Stochastic model of an influenza epidemic with drug resistance

A continuous-time Markov chain (CTMC) model is formulated for an influenza epidemic with drug resistance. This stochastic model is based on an influenza epidemic model, expressed in terms of a system of ordinary differential equations (ODE), developed by Stilianakis, N.I., Perelson, A.S., Hayden, F.G., [1998. Emergence of drug resistance during an influenza epidemic: insights from a mathematical model. J. Inf. Dis. 177, 863-873]. Three different treatments-chemoprophylaxis, treatment after exposure but before symptoms, and treatment after symptoms appear, are considered. The basic reproduction number, R(0), is calculated for the deterministic-model under different treatment strategies. It is shown that chemoprophylaxis always reduces the basic reproduction number. In addition, numerical simulations illustrate that the basic reproduction number is generally reduced with realistic treatment rates. Comparisons are made among the different models and the different treatment strategies with respect to the number of infected individuals during an outbreak. The final size distribution is computed for the CTMC model and, in some cases, it is shown to have a bimodal distribution corresponding to two situations: when there is no outbreak and when an outbreak occurs. Given an outbreak occurs, the total number of cases for the CTMC model is in good agreement with the ODE model. The greatest number of drug resistant cases occurs if treatment is delayed or if only symptomatic individuals are treated.     

Epidemic growth rate and household reproduction number in communities of households,schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.     

Tug-of-war has no borders: it is the missing model in reproductive skew theory

Cooperative breeding often results in unequal reproduction between dominant and subordinate group members. Transactional skew models attempt to predict how unequal reproduction can be before the groups themselves become unstable. A number of variants of transactional models have been developed, with a key difference being whether reproduction is controlled by one party or contested by all. It is shown here that ESS solutions for all situations of contested control over reproduction are given by the original tug-of-war model (TOW). Several interesting results follow. First, TOW can escalate enough to destabilize some types of groups. Particularly vulnerable are those that have low relatedness and gain little from cooperative breeding relative to solitary reproduction. Second, TOW can drastically reduce group productivity and especially the inclusive fitness of dominant individuals. Third, these results contrast strongly with those from variants of TOW models that include concessions to maintain group stability. Such models are shown to be special cases of the general and simpler TOW framework, and to have assumptions that may be biologically suspect. Finally, the overall analysis suggests that there is no mechanism within existing TOW framework that will prevent a costly struggle for reproductive control. Because social species rarely exhibit the high levels of aggression predicted by TOW models, alternative evolutionary mechanisms are considered that can limit conflict and produce more mutually beneficial outcomes. The further development of alternative models to predict patterns of reproductive skew are highly recommended.     

A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance

The evolutionary responses of infectious pathogens often have ruinous consequences for the control of disease spread in the population. Drug resistance is a well-documented instance that is generally driven by the selective pressure of drugs on both the replication of the pathogen within hosts and its transmission between hosts. Management of drug resistance therefore requires the development of treatment strategies that can impede the emergence and spread of resistance in the population. This study evaluates various treatment strategies for influenza infection as a case study by comparing the long-term epidemiological outcomes predicted by deterministic and stochastic versions of a homogeneously mixing (mean-field) model and those predicted by a heterogeneous model that incorporates spatial pair-wise correlation. We discuss the importance of three major parameters in our evaluation: the basic reproduction number, the population level of treatment, and the degree of clustering as a key parameter determining the structure of heterogeneous interactions. The results show that, as a common feature in all models, high treatment levels during the early stages of disease outset can result in large resistant outbreaks, with the possibility of a second wave of infection appearing in the pair-approximation model. Our simulations demonstrate that, if the basic reproduction number exceeds a threshold value, the population-wide spread of the resistant pathogen emerges more rapidly in the pair-approximation model with significantly lower treatment levels than in the homogeneous models. We tested an antiviral strategy that delays the onset of aggressive treatment for a certain amount of time after the onset of the outbreak. The findings indicate that the overall disease incidence is reduced as the degree of clustering increases, and a longer delay should be considered for implementing the large-scale treatment.     

