection by the pre-emptive or ex post provision. Aside from the established voluntary vaccination game, we propose a treatment game model associated with the resistance evolution of antiviral/antibiotic overuse. Moreover, the imperfectness of vaccinations has inevitably led to anti-vaccine behaviour, necessitating a proactive treatment policy. However, under the excessively heavy implementation of treatments such as antiviral medicine, resistant strains emerge. The model explicitly exhibits a dual social dilemma situation, in which the treatment behaviour changes on a local time scale, and the vaccination uptake later evolves on a global time scale. The impact of resistance evolution and the coexistence of dual dilemmas are investigated by the control reproduction number and the social efficiency deficit, respectively. Our investigation might elucidate the substantial impacts of both vaccination and treatment in the framework of epidemic dynamics, and hence suggest the appropriate use of antiviral treatment.

Behavioral incentives in a vaccination-dilemma setting with optional treatment

Social dilemmas are situations wherein individuals choose between selfish interest and common good. One example of this is the vaccination dilemma, in which an individual who vaccinates at a cost protects not only himself but also others by helping maintain a common good called herd immunity. There is, however, a strong incentive to forgo vaccination, thus avoiding the associated cost, all the while enjoying the protection of herd immunity. To analyze behavioral incentives in a vaccination-dilemma setting in which an optional treatment is available to infected individuals, we combined epidemiological and game-theoretic methodologies by coupling a disease-spreading model with treatment and an evolutionary decision-making model. Extensive numerical simulations show that vaccine characteristics are more important in controlling the treatment adoption than the cost of treatment itself. The main effect of the latter is that expensive treatment incentivizes vaccination, which somewhat surprisingly comes at a little cost to society. More surprising is that the margin for a true synergy between vaccine and treatment in reducing the final epidemic size is very small. We furthermore find that society-centered decision making helps protect herd immunity relative to individual-centered decision making, but the latter may be better in establishing a novel vaccine. These results point to useful policy recommendations as well as to intriguing future research directions.

Modelling epidemics with fractional-dose vaccination in response to limited vaccine supply

The control strategies of emergency infectious diseases are constrained by limited medical resources. The fractional dose vaccination strategy as one of feasible strategies was proposed in response to global shortages of vaccine stockpiles. Although a variety of epidemic models have been developed under the circumstances of limited resources in treatment, few models particularly investigated vaccination strategies in resource-limited settings. In this paper, we develop a two-group SIR model with incorporation of proportionate mixing patterns and n-fold fractional dose vaccination related parameters to evaluate the efficiency of fractional dose vaccination on disease control at the population level. The existence and uniqueness of the final size of the two-group SIR epidemic model, the formulation of the basic reproduction number and the relationship between them are established. Moreover, numerical simulations are performed based on this two-group vector-free model to investigate the effectiveness of n-fold fractional dose vaccination by using the emergency outbreaks of yellow fever in Angola in 2016. By employing linear and nonlinear dose-response relationships, we compare the resulting fluctuations of four characteristics of the epidemics, which are the outbreak size, the peak time of the outbreak, the basic reproduction number and the infection attack rate (IAR). For both types of dose-response relationships, dose-fractionation takes positive effects in lowering the outbreak size, delay the peak time of the outbreak, reducing the basic reproduction number and the IAR of yellow fever only when the vaccine efficacy is high enough. Moreover, five-fold fractional dose vaccination strategy may not be the optimal vaccination strategy as proposed by the World Health Organization if the dose-response relationship is nonlinear.

Delayed information induces oscillations in a dynamical model for infectious disease

