Epidemic models with heterogeneous mixing and treatment

The impact of corona populism: Empirical evidence from Austria and theory

I study the co-evolution between public opinion and party policy in situations of crises by investigating a policy U-turn of a major Austrian right-wing party (FPÖ) during the Covid-19 pandemic. My analysis suggests the existence of both i) a "Downsian" effect, which causes voters to adapt their party preferences based on policy congruence and ii) a "party identification" effect, which causes partisans to realign their policy preferences based on "their" party's platform. Specifically, I use individual-level panel data to show that i) "corona skeptical" voters who did not vote for the FPÖ in the pre-Covid-19 elections of 2019 were more likely to vote for the party after it embraced "corona populism", and ii) beliefs of respondents who declared that they voted for the FPÖ in 2019 diverged from the rest of the population in three out of four health-related dimensions only after the turn, causing them to underestimate the threat posed by Covid-19 compared to the rest of the population. Using aggregate-level panel data, I study whether the turn has produced significant behavioral differences which could be observed in terms of reported cases and deaths per capita. Paradoxically, after the turn the FPÖ vote share is significantly positively correlated with deaths per capita, but not with the reported number of infections. I hypothesize that this can be traced back to a self-selection bias in testing, which causes a correlation between the number of "corona skeptics" and the share of unreported cases after the turn. I find empirical support for this hypothesis in individual-level data from a Covid-19 prevalence study that involves information about participants' true vs. reported infection status. I finally study a simple heterogeneous mixing epidemiological model and show that a testing bias can indeed explain the apparent paradox of an increase in deaths without an increase in reported cases. My results can, among others, be used to enrich formal analyses regarding the co-evolution between voter and party behavior.

An age-of-infection model with both symptomatic and asymptomatic infections

We formulate a general age-of-infection epidemic model with two pathways: the symptomatic infections and the asymptomatic infections. We then calculate the basic reproduction number [Formula: see text] and establish the final size relation. It is shown that the ratio of accumulated counts of symptomatic patients and asymptomatic patients is determined by the symptomatic ratio f which is defined as the probability of eventually becoming symptomatic after being infected. We also formulate and study a general age-of-infection model with disease deaths and with two infection pathways. The final size relation is investigated, and the upper and lower bounds for final epidemic size are given. Several numerical simulations are performed to verify the analytical results.

Implications for infectious disease models of heterogeneous mixing on control thresholds

Mixing among sub-populations, as well as heterogeneity in characteristics affecting their reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. In this overview, we apply a linear algebraic approach to re-derive some well-known results pertaining to preferential within- and proportionate among-group contacts in compartmental models of pathogen transmission. We give results for the meta-population effective reproduction number ([Formula: see text]) assuming different levels of vaccination in the sub-populations. Specifically, we unpack the dependency of [Formula: see text] on the fractions of contacts reserved for individuals within one's own subgroup and, by obtaining implicit expressions for the partial derivatives of [Formula: see text], we show that these increase as this preferential-mixing fraction increases in any sub-population.

Modeling the role of land conversion on the spread of an epizootic disease

Land conversion and the resulting contact between domesticated and wild species has arguably been the single largest contributor to the emergence of novel epizootic and zoonotic diseases in the past century. An unintended consequence of these interactions is zoonotic or epizootic disease spillovers from wild species to humans and their domesticates. Disease spillovers are edge effects of land conversion and are sensitive to the size and shape of converted areas. We combine spatial metrics from landscape ecology with theoretical epidemiological models to understand how the size and shape of land conversion affect epizootic and zoonotic disease transmission of single and two species populations. We show that the less compact the converted area, and the greater the depth of the contact zone, the more rapidly will an introduced disease spread through the domesticated population.

Asymptotic Analysis of Optimal Vaccination Policies

Targeted vaccination policies can have a significant impact on the number of infections and deaths in an epidemic. However, optimising such policies is complicated, and the resultant solution may be difficult to explain to policy-makers and to the public. The key novelty of this paper is a derivation of the leading-order optimal vaccination policy under multi-group susceptible-infected-recovered dynamics in two different cases. Firstly, it considers the case of a small vulnerable subgroup in a population and shows that (in the asymptotic limit) it is optimal to vaccinate this group first, regardless of the properties of the other groups. Then, it considers the case of a small vaccine supply and transforms the optimal vaccination problem into a simple knapsack problem by linearising the final size equations. Both of these cases are then explored further through numerical examples, which show that these solutions are also directly useful for realistic parameter values. Moreover, the findings of this paper give some general principles for optimal vaccination policies which will help policy-makers and the public to understand the reasoning behind optimal vaccination programs in more generic cases.

