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As the SARS-CoV-2 trajectory continues, the longer-term immuno-epidemiology of COVID-19, the dynamics of Long COVID, and the impact of escape variants are important outstanding questions. We examine these remaining uncertainties with a simple modelling framework that accounts for multiple (antigenic) exposures via infection or vaccination. If immunity (to infection or Long COVID) accumulates rapidly with the valency of exposure, we find that infection levels and the burden of Long COVID are markedly reduced in the medium term. More pessimistic assumptions on host adaptive immune responses illustrate that the longer term burden of COVID-19 may be elevated for years to come. However, we also find that these outcomes could be mitigated by the eventual introduction of a vaccine eliciting robust (i.e. durable, transmission-blocking and/or ‘evolution-proof’) immunity. Overall, our work stresses the wide range of future scenarios that still remain, the importance of collecting real world epidemiological data to identify likely outcomes, and the crucial need for the development of a highly effective transmission-blocking, durable, and broadly-protective vaccine.

Individual and societal reactions to an ongoing pandemic can lead to social dilemmas: In some cases, each individual is tempted to not follow an intervention, but for the whole society it would be best if they did. Now that in most countries the extent of regulations to reduce SARS-CoV-2 transmission is very small, interventions are driven by individual decision-making. Assuming that individuals act in their best own interest, we propose a framework in which this situation can be quantified, depending on the protection the intervention provides to a user and to others, the risk of getting infected, and the costs of the intervention. We discuss when a tension between individual and societal benefits arises and which parameter comparisons are important to distinguish between different regimes of intervention use.

COVID-19 nonpharmaceutical interventions (NPIs), including mask wearing, have proved highly effective at reducing the transmission of endemic infections. A key public health question is whether NPIs could continue to be implemented long term to reduce the ongoing burden from endemic pathogens. Here, we use epidemiological models to explore the impact of long-term NPIs on the dynamics of endemic infections. We find that the introduction of NPIs leads to a strong initial reduction in incidence, but this effect is transient: As susceptibility increases, epidemics return while NPIs are in place. For low R 0 infections, these return epidemics are of reduced equilibrium incidence and epidemic peak size. For high R 0 infections, return epidemics are of similar magnitude to pre-NPI outbreaks. Our results underline that managing ongoing susceptible buildup, e.g., with vaccination, remains an important long-term goal.

The analysis of the statistics of random walks undertaken by passive particles in complex media has important implications in a number of areas including pathogen transport and drug delivery. In several systems in which heterogeneity is important, the distribution of particle step-sizes has been found to be exponential in nature, as opposed to the Gaussian distribution associated with Brownian motion. Here, we first develop a theoretical framework to study a simplified version of this problem: the motion of passive tracers in a range of sub-environments with different viscosity. We show that in the limit of a large number of equi-distributed sub-environments spanning a broad viscosity range, an exact analytical expression for the underlying particle step-size distribution can be derived, which approaches an exponential distribution when step sizes are small. We then validate this using a simple experimental system of glycerol-water mixtures, in which the volume fraction of glycerol is systematically varied. Overall, the assumption of exponentially distributed step sizes may substantially over-estimate the incidence of large steps in heterogeneous systems, with important implications in the analysis of various biophysical processes.

Understanding viral evolution depends on a synthesis of evolutionary biology and immuno-epidemiology

Imperfect future immunity Humans are infected by several seasonal and cross-reacting coronaviruses. None provokes fully protective immunity, and repeat infections are the norm. Vaccines tend to be less efficient than natural infections at provoking immunity, and there are risks of adverse cross-reactions. Saad-Roy et al. used a series of simple models for a variety of immune scenarios to envisage immunological futures for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) with and without vaccines. The model outcomes show that our imperfect knowledge about the imperfect coronavirus immune landscape can give rise to diverging scenarios ranging from recurring severe epidemics to elimination. It is critical that we accurately characterize immune responses to SARS-CoV-2 for translation into managing disease control. Science, this issue p. 811, Modeling shows how essential it is to improve precise understanding of immune responses to SARS-CoV-2., The future trajectory of the coronavirus disease 2019 (COVID-19) pandemic hinges on the dynamics of adaptive immunity against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2); however, salient features of the immune response elicited by natural infection or vaccination are still uncertain. We use simple epidemiological models to explore estimates for the magnitude and timing of future COVID-19 cases, given different assumptions regarding the protective efficacy and duration of the adaptive immune response to SARS-CoV-2, as well as its interaction with vaccines and nonpharmaceutical interventions. We find that variations in the immune response to primary SARS-CoV-2 infections and a potential vaccine can lead to markedly different immune landscapes and burdens of critically severe cases, ranging from sustained epidemics to near elimination. Our findings illustrate likely complexities in future COVID-19 dynamics and highlight the importance of immunological characterization beyond the measurement of active infections for adequately projecting the immune landscape generated by SARS-CoV-2 infections.