During disease outbreak, it has been observed that information about the disease prevalence induces the individual’s behavioral changes. This information is usually assumed to be generated by the density of infective individuals and active mass media. The delay in reporting of these infective individuals may have its impact on generated information. Hence, to study the impact of delay on information generation, and therefore on the disease dynamics, a delay differential equation model is proposed and analyzed. The dynamics of information with delay effect is also modeled by a separate rate equation. Model analysis is performed and a unique infected equilibrium is obtained when the basic reproduction number ([Formula: see text]) is greater than one, whereas the disease free equilibrium always exists. When [Formula: see text], the disease free equilibrium is found to be locally stable independent of delay effect. The unique infected equilibrium is found to be locally stable till delay reaches a threshold value. The global stability of the unique infected equilibrium is also established under some parametric conditions by constructing a suitable Lyapunov function. The occurrence of Hopf bifurcation is observed when the delay in information crosses the threshold value. Analytically, the direction and stability of bifurcating periodic solutions is established. Further, we observed the occurrence of Hopf-Hopf bifurcation at two different delays. At first delay threshold, the endemic equilibrium loses its stability and produces periodic oscillations via Hopf bifurcation. It further regains its stability at second delay threshold via another Hopf bifurcation. Hence, the delay effect on information shows possibility of stability switches. Numerical experiments are carried out to support the obtained analytical results. Our study infers that the disease will show persistent oscillations if there is a significant time lag in reporting of infective after the disease outbreak. Thus, the delay in dissemination of information shows rich and complex dynamics in the model and provides important insights. We also observe numerically that the saturation in information plays a significant role on stability of infected equilibrium in presence of delay.

Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate

This paper is devoted to studying the dynamics of a susceptible-infective-latent-infective (SILI) epidemic model that is subject to the combined effects of environmental noise and intervention strategy. We extend the classical SILI epidemic model from a deterministic framework to a stochastic one. For the deterministic case, the global stability analysis of the solution is carried out in terms of the basic reproduction number. For the stochastic case, sufficient conditions for the extinction of diseases are obtained. Then, the existence of stationary distribution and asymptotic behavior of the solution are further studied to illustrate the cycling phenomena of recurrent diseases. Numerical simulations are conducted to verify these analytical results. It is shown that both stochastic noise and intervention strategy contribute to the control of diseases.

Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel

In the present study, the fractional version with respect to the Atangana-Baleanu fractional derivative operator in the caputo sense (ABC) of the two-strain epidemic mathematical model involving two vaccinations has extensively been analyzed. Furthermore, using the fixed-point theory, it has been shown that the solution of the proposed fractional version of the mathematical model does not only exist but is also the unique solution under some conditions. The original mathematical model consists of six first order nonlinear ordinary differential equations, thereby requiring a numerical treatment for getting physical interpretations. Likewise, its fractional version is not possible to be solved by any existing analytical method. Therefore, in order to get the observations regarding the output of the model, it has been solved using a newly developed convergent numerical method based on the Atangana-Baleanu fractional derivative operator in the caputo sense. To believe upon the results obtained, the fractional order α has been allowed to vary between ( 0 , 1 ] , whereupon the physical observations match with those obtained in the classical case, but the fractional model has persisted all the memory effects making the model much more suitable when presented in the structure of fractional order derivatives for ABC. Finally, the fractional forward Euler method in the classical caputo sense has been used to illustrate the better performance of the numerical method obtained via the Atangana-Baleanu fractional derivative operator in the caputo sense.

Threshold dynamics of the Transmission of Antibiotic-Resistant Infections

Despite numerous studies conducted, multidrug-resistant infections such as methicillin-resistant Staphylococcus aureus (MRSE) and vancomycin-resistant enterococci (VRE) are still increasing in hospitals, and yet continued to be the important challenges of worldwide health. Mathematical modeling gives an insightful information in policy and decision making to control the transmission and spread of these infections globally. We formulated and analyse a mathematical model to characterise the transmission co-dynamics of hospital-acquired MRSE (HA-MRSA) and community-acquired MRSE (CA-MRSA) and to investigate the long run competitiveness of the two strains in hospital. Numerical simulations are carried out to explore the basic reproduction numbers for the two strains so as to determine the dominant strain in the future in hospital setting. Under some conditions, invasion reproduction numbers are also applied to determine the uniform persistence of the two strains. We further performed sensitivity analysis to examine the influence of model parameters on the transmission and spread of the the strains, thereby determine the effective intervention strategies that reduce the overflow of the infections in hospital setting. To support theoretical findings qualitatively, graphical representations are provided.