Mathematical Analysis of Epidemic Models with Treatment in Heterogeneous Networks

In this paper, we formulate two different network-based epidemic models to investigate the effect of partly effective treatment on disease dynamics. The first network model represents the individuals with heterogeneous number of contacts in a population as choosing a new partner at each moment, whereas the second one assumes the individuals have fixed or stable neighbors. The basic reproduction number [Formula: see text] is computed for each model, using the next generation matrix method. In particular, the critical treatment rate is defined for the model, above which the disease can be eliminated through the treatment. The final epidemic size relations are derived, and the solvability of these implicit equations is studied. In particular, a unique solution of the implicit equation for the final epidemic size is determined, and by rewriting the implicit equation as a suitable fixed point problem, it is proved that the iteration of the fixed point problem converges to the unique solution. Stochastic simulations and numerical simulations, including in comparison with the model outputs and the joint influence of network topology and treatment on the final epidemic size, are conducted to illustrate the theoretical results.

Effect of Human Mobility on the Spatial Spread of Airborne Diseases: An Epidemic Model with Indirect Transmission

We extended a class of coupled PDE-ODE models for studying the spatial spread of airborne diseases by incorporating human mobility. Human populations are modeled with patches, and a Lagrangian perspective is used to keep track of individuals' places of residence. The movement of pathogens in the air is modeled with linear diffusion and coupled to the SIR dynamics of each human population through an integral of the density of pathogens around the population patches. In the limit of fast diffusion pathogens, the method of matched asymptotic analysis is used to reduce the coupled PDE-ODE model to a nonlinear system of ODEs for the average density of pathogens in the air. The reduced system of ODEs is used to derive the basic reproduction number and the final size relation for the model. Numerical simulations of the full PDE-ODE model and the reduced system of ODEs are used to assess the impact of human mobility, together with the diffusion of pathogens on the dynamics of the disease. Results from the two models are consistent and show that human mobility significantly affects disease dynamics. In addition, we show that an increase in the diffusion rate of pathogen leads to a lower epidemic.

Effect of Non-homogeneous Mixing and Asymptomatic Individuals on Final Epidemic Size and Basic Reproduction Number in a Meta-Population Model

To uncover the effective interventions during the pandemic period, a novel mathematical model, which incorporates separate compartments for incubation and asymptomatic individuals, has been developed in this paper. On the basis of a general mixing, final size relation and next-generation matrix are derived for a meta-population model by introducing the matrix blocking. The final size (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}F) and the basic reproduction number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0}$$\end{document}R0) are no longer a simple monotonous relationship. The analytical results of heterogeneity illustrate that activity is more sensitive than the others. And the proportion of asymptomatic individuals is a key factor for final epidemic size compared to the regulatory factor. Furthermore, the impact of preferential contact level on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}F and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0}$$\end{document}R0 is comparatively complex. The isolation can effectively reduce the final size, which further verifies its effectiveness. When vaccination is considered, the mixing methods maybe influence the doses of vaccination used and its effective. Moreover, using the present predictive model, we can provide the valuable reference about identifying the ideal strategies to curb the pandemic disease.

On the basic reproduction number in semi-Markov switching networks

Basic reproduction number R0 in network epidemic dynamics is studied in the case of stochastic regime-switching networks. For generality, the dependence between successive networks is considered to follow a continuous time semi-Markov chain. R0 is the weighted average of the basic reproduction numbers of deterministic subnetworks. Its position with respect to 1 can determine epidemic persistence or extinction in theories and simulations.

Mathematical modeling of bird flu with vaccination and treatment for the poultry farms

A deterministic six-compartmental model was developed based on the progression of the disease in poultry, the epidemiological status of the individuals, and intervention measures. The Runge-Kutta method is applied to calculate the variables of the system of equations of the proposed model. The evolution of the epidemic provides some results, such as reproduction number, vaccine efficiency, and antiviral treatment. Numerical results show that the outbreak sizes known as the infected curves increase and decrease with the vaccine limitation rate and treatment rate, respectively, for a specific transmission rate. The calculated results of the reproduction number indicate that avian influenza would spread when vaccine efficiency is less than 70%, and the primary reproduction number is greater than 1. Finally, the disease-free equilibrium of the model is found locally and globally asymptotically stable for R₀ < 1.

Assessing the impact of adherence to Non-pharmaceutical interventions and indirect transmission on the dynamics of COVID-19: a mathematical modelling study

Adherence to public health policies such as the non-pharmaceutical interventions implemented against COVID-19 plays a major role in reducing infections and controlling the spread of the diseases. In addition, understanding the transmission dynamics of the disease is also important in order to make and implement efficient public health policies. In this paper, we developed an SEIR-type compartmental model to assess the impact of adherence to COVID-19 non-pharmaceutical interventions and indirect transmission on the dynamics of the disease. Our model considers both direct and indirect transmission routes and stratifies the population into two groups: those that adhere to COVID-19 non-pharmaceutical interventions (NPIs) and those that do not adhere to the NPIs. We compute the control reproduction number and the final epidemic size relation for our model and study the effect of different parameters of the model on these quantities. Our results show that there is a significant benefit in adhering to the COVID-19 NPIs.