COVID-19 has shown that hurdles can be overcome

Vaccination provides a powerful tool for mitigating and controlling the COVID-19 pandemic. However, a number of factors reduce these potential benefits. The first problem arises from heterogeneities in vaccine supply and uptake: from global inequities in vaccine distribution, to local variations in uptake derived from vaccine hesitancy. The second complexity is biological: though several COVID-19 vaccines offer substantial protection against infection and disease, ‘breakthrough’ reinfection of vaccinees (and subsequent retransmission from these individuals) can occur, driven especially by new viral variants. Here, using a simple epidemiological model, we show that the combination of infection of remaining susceptible individuals and breakthrough infections of vaccinees can have significant effects in promoting infection of invading variants, even when vaccination rates are high and onward transmission from vaccinees relatively weak. Elaborations of the model show how heterogeneities in immunity and mixing between vaccinated and unvaccinated sub-populations modulate these effects, underlining the importance of quantifying these variables. Overall, our results indicate that high vaccination coverage still leaves no room for complacency if variants are circulating that can elude immunity, even if this happens at very low rates.

Vaccines provide powerful tools to mitigate the enormous public health and economic costs that the ongoing SARS-CoV-2 pandemic continues to exert globally, yet vaccine distribution remains unequal between countries. To examine the potential epidemiological and evolutionary impacts of ‘vaccine nationalism’, we extend previous models to include simple scenarios of stockpiling. In general, we find that stockpiling vaccines by countries with high availability leads to large increases in infections in countries with low vaccine availability, the magnitude of which depends on the strength and duration of natural and vaccinal immunity. Additionally, a number of subtleties arise when the populations and transmission rates in each country differ depending on evolutionary assumptions and vaccine availability. Furthermore, the movement of infected individuals between countries combined with the possibility of increases in viral transmissibility may greatly magnify local and combined infection numbers, suggesting that countries with high vaccine availability must invest in surveillance strategies to prevent case importation. Dose-sharing is likely a high-return strategy because equitable allocation brings non-linear benefits and also alleviates costs of surveillance (e.g. border testing, genomic surveillance) in settings where doses are sufficient to maintain cases at low numbers. Across a range of immunological scenarios, we find that vaccine sharing is also a powerful tool to decrease the potential for antigenic evolution, especially if infections after the waning of natural immunity contribute most to evolutionary potential. Overall, our results stress the importance of equitable global vaccine distribution.

Seasonal influenza causes a substantial public health burden, as well as being a key substrate for pandemic emergence. Future climatic and demographic changes may alter both the magnitude, frequency and timing of influenza epidemics and the prospects for pathogen evolution, however, these issues have not been addressed systematically. Here, we use a parsimonious influenza model, grounded in theoretical understanding of the link between climate, demography and transmission to project future changes globally. We find that climate change generally acts to reduce the intensity of influenza epidemics as specific humidity increases. However, this reduction in intensity is accompanied by increased seasonal epidemic persistence with latitude, which may increase suitability for year-round local influenza evolution. Using a range of population growth scenarios, we find that the number of global locations with high evolution suitability may double by 2050. High population growth in tropical Africa could thus make this region a locus of novel strain emergence, shifting the current focus from South East Asia.