Mathematical Analysis of an Epidemic System in the Presence of Optimal Control and Population Dispersal

In this paper, we proposed and analyzed a susceptible-infected-recovered (SIR) type epidemic model to investigate the effect of transport-related infectious diseases namely tuberculosis, measles, rubella, influenza, sexually transmitted diseases, etc. The existence and stability criteria of both the diseases include free equilibrium point and endemic equilibrium point which are established and the threshold parametric condition for which the system passes through a transcritical bifurcation is also obtained. Optimal control strategy for control parameters is formulated and solved both theoretically and numerically. Lastly, we not only illustrate our theoretical results through graphical illustrations but also computer simulation is used to show that our model would be a good model to study the SARS epidemic in 2003.

Analysis and Numerical Simulations of a Stochastic SEIQR Epidemic System with Quarantine-Adjusted Incidence and Imperfect Vaccination

This paper considers a high-dimensional stochastic SEIQR (susceptible-exposed-infected-quarantined-recovered) epidemic model with quarantine-adjusted incidence and the imperfect vaccination. The main aim of this study is to investigate stochastic effects on the SEIQR epidemic model and obtain its thresholds. We first obtain the sufficient condition for extinction of the disease of the stochastic system. Then, by using the theory of Hasminskii and the Lyapunov analysis methods, we show there is a unique stationary distribution of the stochastic system and it has an ergodic property, which means the infectious disease is prevalent. This implies that the stochastic disturbance is conducive to epidemic diseases control. At last, computer numerical simulations are carried out to illustrate our theoretical results.

Analysis of a fuzzy epidemic model with saturated treatment and disease transmission

In this paper, we describe an SIS epidemic model where both the disease transmission rate and treatment function are considered in saturated forms. The dynamical behavior of the system is analyzed. The system is customized by considering the disease transmission rate and treatment control as fuzzy numbers and then fuzzy expected value of the infected individuals is determined. The fuzzy basic reproduction number is investigated and a threshold condition of pathogen is derived at which the system undergoes a backward bifurcation.

GLOBAL DYNAMICS OF AN SVIR EPIDEMIOLOGICAL MODEL WITH INFECTION AGE AND NONLINEAR INCIDENCE

In this paper, an SVIR epidemiological model with infection age (time elapsed since the infection) and nonlinear incidence is studied. In the model, in order to reflect the dependence of disease progress on the infection age, the infected individual is structured by the infection age, and transmission and removal rates are assumed to depend on the infection age. By analyzing corresponding characteristic equations, the local stability of each of steady states of the model is established. It is proved that the semi-flow generated by this system is asymptotically smooth, and if the basic reproduction number is greater than unity, the system is uniformly persistent. By using Lyapunov functional and LaSalle’s invariance principle, the global dynamics of the model is investigated. It is shown that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable and hence the disease dies out; and if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease persists. Numerical simulations are carried out to illustrate the main analytic results.

An agent-based model simulation of influenza interactions at the host level: insight into the influenza-related burden of pneumococcal infections

BACKGROUND

Host-level influenza virus-respiratory pathogen interactions are often reported. Although the exact biological mechanisms involved remain unelucidated, secondary bacterial infections are known to account for a large part of the influenza-associated burden, during seasonal and pandemic outbreaks. Those interactions probably impact the microorganisms' transmission dynamics and the influenza public health toll. Mathematical models have been widely used to examine influenza epidemics and the public health impact of control measures. However, most influenza models overlooked interaction phenomena and ignored other co-circulating pathogens.

METHODS

Herein, we describe a novel agent-based model (ABM) of influenza transmission during interaction with another respiratory pathogen. The interacting microorganism can persist in the population year round (endemic type, e.g. respiratory bacteria) or cause short-term annual outbreaks (epidemic type, e.g. winter respiratory viruses). The agent-based framework enables precise formalization of the pathogens' natural histories and complex within-host phenomena. As a case study, this ABM is applied to the well-known influenza virus-pneumococcus interaction, for which several biological mechanisms have been proposed. Different mechanistic hypotheses of interaction are simulated and the resulting virus-induced pneumococcal infection (PI) burden is assessed.

RESULTS

This ABM generates realistic data for both pathogens in terms of weekly incidences of PI cases, carriage rates, epidemic size and epidemic timing. Notably, distinct interaction hypotheses resulted in different transmission patterns and led to wide variations of the associated PI burden. Interaction strength was also of paramount importance: when influenza increased pneumococcus acquisition, 4-27% of the PI burden during the influenza season was attributable to influenza depending on the interaction strength.