Dynamic analysis of a SIQR epidemic model considering the interaction of environmental differences

Infectious diseases have a devastating impact on individual health and social development. Different external environmental factors will affect the scale and the speed of disease outbreaks, such as sanitary conditions and policy interventions. In this paper, we attempt to establish a dual-system susceptible-infectious-quarantine-recovered model with different environmental impacts to explore it. For the deterministic model, we calculate the basic reproduction number, simultaneously, and investigate the local asymptotic stability of the disease-free equilibrium and the endemic equilibrium. Numerical simulations and theoretical analyses verify the conclusion of this paper. It is a surprise that there is a great probability of finding a quarantine inflection point, which can effectively control the scale of infection outbreak when two different systems of infection rate and recovery rate are determined.

Commuting in metapopulation epidemic modeling

The COVID-19 pandemic led authorities all over the world to imposing travel restrictions both on a national and on an international scale. Understanding the effect of such restrictions requires analysis of the role of commuting and calls for a metapopulation modeling that incorporates both local, intra-community infection and population exchange between different locations. Standard metapopulation models are formulated as markovian processes, and as such they do not label individuals according to their original location. However, commuting from home to work and backwards (reverse commuting) is the main pattern of transportation. Thus, it is important to be able to accurately model the effect of commuting on epidemic spreading. In this study we develop a methodology for modeling bidirectional commuting of individuals, without keeping track of each individual separately and with no need of proliferation of number of compartments beyond those defined by the epidemiologic model. We demonstrate the method using a city map of the state of Israel. The presented algorithm does not require any special computation resources and it may serve as a basis for intervention strategy examination in various levels of complication and resolution. We show how to incorporate an epidemiological model into a metapopulation commuting scheme while preserving the internal logic of the epidemiological modeling. The method is general and independent on the details of the epidemiological model under consideration.

Effect of population heterogeneity on herd immunity and on vaccination decision making process

We study the influence of population heterogeneity on herd immunity level and on individual's vaccination decision making process. We first formulate the mathematical model in a population with two subgroups, based on different activity levels or different susceptibilities. The herd immunity threshold is derived and discussed. It is calculated that the required vaccine coverage level for herd immunity in a heterogeneous mixing population can be varied significantly. The required vaccine coverage level is lower than the classical herd immunity level, if the vaccine coverage level in the more active group or more susceptible group is higher than the other subgroup. It is suggested that the classical herd immunity levels can be misleading in the process of planning mass vaccination programs. The analysis is further extended to study the population with more subgroups. We then study the formal vaccination games to simulate the process of vaccination decision making, in either homogeneous or heterogeneous mixing populations. It is proved that the Nash equilibrium in the vaccination game is not unique if population heterogeneity is considered. Moreover, herd immunity is not achieved if individuals are solely driven by self-interests.

The Effect of Face Mask Use on COVID-19 Models

We begin with a simple model for the COVID-19 epidemic and add face mask usages and testing and quarantine of infectives. We estimate the effect on the reproduction number and discuss the question of whether the epidemic can be controlled by increased use of face masks.

Evaluating different epidemiological models with the identical basic reproduction number ℛ0

Epidemiological models with the identical basic reproduction number R0 may behave differently on both short time scale and long time scale. In this paper, we compare the predicted final sizes for several deterministic epidemic models and estimate the probabilities of a minor/major outbreak for continuous-time Markov chain (CTMC) models, all epidemic models have the identical R0 . It is proved that the final size predicted by the epidemic model with homogeneous mixing is larger than with heterogeneous mixing. For CTMC models with heterogeneous mixing, the probabilities of a minor outbreak initiated by superspreaders and non-superspreaders are calculated and compared. For both deterministic modelling and stochastic modelling, numerical simulations are performed to support the mathematical analysis.

Resource Allocation in Two-Patch Epidemic Model with State-Dependent Dispersal Behaviors Using Optimal Control

A two-patch epidemic model is considered in order to assess the impact of virtual dispersal on disease transmission dynamics. The two-patch system models the movement of individuals between the two-patches using a residence-time matrix P, where P depends on both residence times and state variables (infected classes). In this work, we employ this approach to a general two-patch SIR model in order to investigate the effect of state dependent dispersal behaviors on the disease dynamics. Furthermore, optimal control theory is employed to identify and evaluate patch-specific control measures aimed at reducing disease prevalence at a minimal cost. Optimal policies are computed under various dispersal scenarios (depending on the different residence-time matrix configurations). Our results suggest there is a reduction of the outbreak and the proportion of time spent by individuals in a patch exhibits less fluctuations in the presence of patch-specific optimal controls. Furthermore, the optimal strategies for each patch differ depending on the type of dispersal behavior and the different infection rate in a patch. In all of our results, we obtain that the optimal strategies reduce the number of infections per patch.