One dose or two? For two-dose vaccines against severe acute respiratory syndrome coronavirus 2, some jurisdictions have decided to delay the second dose to rapidly get the vaccine into more people. The consequences of deviating from manufacturer-prescribed dosing regimens are unknown but will depend on the strength of immune responses to the vaccines. Saad-Roy et al. took a modeling approach to tackling the inevitable uncertainties facing vaccine rollout. The authors found that although one-dose strategies generally reduce infections in the short term, in the long term, the outcome depends on immune robustness. A one-dose strategy may increase the potential for antigenic evolution if immune responses are suboptimal and the virus continues to replicate in some vaccinated people, potentially leading to immune-escape mutations. It is critical to gather serological data from vaccinated people and, to avoid negative outcomes, to ramp up vaccination efforts worldwide. Science, this issue p. 363, A one-dose vaccine policy may increase the potential for viral antigenic evolution in some scenarios., INTRODUCTION As the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic continues, the deployment of safe and effective vaccines presents a key intervention for mitigating disease severity and spread. Numerous logistical challenges and shortages have emerged alongside the international distribution of approved vaccines. In response, several countries have chosen to delay the second dose in an effort to increase the number of individuals receiving at least one dose. A key question then becomes how the timing of delivery of the second dose will affect future epidemiological and evolutionary outcomes. RATIONALE We build on an existing immuno-epidemiological framework that assumes that, without vaccination, individual immunity after recovery from primary infection may eventually wane, leading to (a potentially reduced) susceptibility to secondary infections. To explore epidemiological outcomes, we extend the model to incorporate two vaccinated classes, corresponding to individuals who have received either one dose or two doses of a SARS-CoV-2 vaccine. As with natural immunity, we allow for one- or two-dose vaccinal immunity to wane, and we consider a continuous spectrum for the interdose period between vaccines. To reflect the increase in available doses resulting from a delayed second dose, we model the rate of administration of the first dose as an increasing function of the interdose period. We then consider evolutionary outcomes by coupling this framework to a simple phylodynamic model for potential viral adaptation under different evolutionary scenarios, each with its own assumptions regarding viral abundance and within-host selection for the different partially susceptible classes. RESULTS We find that delaying second vaccine doses reduces COVID-19 infections in the short term by increasing the proportion of immune individuals. In the longer term, however, both the infection burden and the relative potential for viral adaptation are highly dependent on the robustness of natural or vaccinal immune responses. Notably, we find that even if immunity conferred by a single vaccine dose is poor, starting with a one-dose policy early on to increase the number of individuals immunized and then switching to the manufacturer-recommended two-dose regime as vaccine capacity increases can mitigate potential negative longer-term epidemiological and evolutionary outcomes. This mitigation can also be achieved by ramping up overall vaccination rates as availability improves. CONCLUSION The deployment of SARS-CoV-2 vaccines will strongly shape postpandemic epidemiological trajectories and characteristics of accumulated population immunity. Our models show that the combination of different vaccine dosing regimes and variations in the robustness of natural and vaccinal immunity may result in a wide range of potential epidemiological and evolutionary outcomes in the medium term. It is therefore imperative to determine the strength and duration of clinical protection and transmission-blocking immunity through careful clinical evaluations in order to enforce sound public policies. In places where vaccine deployment is delayed and vaccination rates are low, our results stress the subsequent negative epidemiological and evolutionary impacts that may emerge. Particularly because these consequences (for example, the evolution of new variants) could emerge as global problems, there is an urgent need for global equity in vaccine distribution and deployment. The relative robustness of one- or two-dose vaccinal immunity and natural immunity shape future epidemiological and evolutionary outcomes for SARS-CoV-2. An immuno-epidemiological model (left) coupled with a phylodynamic model (middle) is used to explore projections for COVID-19 infection burden and immune landscapes (top right) and potential rates of SARS-CoV-2 viral adaptation (bottom right) in the medium term. The accompanying online interactive application (http://grenfelllab.princeton.edu/sarscov2vaccine) can be used to explore these projections for a broad range of model parameters., Given vaccine dose shortages and logistical challenges, various deployment strategies are being proposed to increase population immunity levels to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Two critical issues arise: How timing of delivery of the second dose will affect infection dynamics and how it will affect prospects for the evolution of viral immune escape via a buildup of partially immune individuals. Both hinge on the robustness of the immune response elicited by a single dose as compared with natural and two-dose immunity. Building on an existing immuno-epidemiological model, we find that in the short term, focusing on one dose generally decreases infections, but that longer-term outcomes depend on this relative immune robustness. We then explore three scenarios of selection and find that a one-dose policy may increase the potential for antigenic evolution under certain conditions of partial population immunity. We highlight the critical need to test viral loads and quantify immune responses after one vaccine dose and to ramp up vaccination efforts globally.