Approximating the long-term behaviour of a model for parasitic infection

In a companion paper two stochastic models, useful for the initial behaviour of a parasitic infection, were introduced. Now we analyse the long term behaviour. First a law of large numbers is proved which allows us to analyse the deterministic analogues of the stochastic models. The behaviour of the deterministic models is analogous to the stochastic models in that again three basic reproduction ratios are necessary to fully describe the information needed to separate growth from extinction. The existence of stationary solutions is shown in the deterministic models, which can be used as a justification for simulation of quasi-equilibria in the stochastic models. Host-mortality is included in all models. The proofs involve martingale and coupling methods.     

Antiviral resistance and the control of pandemic influenza: the roles of stochasticity, evolution and model details

Antiviral drugs, most notably the neuraminidase inhibitors, are an important component of control strategies aimed to prevent or limit any future influenza pandemic. The potential large-scale use of antiviral drugs brings with it the danger of drug resistance evolution. A number of recent studies have shown that the emergence of drug-resistant influenza could undermine the usefulness of antiviral drugs for the control of an epidemic or pandemic outbreak. While these studies have provided important insights, the inherently stochastic nature of resistance generation and spread, as well as the potential for ongoing evolution of the resistant strain have not been fully addressed. Here, we study a stochastic model of drug resistance emergence and consecutive evolution of the resistant strain in response to antiviral control during an influenza pandemic. We find that taking into consideration the ongoing evolution of the resistant strain does not increase the probability of resistance emergence; however, it increases the total number of infecteds if a resistant outbreak occurs. Our study further shows that taking stochasticity into account leads to results that can differ from deterministic models. Specifically, we find that rapid and strong control cannot only contain a drug sensitive outbreak, it can also prevent a resistant outbreak from occurring. We find that the best control strategy is early intervention heavily based on prophylaxis at a level that leads to outbreak containment. If containment is not possible, mitigation works best at intermediate levels of antiviral control. Finally, we show that the results are not very sensitive to the way resistance generation is modeled.     

Bringing consistency to simulation of population models--Poisson simulation as a bridge between micro and macro simulation

Population models concern collections of discrete entities such as atoms, cells, humans, animals, etc., where the focus is on the number of entities in a population. Because of the complexity of such models, simulation is usually needed to reproduce their complete dynamic and stochastic behaviour. Two main types of simulation models are used for different purposes, namely micro-simulation models, where each individual is described with its particular attributes and behaviour, and macro-simulation models based on stochastic differential equations, where the population is described in aggregated terms by the number of individuals in different states. Consistency between micro- and macro-models is a crucial but often neglected aspect. This paper demonstrates how the Poisson Simulation technique can be used to produce a population macro-model consistent with the corresponding micro-model. This is accomplished by defining Poisson Simulation in strictly mathematical terms as a series of Poisson processes that generate sequences of Poisson distributions with dynamically varying parameters. The method can be applied to any population model. It provides the unique stochastic and dynamic macro-model consistent with a correct micro-model. The paper also presents a general macro form for stochastic and dynamic population models. In an appendix Poisson Simulation is compared with Markov Simulation showing a number of advantages. Especially aggregation into state variables and aggregation of many events per time-step makes Poisson Simulation orders of magnitude faster than Markov Simulation. Furthermore, you can build and execute much larger and more complicated models with Poisson Simulation than is possible with the Markov approach.     