A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission

We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.

The role of HIV viral load in mathematical models of HIV transmission and treatment: a review

Introduction

HIV viral load (VL) is accepted as a key biomarker in HIV transmission and pathogenesis. This paper presents a review of the role of VL testing in mathematical models for HIV prevention and treatment.

Methods

A search for simulation models of HIV was conducted in PubMed, yielding a total of 1210 studies. Publications before the year 2000, studies involving animals and analyses that did not use mathematical simulations were excluded. The full text of eligible articles was sourced and information about the intervention and population being modelled, type of modelling approach and disease monitoring strategy was extracted.

Results and discussion

A total of 279 studies related to HIV simulation models were included in the review, though only 17 (6%) included consideration of VL or VL testing and were evaluated in detail. Within the studies that included assessment of VL, routine monitoring was the focus, and usually in comparison to alternate monitoring strategies such as clinical or CD4 count-based monitoring. The majority of remaining models focus on the impact or delivery of antiretroviral therapy (n=68; 27%), pre-exposure prophylaxis (n=28; 11%) and/or HIV testing (n=24; 9%) on population estimates of HIV epidemiology and exclude consideration of VL. Few studies investigate or compare alternate VL monitoring frequencies, and only a small number of studies overall (3%) include consideration of vulnerable population groups such as pregnant women or infants.

Conclusions

There are very few simulations of HIV treatment or prevention that include VL measures, despite VL being recognised as the key determinant of both transmission and treatment outcomes. With growing emphasis on VL monitoring as key tool for population-level HIV control, there is a clear need for simulations of HIV epidemiology based on VL.

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.

Prioritization of vaccine strategy using an age-dependent mathematical model for 2009 A/H1N1 influenza in the Republic of Korea

We developed a mathematical model of the 2009 A/H1N1 influenza epidemic in the Republic of Korea by considering five age groups and suggested the best way to prioritize an age-dependent vaccination strategy for mitigating the epidemic. An age-structured SEIAR influenza model was constructed based on the laboratory confirmed data obtained from the Korea Centers for Disease Control and Prevention (KCDC). The estimated transmission matrix captured one of the main characteristics of the 2009 A/H1N1 influenza, the transmission rate of which is high among young people, unlike that of seasonal influenza. We investigated the impact of age-dependent vaccination priority on the transmission dynamics of the 2009 A/H1N1 influenza and evaluated the Korean government vaccination policy when the vaccination started being administered 90 days (or 120 days) after the onset of the outbreak. We found that the government's age priority vaccination policy (Group 2, Group 1, Group 5, Group 4, and Group 3 in order) was a good strategy for reducing 62.06% of the cumulative cases when the vaccination was applied 90 days after the onset of the outbreak, while the proposed model's best suggestion (Group 2, Group 1, Group 3, Group 4, and Group 5 in order) showed 64.52% reduction. Furthermore, we studied the region-specific vaccination policy. For instance, the best age-priority of vaccination in Gwangwon province showed a different order (Group 3, Group 1, Group 2, Group 4, and Group 5 in order) and it reduced the incidence by 58.1%, which is 5.54% higher than that of the 2009 Korean government policy.

N-intertwined SIS epidemic model with Markovian switching

Epidemic dynamics is often subject to environmental noise and uncertainty. In this paper, we investigate the effect of color noise on the spread of epidemic in complex networks, which is modeled by stochastic switched differential equations based on the [Formula: see text]-intertwined SIS model using a continuous time finite-state Markov chain. Applying Lyapunov functions, we prove that the model has a unique global positive solution and establish sufficient conditions for stochastic extinction and permanence of the epidemic. We also show that the solution is stochastically ultimately bounded and the variance of the solution is bounded too. Furthermore, we discuss the limit of the time average of the solution. Finally, numerical simulations are carried out to illustrate our theoretical results.