SARS-CoV-2 is an international public health emergency; high transmissibility and morbidity and mortality can result in the virus overwhelming health systems. Combinations of social distancing, and test, trace, and isolate strategies can reduce the number of new infections per infected individual below 1, thus driving declines in case numbers, but may be both challenging and costly. These interventions must also be maintained until development and (now likely) mass deployment of a vaccine (or therapeutics), since otherwise, many susceptible individuals are still at risk of infection. We use a simple analytical model to explore how low levels of infection, combined with vaccination, determine the trajectory to community immunity. Understanding the repercussions of the biological characteristics of the viral life cycle in this scenario is of considerable importance. We provide a simple description of this process by modelling the scenario where the effective reproduction number R eff is maintained at 1. Since the additional complexity imposed by the strength and duration of transmission-blocking immunity is not yet clear, we use our framework to probe the impact of these uncertainties. Through intuitive analytical relations, we explore how the necessary magnitude of vaccination rates and mitigation efforts depends crucially on the durations of natural and vaccinal immunity. We also show that our framework can encompass seasonality or preexisting immunity due to epidemic dynamics prior to strong mitigation measures. Taken together, our simple conceptual model illustrates the importance of individual and vaccinal immunity for community immunity, and that the quantification of individuals immunized against SARS-CoV-2 is paramount.

Uncertainty in the immune response to SARS-CoV-2 may have implications for future outbreaks. We use simple epidemiological models to explore estimates for the magnitude and timing of future Covid-19 cases given different impacts of the adaptive immune response to SARS-CoV-2 as well as its interaction with vaccines and nonpharmaceutical interventions. We find that variations in the immune response to primary SARS-CoV-2 infections and a potential vaccine can lead to dramatically different immunity landscapes and burdens of critically severe cases, ranging from sustained epidemics to near elimination. Our findings illustrate likely complexities in future Covid-19 dynamics, and highlight the importance of immunological characterization be-yond the measurement of active infections for adequately characterizing the immune landscape generated by SARS-CoV-2 infections.

A stochastic model for Bovine Babesiosis (BB) including ticks, and both juvenile and adult cattle is developed. This model is formulated by a system of continuous-time Markov chains (CTMCs) that is derived based on an extension of the deterministic ordinary differential equation model developed by Saad-Roy et al. (Bull Math Biol 77:514-547, 2015). The nonlinear CTMC model is approximated by a multitype branching process, giving a theoretical estimate of the probability of an outbreak of BB. Unlike the deterministic dynamics where the basic reproduction number is a sharp threshold parameter, the stochastic model indicates that there is always a positive probability of disease extinction within the cattle population. For parameter values from Colombia data, conditional probability distributions are numerically obtained for the time to disease extinction or outbreak, and are found to depend on the host type at the initiation of infection. The models with and without the inclusion of juvenile cattle are compared, and our result highlights that neglecting juvenile bovine in the models may lead to faulty predictions of critical disease statistics: particularly, it may underestimate the risk of infection. Endemic disease prevalence in adult cattle is examined for certain parameter values in the corresponding deterministic model. Notably, with long-lasting immunity, increased tick to juvenile infectivity decreases the proportion of infectious adults.

Pathogens exhibit a rich variety of life history strategies, shaped by natural selection. An important pathogen life history characteristic is the propensity to induce an asymptomatic yet productive (transmissive) stage at the beginning of an infection. This characteristic is subject to complex trade-offs, ranging from immunological considerations to population-level social processes. We aim to classify the evolutionary dynamics of such asymptomatic behavior of pathogens (hereafter “latency”) in order to unify epidemiology and evolution for this life history strategy. We focus on a simple epidemiological model with two infectious stages, where hosts in the first stage can be partially or fully asymptomatic. Immunologically, there is a trade-off between transmission and progression in this first stage. For arbitrary trade-offs, we derive different conditions that guarantee either at least one evolutionarily stable strategy (ESS) at zero, some, or maximal latency of the first stage or, perhaps surprisingly, at least one unstable evolutionarily singular strategy. In this latter case, there is bistability between zero and nonzero (possibly maximal) latency. We then prove the uniqueness of interior evolutionarily singular strategies for power-law and exponential trade-offs: Thus, bistability is always between zero and maximal latency. Overall, previous multistage infection models can be summarized with a single model that includes evolutionary processes acting on latency. Since small changes in parameter values can lead to abrupt transitions in evolutionary dynamics, appropriate disease control strategies could have a substantial impact on the evolution of first-stage latency.