Analysis and simulation of a stochastic, discrete-individual model of STD transmission with partnership concurrency

Deterministic differential equation models indicate that partnership concurrency and non-homogeneous mixing patterns play an important role in the spread of sexually transmitted infections. Stochastic discrete-individual simulation studies arrive at similar conclusions, but from a very different modeling perspective. This paper presents a stochastic discrete-individual infection model that helps to unify these two approaches to infection modeling. The model allows for both partnership concurrency, as well as the infection, recovery, and reinfection of an individual from repeated contact with a partner, as occurs with many mucosal infections. The simplest form of the model is a network-valued Markov chain, where the network's nodes are individuals and arcs represent partnerships. Connections between the differential equation and discrete-individual approaches are constructed with large-population limits that approximate endemic levels and equilibrium probability distributions that describe partnership concurrency. A more general form of the discrete-individual model that allows for semi-Markovian dynamics and heterogeneous contact patterns is implemented in simulation software. Analytical and simulation results indicate that the basic reproduction number R(0) increases when reinfection is possible, and the epidemic rate of rise and endemic levels are not related by 1-1/R(0), when partnerships are not point-time processes.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?(0) can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ?(j), j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

Stochastic epidemic models: A survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.     

The critical vaccination fraction for heterogeneous epidemic models

Given a population with m heterogeneous subgroups, a method is developed for determining minimal vaccine allocations to prevent an epidemic by setting the reproduction number to 1. The framework is sufficiently general to apply to several epidemic situations, such as SIR, SEIR and SIS models with vital dynamics. The reproduction number is the largest eigenvalue of the linearized system round the local point of equilibrium of the model. Using the Perron-Frobenius theorem, an exact method for generating solutions is given and the threshold surface of critical vaccine allocations is shown to be a compact, connected subset of a regular (m-1)-dimensional manifold. Populations with two subgroups are examined in full. The threshold curves are either hyperbolas or straight lines. Explicit conditions are given as to when threshold elimination is achievable by vaccinating just one or two groups in a multi-group population and expressions for the critical coverage are derived. Specific reference is made to an influenza A model. Separable or proportionate mixing is also treated. Conditions are conjectured for convexity of the threshold surface and the problem of minimizing the amount of vaccine used while remaining on the threshold surface is discussed.     

FluTE,a Publicly Available Stochastic Influenza Epidemic Simulation Model

Mathematical and computer models of epidemics have contributed to our understanding of the spread of infectious disease and the measures needed to contain or mitigate them. To help prepare for future influenza seasonal epidemics or pandemics, we developed a new stochastic model of the spread of influenza across a large population. Individuals in this model have realistic social contact networks, and transmission and infections are based on the current state of knowledge of the natural history of influenza. The model has been calibrated so that outcomes are consistent with the 1957/1958 Asian A(H2N2) and 2009 pandemic A(H1N1) influenza viruses. We present examples of how this model can be used to study the dynamics of influenza epidemics in the United States and simulate how to mitigate or delay them using pharmaceutical interventions and social distancing measures. Computer simulation models play an essential role in informing public policy and evaluating pandemic preparedness plans. We have made the source code of this model publicly available to encourage its use and further development.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Structural and Practical Identifiability Analysis of Zika Epidemiological Models

The Zika virus (ZIKV) epidemic has caused an ongoing threat to global health security and spurred new investigations of the virus. Use of epidemiological models for arbovirus diseases can be a powerful tool to assist in prevention and control of the emerging disease. In this article, we introduce six models of ZIKV, beginning with a general vector-borne model and gradually including different transmission routes of ZIKV. These epidemiological models use various combinations of disease transmission (vector and direct) and infectious classes (asymptomatic and pregnant), with addition to loss of immunity being included. The disease-induced death rate is omitted from the models. We test the structural and practical identifiability of the models to find whether unknown model parameters can uniquely be determined. The models were fit to obtain time-series data of cumulative incidences and pregnant infections from the Florida Department of Health Daily Zika Update Reports. The average relative estimation errors (AREs) were computed from the Monte Carlo simulations to further analyze the identifiability of the models. We show that direct transmission rates are not practically identifiable; however, fixed recovery rates improve identifiability overall. We found ARE is low for each model (only slightly higher for those that account for a pregnant class) and help to confirm a reproduction number greater than one at the start of the Florida epidemic. Basic reproduction number, \(\mathcal {R}_0\), is an epidemiologically important threshold value which gives the number of secondary cases generated by one infected individual in a totally susceptible population in duration of infectiousness. Elasticity of the reproduction numbers suggests that the mosquito-to-human ratio, mosquito life span and biting rate have the greatest potential for reducing the reproduction number of Zika, and therefore, corresponding control measures need to be focused on.     