Influence of non-homogeneous mixing on final epidemic size in a meta-population model

In meta-population models for infectious diseases, the basic reproduction number R0 can be as much as 70% larger in the case of preferential mixing than that in homogeneous mixing [J.W. Glasser, Z. Feng, S.B. Omer, P.J. Smith, and L.E. Rodewald, The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: A modelling study, Lancet ID 16 (2016), pp. 599-605. doi: 10.1016/S1473-3099(16)00004-9 ]. This suggests that realistic mixing can be an important factor to consider in order for the models to provide a reliable assessment of intervention strategies. The influence of mixing is more significant when the population is highly heterogeneous. In this paper, another quantity, the final epidemic size ( F ) of an outbreak, is considered to examine the influence of mixing and population heterogeneity. Final size relation is derived for a meta-population model accounting for a general mixing. The results show that F can be influenced by the pattern of mixing in a significant way. Another interesting finding is that, heterogeneity in various sub-population characteristics may have the opposite effect on R0 and F .

Introduction: A Prelude to Mathematical Epidemiology

Recorded history continuously documents the invasion of populations by infectious agents, some causing many deaths before disappearing, others reappearing in invasions some years later in populations that have acquired some degree of immunity, due to prior exposure to related infectious pathogens. The “Spanish” flu epidemic of 1918–1919 exemplifies the devastating impact of relatively rare pandemics; this one was responsible for about 50,000,000 deaths worldwide, while on the mild side of the spectrum we experience annual influenza seasonal epidemics that cause roughly 35,000 deaths in the USA each year.

Epidemic models with heterogeneous mixing and indirect transmission

We develop an age of infection model with heterogeneous mixing in which indirect pathogen transmission is considered as a good way to describe contact that is usually considered as direct and we also incorporate virus shedding as a function of age of infection. The simplest form of SIRP epidemic model is introduced and it serves as a basis for the age of infection model and a 2-patch SIRP model where the risk of infection is solely dependent on the residence times and other environmental factors. The computation of the basic reproduction number [Formula: see text], the initial exponential growth rate and the final size relation is done and by mathematical analysis, we study the impact of patches connection and use the final size relation to analyse the ability of disease to invade over a short period of time.

Epidemics in Competition: Partial Cross-Immunity

The competition between two pathogen strains during the course of an epidemic represents a fundamental step in the early evolution of emerging diseases as well as in the antigenic drift process of influenza. The outcome of the competition, however, depends not only on the epidemic properties of the two strains but also on the timing and size of the introduction, characteristics that are poorly captured by deterministic mean-field epidemic models. We describe those aspects of the competition that can be determined from the mean-field models giving the range of possible final sizes of susceptible hosts and cumulated attack rates that could be observed after an epidemic with two cross-reacting strains. In the limit where the size of the initial infection goes to zero, the possible outcomes lie on a (one dimensional) curve in the outcome space.

Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission

A model of an epidemic outbreak incorporating multiple subgroups of susceptible and infected individuals is investigated. The asymptotic behavior of the model is analyzed and it is proved that the infected classes all converge to 0. A computational algorithm is developed for the cumulative final size of infected individuals over the course of the epidemic. The results are applied to the SARS epidemic in Singapore in 2003, where it is shown that the two-peak evolution of the infected population can be attributed to a two-group formulation of transmission.

Mathematical epidemiology: Past, present, and future

We give a brief outline of some of the important aspects of the development of mathematical epidemiology.

Assessing the Potential Impact of Hormonal-Based Contraceptives on HIV Transmission Dynamics Among Heterosexuals

HIV susceptibility linked to hormonal contraception (HC) has been studied before, but with mixed results. Reports from some of the recent findings have prompted the World Health Organisation to encourage women who use HC to concurrently use condoms in order to prevent HIV infection in the light of possible increased HIV risk of infection associated with hormone-based contraceptives. A two-sex HIV model classifying women into three risk groups consisting of individuals who use condoms, natural methods, and hormone-based contraceptives is formulated and analysed to assess the possible effects of various birth control strategies on the transmission dynamics of the disease. Our model results showed that women who use HC could be key drivers of the epidemic and that their increased infectivity may be critical in driving the epidemic. Women who use hormone-based contraceptives potentially act as a core group from which men get infected and in turn transmit the disease to other population groups. We fitted the model to HIV prevalence data for Zimbabwe reported by UNAIDS and Zimbabwe Ministry of Health and Child Care and used the model fit to project HIV prevalence. Predictions using HIV data for Zimbabwe suggest that a hypothesised increase in susceptibility and infectivity of two-, three-, and fourfold would result in a 25, 50, and 100% increase in baseline HIV prevalence projection, respectively, thus suggesting possible increased disease burden even in countries reporting plausible HIV prevalence declines. Although a possible causal relationship between HIV susceptibility and HC use remains subject of continuing scientific probe, its inclusion as part of birth control strategy has been shown in this study, to possibly increase HIV transmission. If proven, HC use may potentially explain the inordinate spread of HIV within the sub-Saharan Africa region and therefore compel for urgent assessment with a view to reorienting birth control methods in use in settings with generalised epidemics.