Zika virus is a human disease that may lead to neurological disorders in affected individuals, and may be transmitted vectorially (by mosquitoes) or sexually. A mathematical model of Zika virus transmission is formulated, taking into account mosquitoes, sexually active males and females, inactive individuals, and considering both vector transmission and sexual transmission from infectious males to susceptible females. Basic reproduction numbers are computed, and disease control strategies are evaluated. The effect of the incidence function used to model sexual transmission from infectious males to susceptible females is investigated. It is proved that for such functions that are sublinear, if the basic reproduction [Formula: see text], then the disease dies out and [Formula: see text] is a sharp threshold. Moreover, under certain conditions on model parameters and assuming mass action incidence for sexual transmission, it is proved that if [Formula: see text], there exists a unique endemic equilibrium that is globally asymptotically stable. However, under nonlinear incidence, it is shown that for certain functions backward bifurcation and Hopf bifurcation may occur, giving rise to subthreshold equilibria and periodic solutions, respectively. Numerical simulations for various parameter values are displayed to illustrate these behaviours.

Hanage and Russell conjecture [see also ([ 1 ][1])] that intermediate levels of immunity may sufficiently restrict within-host viral population size and thus limit adaptive mutations. The evolutionary model in our Research Article encompasses a wide range of scenarios, including the optimistic one Hanage and Russell propose (see scenario 1 in Fig. 4A of the Research Article). However, we also argued that uncertainties in immunodynamics, and in particular evolutionary dynamics of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), dominate our ability to project key scenarios. We therefore produced generic models in an effort to encompass this variability while providing a useful framework for considering possible future outcomes. We would also like to clarify that the calculations for evolutionary potential in our Research Article involve only infections after natural or vaccinal immunity has waned [i.e., I S (secondary infection after waned natural immunity), I S1 (infection after waned one-dose vaccinal immunity), and I S2 (infection after waned two-dose vaccinal immunity), see Fig. 1]. Our accompanying interactive online application allows for the exploration of the effect of an even wider range of parameters on immunological and epidemiological outcomes. We stress that the understanding of viral phylodynamics (the intersection of the epidemiological and evolutionary dynamics of pathogens) is still in its infancy, especially for novel pathogens like SARS-CoV-2. We therefore believe that exploring the resulting uncertainty is key. Understanding the dynamics of different classes of individuals experiencing infections after the waning of natural or vaccinal immunity will be crucial for teasing out drivers of viral immune escape (i.e., the ability of a virus to evolve to evade host immune factors); this is at the heart of our model. Along these lines, and as we highlight in the Research Article, an important area of future work will be to develop phylodynamic models with explicit within-host dynamics ([ 2 ][2]). The other major source of uncertainties is in epidemiological outcomes. In their Letter, Hanage and Russell emphasize the likely epidemiological importance of widespread vaccine deployment. This echoes a conclusion of our Research Article, wherein we stress that short-term dose sparing deployment of a vaccine reduces infections and buys much time in public health planning. However, we also stressed the range of uncertainties that may modulate the longer-term outcomes of this strategy; in particular, less robust immunity could lead to more complex epidemiological and evolutionary outcomes. 1. [↵][3]1. S. Cobey, 2. D. B. Larremore, 3. Y. H. Grad, 4. M. Lipsitch , “Concerns about SARS-CoV-2 evolution should not hold back efforts to expand vaccination,” preprint, Harvard University (2021); . 2. [↵][4]1. D. H. Morris et al ., eLife 9, e62105 (2020). [OpenUrl][5][CrossRef][6][PubMed][7] [1]: #ref-1 [2]: #ref-2 [3]: #xref-ref-1-1 "View reference 1 in text" [4]: #xref-ref-2-1 "View reference 2 in text" [5]: {openurl}?query=rft.jtitle%253DeLife%26rft.volume%253D9%26rft.spage%253De62105%26rft_id%253Dinfo%253Adoi%252F10.7554%252FeLife.62105%26rft_id%253Dinfo%253Apmid%252F33174838%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx [6]: /lookup/external-ref?access_num=10.7554/eLife.62105&link_type=DOI [7]: /lookup/external-ref?access_num=33174838&link_type=MED&atom=%2Fsci%2F372%2F6540%2F354.2.atom