Dependence of epidemic and population velocities on basic parameters.

This paper describes the use of linear deterministic models for examining the spread of population processes, discussing their advantages and limitations. Their main advantages are that their assumptions are relatively transparent and that they are easy to analyze, yet they generally give the same velocity as more complex linear stochastic and nonlinear deterministic models. Their simplicity, especially if we use the elegant reproduction and dispersal kernel formulation of Diekmann and van den Bosch et al., allows us greater freedom to choose a biologically realistic model and greatly facilitates examination of the dependence of conclusions on model components and of how these are incorporated into the model and fitted from data. This is illustrated by consideration of a range of examples, including both diffusion and dispersal models and by discussion of their application to both epidemic and population dynamic problems. A general limitation on fitting models results from the poor accuracy of most ecological data, especially on dispersal distances. Confirmation of a model is thus rarely as convincing as those cases where we can clearly reject one. We also need to be aware that linear models provide only an upper bound for the velocity of more realistic nonlinear stochastic models and are almost wholly inadequate when it comes to modeling more complex aspects such as the transition to endemicity and endemic patterns. These limitations are, however, to a great extent shared by linear stochastic and nonlinear deterministic models.     

Modeling optimal treatment strategies in a heterogeneous mixing model

Background

Many mathematical models assume random or homogeneous mixing for various infectious diseases. Homogeneous mixing can be generalized to mathematical models with multi-patches or age structure by incorporating contact matrices to capture the dynamics of the heterogeneously mixing populations. Contact or mixing patterns are difficult to measure in many infectious diseases including influenza. Mixing patterns are considered to be one of the critical factors for infectious disease modeling.

Methods

A two-group influenza model is considered to evaluate the impact of heterogeneous mixing on the influenza transmission dynamics. Heterogeneous mixing between two groups with two different activity levels includes proportionate mixing, preferred mixing and like-with-like mixing. Furthermore, the optimal control problem is formulated in this two-group influenza model to identify the group-specific optimal treatment strategies at a minimal cost. We investigate group-specific optimal treatment strategies under various mixing scenarios.

Results

The characteristics of the two-group influenza dynamics have been investigated in terms of the basic reproduction number and the final epidemic size under various mixing scenarios. As the mixing patterns become proportionate mixing, the basic reproduction number becomes smaller; however, the final epidemic size becomes larger. This is due to the fact that the number of infected people increases only slightly in the higher activity level group, while the number of infected people increases more significantly in the lower activity level group. Our results indicate that more intensive treatment of both groups at the early stage is the most effective treatment regardless of the mixing scenario. However, proportionate mixing requires more treated cases for all combinations of different group activity levels and group population sizes.

Conclusions

Mixing patterns can play a critical role in the effectiveness of optimal treatments. As the mixing becomes more like-with-like mixing, treating the higher activity group in the population is almost as effective as treating the entire populations since it reduces the number of disease cases effectively but only requires similar treatments. The gain becomes more pronounced as the basic reproduction number increases. This can be a critical issue which must be considered for future pandemic influenza interventions, especially when there are limited resources available.

     

Extending the type reproduction number to infectious disease control targeting contacts between types

A new quantity called the target reproduction number is defined to measure control strategies for infectious diseases with multiple host types such as waterborne, vector-borne and zoonotic diseases. The target reproduction number includes as a special case and extends the type reproduction number to allow disease control targeting contacts between types. Relationships among the basic, type and target reproduction numbers are established. Examples of infectious disease models from the literature are given to illustrate the use of the target reproduction number.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?₀ can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ? _j , j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely “take,” “degree,” “duration,” “infectiousness,” and “progression,” and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.