Toward a Realistic Modeling of Epidemic Spreading with Activity Driven Networks

Models of epidemic spreading are widely used to predict the evolution of an outbreak, test specific intervention scenarios, and steer interventions in the field. Compartmental models are the most common class of models. They are very effective for qualitative analysis, but they rely on simplifying assumptions, such as homogeneous mixing and time scale separation. On the other end of the spectrum, detailed agent-based models, based on realistic mobility pattern models, provide extremely accurate predictions. However, these models require significant computing power and are not suitable for analytical treatment. Our research aims at bridging the gap between these two approaches, toward time-varying network models that are sufficiently accurate to make predictions for real-world applications, while being computationally affordable and amenable to analytical treatment. We leverage the novel paradigm of activity driven networks (ADNs), a particular type of time-varying network that accounts for inherent inhomogeinities within a population. Starting from the basic incarnation of ADNs, we expand on the framework to include behavioral factors triggered by health status and spreading awareness. The enriched paradigm is then utilized to model the 2014–2015 Ebola Virus Disease (EVD) spreading in Liberia, and perform a what-if analysis on the timely application of sanitary interventions in the field. Finally, we propose a new formulation, which is amenable to analytical treatment, beyond the mere computation of the epidemic threshold.

Uniqueness of Nash equilibrium in vaccination games

One crucial condition for the uniqueness of Nash equilibrium set in vaccination games is that the attack ratio monotonically decreases as the vaccine coverage level increasing. We consider several deterministic vaccination models in homogeneous mixing population and in heterogeneous mixing population. Based on the final size relations obtained from the deterministic epidemic models, we prove that the attack ratios can be expressed in terms of the vaccine coverage levels, and also prove that the attack ratios are decreasing functions of vaccine coverage levels. Some thresholds are presented, which depend on the vaccine efficacy. It is proved that for vaccination games in homogeneous mixing population, there is a unique Nash equilibrium for each game.

The Impact of Population, Contact, and Spatial Heterogeneity on Epidemic Model Predictions

Our objective was to evaluate the effect that complexity in the form of different levels of spatial, population, and contact heterogeneity has in the predictions of a mechanistic epidemic model. A model that simulates the spatiotemporal spread of infectious diseases between animal populations was developed. Sixteen scenarios of foot-and-mouth disease infection in cattle were analyzed, involving combinations of the following factors: multiple production-types (PT) with heterogeneous contact and population structure versus single PT, random versus actual spatial distribution of population units, high versus low infectivity, and no vaccination versus preemptive vaccination. The epidemic size and duration was larger for scenarios with multiple PT versus single PT. Ignoring the actual unit locations did not affect the epidemic size in scenarios with multiple PT/high infectivity, but resulted in smaller epidemic sizes in scenarios using multiple PT/low infectivity. In conclusion, when modeling fast-spreading epidemics, knowing the actual locations of population units may not be as relevant as collecting information on population and contact heterogeneity. In contrast, both population and spatial heterogeneity might be important to model slower spreading epidemic diseases. Our findings can be used to inform data collection and modeling efforts to inform health policy and planning.

A network model for Ebola spreading

The availability of accurate models for the spreading of infectious diseases has opened a new era in management and containment of epidemics. Models are extensively used to plan for and execute vaccination campaigns, to evaluate the risk of international spreadings and the feasibility of travel bans, and to inform prophylaxis campaigns. Even when no specific therapeutical protocol is available, as for the Ebola Virus Disease (EVD), models of epidemic spreading can provide useful insight to steer interventions in the field and to forecast the trend of the epidemic. Here, we propose a novel mathematical model to describe EVD spreading based on activity driven networks (ADNs). Our approach overcomes the simplifying assumption of homogeneous mixing, which is central to most of the mathematically tractable models of EVD spreading. In our ADN-based model, each individual is not bound to contact every other, and its network of contacts varies in time as a function of an activity potential. Our model contemplates the possibility of non-ideal and time-varying intervention policies, which are critical to accurately describe EVD spreading in afflicted countries. The model is calibrated from field data of the 2014 April-to-December spreading in Liberia. We use the model as a predictive tool, to emulate the dynamics of EVD in Liberia and offer a one-year projection, until December 2015. Our predictions agree with the current vision expressed by professionals in the field, who consider EVD in Liberia at its final stage. The model is also used to perform a what-if analysis to assess the efficacy of timely intervention policies. In particular, we show that an earlier application of the same intervention policy would have greatly reduced the number of EVD cases, the duration of the outbreak, and the infrastructures needed for the implementation of the intervention.

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.