Despite vast improvements in global vaccination coverage during the last decade, there is a growing trend in vaccine hesitancy and/or refusal globally. This has implications for the acceptance and coverage of a potential vaccine against COVID-19. In the United States, the number of children exempt from vaccination for “philosophical belief-based” non-medical reasons increased in 12 of the 18 states that allowed this policy from 2009 to 2017 (1). Meanwhile, the overuse and misuse of antibiotics, especially in young children, have led to increasing rates of drug resistance that threaten our ability to treat infectious diseases. Vaccine hesitancy and antibiotic overuse exist side-by-side in the same population of young children, and it is unclear why one modality (antibiotics) is universally seen as safe and effective, while the other (vaccines) is seen as potentially hazardous by some. In this review, we consider the drivers shaping the use of vaccines and antibiotics in the context of three factors: individual incentives, risk perceptions, and social norms and group dynamics. We illustrate how these factors contribute to the societal and individual costs of vaccine underuse and antimicrobial overuse. Ultimately, we seek to understand these factors that are at the nexus of infectious disease epidemiology and social science to inform policy-making.

Pathogens evolve different life-history strategies, which depend in part on differences in their host populations. A central feature of hosts is their population structure (e.g. spatial). Additionally, hosts themselves can exhibit different degrees of symptoms when newly infected; this latency is a key life-history property of pathogens. With an evolutionary-epidemiological model, we examine the role of population structure on the evolutionary dynamics of latency. We focus on specific power-law-like formulations for transmission and progression from the first infectious stage as a function of latency, assuming that the across-group to within-group transmission ratio increases if hosts are less symptomatic. We find that simple population heterogeneity can lead to local evolutionarily stable strategies (ESSs) at zero and infinite latency in situations where a unique ESS exists in the corresponding homogeneous case. Furthermore, there can exist more than one interior evolutionarily singular strategy. We find that this diversity of outcomes is due to the (possibly slight) advantage of across-group transmission for pathogens that produce fewer symptoms in a first infectious stage. Thus, our work reveals that allowing individuals without symptoms to travel can have important unintended evolutionary effects and is thus fundamentally problematic in view of the evolutionary dynamics of latency.

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number [Formula: see text] is calculated, and existence of a unique endemic equilibrium is established for [Formula: see text] above the threshold value 1. Using data from the literature, elasticity indices for [Formula: see text] and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if [Formula: see text]. For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with [Formula: see text] and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.

Syphilis is caused by the bacterium Treponema pallidum subspecies pallidum, and is a sexually transmitted disease with multiple stages. A model of transmission of syphilis in an MSM population (there has recently been a resurgence of syphilis in such populations) that includes infection stages and treatment is formulated as a system of ordinary differential equations. The control reproduction number is calculated, and it is proved that if this threshold parameter is below one, syphilis dies out; otherwise, if it is greater than one, it is shown that there exists a unique endemic equilibrium and that for certain special cases, this equilibrium is globally asymptotically stable. Using data from the literature on MSM populations, numerical methods are used to determine the variation and robustness of the control reproduction number with respect to the model parameters, and to determine adequate treatment rates for syphilis eradication. By assuming a closed population and no return to susceptibility, an epidemic model is obtained. Final outbreak sizes are numerically determined for various parameter values, and its variation and robustness to parameter value changes is also investigated. Results quantify the importance of early treatment for syphilis control.