CONCLUSIONS

This open-source ABM provides new opportunities to investigate influenza interactions from a theoretical point of view and could easily be extended to other pathogens. It provides a unique framework to generate in silico data for different scenarios and thereby test mechanistic hypotheses.

Mathematical model of transmission dynamics and optimal control strategies for 2009 A/H1N1 influenza in the Republic of Korea

A mathematical model for the transmission dynamics of the 2009 A/H1N1 influenza epidemic in the Republic of Korea is developed. The simulation period is separated into three consecutive periods based on the government's intervention strategies: the nonpharmaceutical strategy is used during Period 1. The nonpharmaceutical and antiviral strategies are executed during Period 2 and the vaccine strategy is added during Period 3. During Period 1, we estimate the reduction in the transmission rate due to the government's intervention policies as a difference between the data-fitted and uncontrolled transmission rate that is derived from the basic reproductive number, R₀, of the model without intervention. This quantified reduced transmission rate is used as an upperbound of the nonpharmaceutical control for studying optimal control strategies, which is a new approach for determining the realistic upperbound of control. In this study, we also explore the real-time prediction of incidence using the mathematical model during the early stage of the epidemic. We investigate the impact of vaccination coverage and timing with respect to the cumulative incidence. The result implies that early vaccination plays a significant role for preventing the epidemic.

Complex Dynamics of an SIR Epidemic Model with Saturated Incidence Rate and Treatment

This paper describes a traditional SIR type epidemic model with saturated infection rate and treatment function. The dynamics of the model is studied from the point of view of stability and bifurcation. Basic reproduction number is obtained and it is shown that the model system may possess a backward bifurcation. The global asymptotic stability of the endemic equilibrium is studied with the help of a geometric approach. Optimal control problem is formulated and solved. Some numerical simulation works are carried out to validate our analytical results.

Dynamical Behavior of an Epidemic Model in a Fuzzy Transmission

In this paper, we have formulated a simple SIS type epidemic model in the presence of treatment control, and we have discussed the dynamical behavior of the system. The system is modified by considering both the disease transmission rate and the treatment function as fuzzy numbers, and also the fuzzy expected value of the infected individuals is calculated. Furthermore, the fuzzy basic reproduction number is investigated and a threshold condition of pathogen is obtained at which the system undergoes a transcritical bifurcation.

Assessment of intensive vaccination and antiviral treatment in 2009 influenza pandemic in Korea

OBJECTIVES

We characterized and assessed public health measures, including intensive vaccination and antiviral treatment, implemented during the 2009 influenza pandemic in the Republic of Korea.

METHODS

A mathematical model for the 2009 influenza pandemic is formulated. The transmission rate, the vaccination rate, the antiviral treatment rate, and the hospitalized rate are estimated using the least-squares method for the 2009 data of the incidence curves of the infected, vaccinated, treated, and hospitalized.

RESULTS

The cumulative number of infected cases has reduced significantly following the implementation of the intensive vaccination and antiviral treatment. In particular, the intensive vaccination was the most critical factor that prevented severe outbreak.

CONCLUSION

We have found that the total infected proportion would increase by approximately six times under the half of vaccination rates.

Dynamic behavior for an SIRS model with nonlinear incidence rate and treatment

This paper considers an SIRS model with nonlinear incidence rate and treatment. It is assumed that susceptible and infectious individuals have constant immigration rates. We investigate the existence of equilibrium and prove the global asymptotical stable results of the endemic equilibrium. We then obtained that the model undergoes a Hopf bifurcation and existences a limit cycle. Some numerical simulations are given to illustrate the analytical results.