SIS and SIR Epidemic Models Under Virtual Dispersal

We develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general n-patch SIS model whose basic reproduction number [Formula: see text] is computed as a function of a patch residence-time matrix [Formula: see text]. Our analysis implies that the resulting n-patch SIS model has robust dynamics when patches are strongly connected: There is a unique globally stable endemic equilibrium when [Formula: see text], while the disease-free equilibrium is globally stable when [Formula: see text]. Our further analysis indicates that the dispersal behavior described by the residence-time matrix [Formula: see text] has profound effects on the disease dynamics at the single patch level with consequences that proper dispersal behavior along with the local environmental risk can either promote or eliminate the endemic in particular patches. Our work highlights the impact of residence-time matrix if the patches are not strongly connected. Our framework can be generalized in other endemic and disease outbreak models. As an illustration, we apply our framework to a two-patch SIR single-outbreak epidemic model where the process of disease invasion is connected to the final epidemic size relationship. We also explore the impact of disease-prevalence-driven decision using a phenomenological modeling approach in order to contrast the role of constant versus state-dependent [Formula: see text] on disease dynamics.

Influence of sociodemographic characteristics on human mobility [corrected]

Human mobility has been traditionally studied using surveys that deliver snapshots of population displacement patterns. The growing accessibility to ICT information from portable digital media has recently opened the possibility of exploring human behavior at high spatio-temporal resolutions. Mobile phone records, geolocated tweets, check-ins from Foursquare or geotagged photos, have contributed to this purpose at different scales, from cities to countries, in different world areas. Many previous works lacked, however, details on the individuals’ attributes such as age or gender. In this work, we analyze credit-card records from Barcelona and Madrid and by examining the geolocated credit-card transactions of individuals living in the two provinces, we find that the mobility patterns vary according to gender, age and occupation. Differences in distance traveled and travel purpose are observed between younger and older people, but, curiously, either between males and females of similar age. While mobility displays some generic features, here we show that sociodemographic characteristics play a relevant role and must be taken into account for mobility and epidemiological modelization.

One-Health Simulation Modelling: A Case Study of Influenza Spread between Human and Swine Populations using NAADSM

The circulation of zoonotic influenza A viruses including pH1N1 2009 and H5N1 continue to present a constant threat to animal and human populations. Recently, an H3N2 variant spread from pigs to humans and between humans in limited numbers. Accordingly, this research investigated a range of scenarios of the transmission dynamics of pH1N1 2009 virus at the swine-human interface while accounting for different percentages of swine workers initially immune. Furthermore, the feasibility of using NAADSM (North American Animal Disease Spread Model) applied as a one-health simulation model was assessed. The study population included 488 swine herds and 29, 707 households of people within a county in Ontario, Canada. Households were categorized as follows: (i) rural households with swine workers, (ii) rural households without swine workers, and (iii) urban households without swine workers. Forty-eight scenarios were investigated, based on the combination of six scenarios around the transmissibility of the virus at the interface and four vaccination coverage levels of swine workers (0-60%), all under two settings of either swine or human origin of the virus. Outcomes were assessed in terms of stochastic 'die-out' fraction, size and time to peak epidemic day, overall size and duration of the outbreaks. The modelled outcomes indicated that minimizing influenza transmissibility at the interface and targeted vaccination of swine workers had significant beneficial effects. Our results indicate that NAADSM can be used as a framework to model the spread and control of contagious zoonotic diseases among animal and human populations, under certain simplifying assumptions. Further evaluation of the model is required. In addition to these specific findings, this study serves as a benchmark that can provide useful input to a future one-health influenza modelling studies. Some pertinent information gaps were also identified. Enhanced surveillance and the collection of high-quality information for more accurate parameterization of such models are encouraged.

Metapopulation epidemic models with heterogeneous mixing and travel behaviour

Background

Determining the pandemic potential of an emerging infectious disease and how it depends on the various epidemic and population aspects is critical for the preparation of an adequate response aimed at its control. The complex interplay between population movements in space and non-homogeneous mixing patterns have so far hindered the fundamental understanding of the conditions for spatial invasion through a general theoretical framework. To address this issue, we present an analytical modelling approach taking into account such interplay under general conditions of mobility and interactions, in the simplifying assumption of two population classes.

Methods

We describe a spatially structured population with non-homogeneous mixing and travel behaviour through a multi-host stochastic epidemic metapopulation model. Different population partitions, mixing patterns and mobility structures are considered, along with a specific application for the study of the role of age partition in the early spread of the 2009 H1N1 pandemic influenza.

Results

We provide a complete mathematical formulation of the model and derive a semi-analytical expression of the threshold condition for global invasion of an emerging infectious disease in the metapopulation system. A rich solution space is found that depends on the social partition of the population, the pattern of contacts across groups and their relative social activity, the travel attitude of each class, and the topological and traffic features of the mobility network. Reducing the activity of the less social group and reducing the cross-group mixing are predicted to be the most efficient strategies for controlling the pandemic potential in the case the less active group constitutes the majority of travellers. If instead traveling is dominated by the more social class, our model predicts the existence of an optimal across-groups mixing that maximises the pandemic potential of the disease, whereas the impact of variations in the activity of each group is less important.