A universal influenza vaccine (UIV) could considerably alleviate the public health burden of both seasonal and pandemic influenza. Although significant progress has been achieved in clarifying basic immunology and virology relating to UIV, several important questions relating to the dynamics of infection, immunity, and pathogen evolution remain unsolved. In this study, we review these gaps, which span integrative levels, from cellular to global and timescales from molecular events to decades. We argue that they can be best addressed by a tight integration of empirical (laboratory, epidemiological) research and theory and suggest fruitful areas for this synthesis. In particular, quantifying natural and vaccinal limitations on viral transmission are central to this effort.

There is increasing interest in the development of new, ‘universal’ influenza vaccines (UIVs) that––unlike current vaccines––are effective against a broad range of seasonal influenza strains, as well as against novel pandemic viruses. While the existing literature discusses the potential epidemiological benefits of UIVs, it is also important to anticipate their potential unintended population consequences. Using mathematical modelling, we illustrate two such types of adverse consequences. First, by reducing the amount of infection-induced immunity in a population without fully replacing it, a seasonal UIV programme may permit larger pandemics than in the absence of vaccination. Second, the more successful a future UIV programme is in reducing transmission of seasonal influenza, the more vulnerable the population could become to the emergence of a vaccine escape variant. These risks could be mitigated by optimal deployment of any future UIV vaccine: namely, the use of a combined vaccine formulation (incorporating conventional as well as multiple universal antigenic targets) and achieving sufficient population coverage to compensate for any reductions in infection-induced immunity. In the absence of large-scale trials of UIVs, disease-dynamic models can provide helpful, early insights into their potential impact. In future, data from continuing vaccine development will be invaluable in developing robustly predictive modelling approaches.

Human African Trypanosomiasis (HAT) and Nagana in cattle, commonly called sleeping sickness, is caused by trypanosome protozoa transmitted by bites of infected tsetse flies. We present a deterministic model for the transmission of HAT caused by Trypanosoma brucei gambiense between human hosts, cattle hosts and tsetse flies. The model takes into account the growth of the tsetse fly, from its larval stage to the adult stage. Disease in the tsetse fly population is modeled by three compartments, and both the human and cattle populations are modeled by four compartments incorporating the two stages of HAT. We provide a rigorous derivation of the basic reproduction number $ \mathcal{R}_{0} $ . For $ \mathcal{R}_{0}1 $ , HAT persists. Elasticity indices for $ \mathcal{R}_{0} $ with respect to different parameters are calculated with baseline parameter values appropriate for HAT in West Africa; indicating parameters that are important for control strategies to bring $ \mathcal{R}_{0} $ below 1. Numerical simulations with $ \mathcal{R}_0 > 1 $ show values for the infected populations at the endemic equilibrium, and indicate that with certain parameter values, HAT could not persist in the human population in the absence of cattle.

Background There currently is a severe Zika Virus (ZIKV) epidemic in Brazil and other South American countries. Due to international travel, this poses severe public health risk of ZIKV importation to other countries. We estimate the prevalence of ZIKV in an import region by the time a microcephaly case is detected, since microcephaly is presently the most significant indication of ZIKV presence. Methods We establish a mathematical model to describe ZIKV spread from a source region to an import region. This model incorporates both vector transmission (between humans and mosquitoes) and sexual transmission (from males to females). We take account of population structure through a contact network for sexually active individuals. Parameter values of our model are either taken from the literature or estimated from travel data. Results This model gives us the probability distribution of time until detection of the first microcephaly case. Based on current field observations, our results also indicate that the percentage of infected pregnant women that results in fetal abnormalities is more likely to be on the smaller end of the 1%–30% spectrum that is currently hypothesized. Our model predicts that for import regions with at least 250,000 people, on average 1,000–12,000 will have been infected by the time of the first detection of microcephaly, and on average 200–1,500 will be infectious at this time. Larger population sizes do not significantly change our predictions. Conclusions By the first detection of a microcephaly case, a sizable fraction of the population will have been infected by ZIKV. It is thus clear that adequate surveillance, isolation, and quarantine are needed in susceptible import regions to stop the dissemination of a Zika epidemic. Electronic supplementary material The online version of this article (doi:10.1186/s12879-016-2076-z) contains supplementary material, which is available to authorized users.

Bovine Babesiosis in cattle is caused by the transmission of protozoa of Babesia spp. by ticks as vectors. Juvenile cattle ( $$