Optimizing treatment regimes to hinder antiviral resistance in influenza across time scales

The large-scale use of antivirals during influenza pandemics poses a significant selection pressure for drug-resistant pathogens to emerge and spread in a population. This requires treatment strategies to minimize total infections as well as the emergence of resistance. Here we propose a mathematical model in which individuals infected with wild-type influenza, if treated, can develop de novo resistance and further spread the resistant pathogen. Our main purpose is to explore the impact of two important factors influencing treatment effectiveness: i) the relative transmissibility of the drug-resistant strain to wild-type, and ii) the frequency of de novo resistance. For the endemic scenario, we find a condition between these two parameters that indicates whether treatment regimes will be most beneficial at intermediate or more extreme values (e.g., the fraction of infected that are treated). Moreover, we present analytical expressions for effective treatment regimes and provide evidence of its applicability across a range of modeling scenarios: endemic behavior with deterministic homogeneous mixing, and single-epidemic behavior with deterministic homogeneous mixing and stochastic heterogeneous mixing. Therefore, our results provide insights for the control of drug-resistance in influenza across time scales.

Drivers and consequences of influenza antiviral resistant-strain emergence in a capacity-constrained pandemic response

Antiviral agents remain a key component of most pandemic influenza preparedness plans, but there is considerable uncertainty regarding their optimal use. In particular, concerns exist regarding the likelihood of wide-scale distribution to select for drug-resistant variants. We used a model that considers the influence of logistical constraints on diagnosis and drug delivery to consider achievable 'reach' of alternative antiviral intervention strategies targeted at cases of varying severity, with or without pre-exposure prophylaxis of contacts. To identify key drivers of epidemic mitigation and resistance emergence, we used Latin hypercube sampling to explore plausible ranges of parameters describing characteristics of wild type and resistant viruses, along with intervention efficacy, target coverage and distribution capacity. Within our model framework, 'real world' constraints substantially reduced achievable drug coverage below stated targets as the epidemic progressed. In consequence, predictions of both intervention impact and selection for resistance were more modest than earlier work that did not consider such limitations. Definitive containment of transmission was unlikely but, where observed, achieved through early liberal post-exposure prophylaxis of known contacts of treated cases. Predictors of resistant strain dominance were high intrinsic fitness relative to the wild type virus, and early emergence in the course of the epidemic into a largely susceptible population, even when drug use was restricted to severe case treatment. Our work demonstrates the importance of consideration of 'real world' constraints in scenario analysis modeling, and highlights the utility of models to guide surveillance activities in preparedness and response.

Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant Tuberculosis Dynamics

Tuberculosis, an airborne disease affecting almost a third of the world’s population remains one of the major public health burdens globally, and the resurgence of multidrug-resistant tuberculosis in some parts of sub-Saharan Africa calls for concern. To gain insight into its qualitative dynamics at the population level, mathematical modeling which require as inputs key demographic and epidemiological information can fill in gaps where field and lab data are not readily available. A deterministic model for the transmission dynamics of multi-drug resistant tuberculosis to assess the impact of diagnosis, treatment, and health education is formulated. The model assumes that exposed individuals develop active tuberculosis due to endogenous activation and exogenous re-infection. Treatment is offered to all infected individuals except those latently infected with multi-drug resistant tuberculosis. Qualitative analysis using the theory of dynamical systems shows that, in addition to the disease-free equilibrium, there exists a unique dominant locally asymptotically stable equilibrium corresponding to each strain. Numerical simulations suggest that, at the current level of control strategies (with Malawi as a case study), the drug-sensitive tuberculosis can be completely eliminated from the population, thereby reducing multi-drug resistant tuberculosis.

The impact of personal experiences with infection and vaccination on behaviour-incidence dynamics of seasonal influenza

Personal experiences with past infection events, or perceived vaccine failures and complications, are known to drive vaccine uptake. We coupled a model of individual vaccinating decisions, influenced by these drivers, with a contact network model of influenza transmission dynamics. The impact of non-influenzal influenza-like illness (niILI) on decision-making was also incorporated: it was possible for individuals to mistake niILI for true influenza. Our objectives were to (1) evaluate the impact of personal experiences on vaccine coverage; (2) understand the impact of niILI on behaviour-incidence dynamics; (3) determine which factors influence vaccine coverage stability; and (4) determine whether vaccination strategies can become correlated on the network in the absence of social influence. We found that certain aspects of personal experience can significantly impact behaviour-incidence dynamics. For instance, longer term memory for past events had a strong stabilising effect on vaccine coverage dynamics, although it could either increase or decrease average vaccine coverage depending on whether memory of past infections or past vaccine failures dominated. When vaccine immunity wanes slowly, vaccine coverage is low and stable, and infection incidence is also very low, unless the effects of niILI are ignored. Strategy correlations can occur in the absence of imitation, on account of the neighbour-neighbour transmission of infection and history-dependent decision making. Finally, niILI weakens the behaviour-incidence coupling and therefore tends to stabilise dynamics, as well as breaking up strategy correlations. Behavioural feedbacks, and the quality of self-diagnosis of niILI, may need to be considered in future programs adopting "universal" flu vaccines conferring long-term immunity. Public health interventions that focus on reminding individuals about their previous influenza infections, as well as communicating facts about vaccine efficacy and the difference between influenza and niILI, may be an effective way to increase vaccine coverage and prevent unexpected drops in coverage.