Conclusions

The proposed modelling approach introduces a theoretical framework for the study of infectious diseases spread in a population with two layers of heterogeneity relevant for the local transmission and the spatial propagation of the disease. It can be used for pandemic preparedness studies to identify adequate interventions and quantitatively estimate the corresponding required effort, as well as in an emerging epidemic situation to assess the pandemic potential of the pathogen from population and early outbreak data.

A vaccination model for a multi-city system

A modelling approach is used for studying the effects of population vaccination on the epidemic dynamics of a set of n cities interconnected by a complex transportation network. The model is based on a sophisticated mover-stayer formulation of inter-city population migration, upon which is included the classical SIS dynamics of disease transmission which operates within each city. Our analysis studies the stability properties of the Disease-Free Equilibrium (DFE) of the full n-city system in terms of the reproductive number R(0). Should vaccination reduce R(0) below unity, the disease will be eradicated in all n-cities. We determine the precise conditions for which this occurs, and show that disease eradication by vaccination depend on the transportation structure of the migration network in a very direct manner. Several concrete examples are presented and discussed, and some counter-intuitive results found.

Impact of discontinuous treatments on disease dynamics in an SIR epidemic model

We consider an SIR epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we are able to determine the basic reproduction number R₀, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that in the case that an equilibrium is asymptotically stable, the convergence to the equilibrium can actually be achieved in finite time, and we can estimate this time in terms of the model parameters, initial sub-populations and the initial treatment strength. This suggests that from the view point of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we use to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on non-smooth analysis.

Critical immune and vaccination thresholds for determining multiple influenza epidemic waves

Previous influenza pandemics (1918, 1957, and 1968) have all had multiple waves. The 2009 pandemic influenza A (H1N1) (pandemic H1N1) started in April 2009 and was followed, in the United States (US) and temperate Northern Hemisphere, by a second wave during the fall of 2009. The ratio of susceptible and immune individuals in a population at the end of a wave determines the potential and magnitude of a subsequent wave. As influenza vaccines are not completely protective, there was a combined immunity in the population at the beginning of 2010 (due to vaccination and due to previous natural infection), and it was uncertain if this mixture of herd immunity was enough to prevent a third wave of pandemic influenza during the winter of 2010. Motivated by this problem, we developed a mathematical deterministic two-group epidemic model with vaccination and calibrated it for the 2009 pandemic H1N1. Then, applying methods from mathematical epidemiology we developed a scheme that allowed us to determine critical thresholds for vaccine-induced and natural immunity that would prevent the spread of influenza. Finally, we estimated the level of combined immunity in the US during winter 2010. Our results suggest that a third wave was unlikely if the basic reproduction number R(0) were below 1.6, plausible if the original R(0) was 1.6, and likely if the original R(0) was 1.8 or higher. Given that the estimates for the basic reproduction number for pandemic influenza place it in the range between 1.4 and 1.6 (Bacaer and Ait Dads, 2011; Fraser et al., 2009; Munayco et al., 2009; Pourbohloul et al., 2009; Tuite et al., 2010; White et al., 2009; Yang et al., 2009), our approach accurately predicted the absence of a third wave of influenza in the US during the winter of 2010. We also used this scheme to accurately predict the second wave of pandemic influenza in London and the West Midlands, UK during the fall of 2009.

The final size of an epidemic and its relation to the basic reproduction number

We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number R0. There is a unique positive solution giving the size of the epidemic if and only if R0 exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number R0. For other mixing patterns, the relation may be reversed.

Effectiveness analysis of resistance and tolerance to infection

Background

Tolerance and resistance provide animals with two distinct strategies to fight infectious pathogens and may exhibit different evolutionary dynamics. However, few studies have investigated these mechanisms in the case of animal diseases under commercial constraints.

Methods

The paper proposes a method to simultaneously describe (1) the dynamics of transmission of a contagious pathogen between animals, (2) the growth and death of the pathogen within infected hosts and (3) the effects on their performances. The effectiveness of increasing individual levels of tolerance and resistance is evaluated by the number of infected animals and the performance at the population level.

Results

The model is applied to a particular set of parameters and different combinations of values. Given these imputed values, it is shown that higher levels of individual tolerance should be more effective than increased levels of resistance in commercial populations. As a practical example, a method is proposed to measure levels of animal tolerance to bovine mastitis.

Conclusions

The model provides a general framework and some tools to maximize health and performances of a population under infection. Limits and assumptions of the model are clearly identified so it can be improved for different epidemiological settings.