TRANSMISSION DYNAMICS AND OPTIMAL CONTROL OF AN INFLUENZA MODEL WITH QUARANTINE AND TREATMENT

We develop an influenza pandemic model with quarantine and treatment, and analyze the dynamics of the model. Analytical results of the model show that, if basic reproduction number [Formula: see text], the disease-free equilibrium (DFE) is globally asymptotically stable, if [Formula: see text], the disease is uniformly persistent. The model is then extended to assess the impact of three anti-influenza control measures, precaution, quarantine and treatment, by re-formulating the model as an optimal control problem. We focus primarily on controlling disease with a possible minimal the systemic cost. Pontryagin's maximum principle is used to characterize the optimal levels of the three controls. Numerical simulations of the optimality system, using a set of reasonable parameter values, indicate that the precaution measure is more effective in reducing disease transmission than the other two control measures. The precaution measure should be emphasized.

Assessing the effect of non-pharmaceutical interventions on containing an emerging disease

Non-pharmaceutical interventions, such as quarantine, isolation and entry screening, are usually the primary public health measures to control the spread of an emerging infectious disease through a human population. This paper proposes a multi-regional deterministic compartmental model to assess the effectiveness and implications of non-pharmaceutical interventions. The reproduction number is determined as the spectral radius of a nonnegative matrix product. Comparisons are made using the reproduction number, epidemic peaks and cumulative number of infections and mortality as indexes. Simulation results show that quarantine of suspected cases and isolation of cases with symptom are effective in reducing disease burden for multiple regions. Using entry screening strategy leads to a moderate time delay for epidemic peaks, but is of no help for preventing an epidemic breaking out. The study further shows that isolation strategy is always the best choice in the presence or absence of stringent hygiene precautions and should be given priority in combating an emerging epidemic.

Modeling the effects of vaccination and treatment on pandemic influenza

In this paper, we demonstrate the uses of some simple mathematical models for the study of disease dynamics in a pandemic situation with a focus on influenza. These models are employed to evaluate the effectiveness of various control programs via vaccination and antiviral treatment. We use susceptible-, infectious-, recovered-type epidemic models consisting of ordinary differential equations. These models allow us to derive threshold conditions that can be used to assess the effectiveness of vaccine and drug use and to determine disease outcomes. Simulations are helpful for examining the potential consequences of control options under different scenarios. Particularly, results from models with constant parameters and models with time-dependent parameter functions are compared, demonstrating the significant differences in model outcomes. Results suggest that the effectiveness of vaccination and drug treatment can be very sensitive to factors including the time of introduction of the pathogen into the population, the beginning time of control programs, and the levels of control measures. More importantly, in some cases, the benefits of vaccination and antiviral use might be significantly compromised if these control programs are not designed appropriately. Mathematical models can be very useful for understanding the effects of various factors on the spread and control of infectious diseases. Particularly, the models can help identify potential adverse effects of vaccination and drug treatment in the case of pandemic influenza.

Antiviral treatment for pandemic influenza: assessing potential repercussions using a seasonally forced SIR model

When resources are limited, measures to control an incipient influenza pandemic must be carefully considered. Because several months are needed to mass-produce vaccines once a new pandemic strain has been identified, antiviral drugs are often considered the first line of defense in a pandemic situation. Here we use an SIR disease model with periodic transmission rate to assess the efficacy of control strategies via antiviral drug treatment during an outbreak of pandemic influenza. We show that in some situations, and independent of drug-resistance effects, antiviral treatment can have a detrimental impact on the final size of the pandemic. Antiviral treatment also has the potential to increase the size of the major peak of the pandemic, and cause it to occur earlier than it would have if treatment were not used. Our studies suggest that when a disease exhibits periodic patterns in transmission, decisions of public health policy will be particularly important as to how control measures such as drug treatment should be implemented, and to what end (i.e.; towards immediate control of a current epidemic peak, or towards potential delay and/or reduction of an anticipated autumn peak).

Optimal control and sensitivity analysis of an influenza model with treatment and vaccination

We formulate and analyze the dynamics of an influenza pandemic model with vaccination and treatment using two preventive scenarios: increase and decrease in vaccine uptake. Due to the seasonality of the influenza pandemic, the dynamics is studied in a finite time interval. We focus primarily on controlling the disease with a possible minimal cost and side effects using control theory which is therefore applied via the Pontryagin's maximum principle, and it is observed that full treatment effort should be given while increasing vaccination at the onset of the outbreak. Next, sensitivity analysis and simulations (using the fourth order Runge-Kutta scheme) are carried out in order to determine the relative importance of different factors responsible for disease transmission and prevalence. The most sensitive parameter of the various reproductive numbers apart from the death rate is the inflow rate, while the proportion of new recruits and the vaccine efficacy are the most sensitive parameters for the endemic equilibrium point.

A note on the use of optimal control on a discrete time model of influenza dynamics

A discrete time Susceptible - Asymptomatic - Infectious - Treated - Recovered (SAITR) model is introduced in the context of influenza transmission. We evaluate the potential effect of control measures such as social distancing and antiviral treatment on the dynamics of a single outbreak. Optimal control theory is applied to identify the best way of reducing morbidity and mortality at a minimal cost. The problem is solved by using a discrete version of Pontryagin's maximum principle. Numerical results show that dual strategies have stronger impact in the reduction of the final epidemic size.

Optimal antiviral treatment strategies and the effects of resistance

Recent pandemic planning has highlighted the importance of understanding the effect that widespread antiviral use will have on the emergence and spread of resistance. A number of recent studies have determined that if resistance to antiviral medication can evolve, then deploying treatment at a less than maximum rate often minimizes the outbreak size. This finding, however, involves the assumption that treatment levels remain constant during the entire outbreak. Using optimal control theory, we address the question of optimal antiviral use by considering a large class of time-varying treatment strategies. We prove that, contrary to previous results, it is always optimal to treat at the maximum rate provided that this treatment occurs at the right time. In general the optimal strategy is to wait some fixed amount of time and then to deploy treatment at the maximum rate for the remainder of the outbreak. We derive analytical conditions that characterize this optimal amount of delay. Our results show that it is optimal to start treatment immediately when one of the following conditions holds: (i) immediate treatment can prevent an outbreak, (ii) the initial pool of susceptibles is small, or (iii) when the maximum possible rate of treatment is low, such that there is little de novo emergence of resistant strains. Finally, we use numerical simulations to verify that the results also hold under more general conditions.

The dynamics of an epidemic model with targeted antiviral prophylaxis

Due to the increasing risk of drug resistance and side effects with large-scale antiviral use, it has been suggested to provide antiviral drugs only to susceptibles who have had contacts with infectives. This antiviral distribution strategy is referred to as 'targeted antiviral prophylaxis'. The question of how effective this strategy is in infection control is of great public heath interest. In this paper, we formulate an ordinary differential equation model to describe the transmission dynamics of infectious disease with targeted antiviral prophylaxis, and provide the analysis of dynamical behaviours of the model. The control reproduction number R(c) is derived and shown to govern the disease dynamics, and the stability analysis is carried out. The local bifurcation theory is applied to explore the variety of dynamics of the model. Our theoretical results show that the system undergoes two Hopf bifurcations due to the existence of multiple endemic equilibria and the switch of their stability. Numerical results demonstrate that the system may have more complex dynamical behaviours including multiple periodic solutions and a homoclinic orbit. The results of this study suggest that the possibility of complex disease dynamics can be driven by the use of targeted antiviral prophylaxis, and the critical level of prophylaxis which achieves ℛ(c)=1 is not enough to control the prevalence of a disease.