Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE)

One goal of this paper is to give an algorithm for computing a threshold condition for epidemiological systems arising from compartmental deterministic modeling. We calculate a threshold condition T(0) of the parameters of the system such that if T(0)<1 the disease-free equilibrium (DFE) is locally asymptotically stable (LAS), and if T(0)>1, the DFE is unstable. The second objective, by adding some reasonable assumptions, is to give, depending on the model, necessary and sufficient conditions for global asymptotic stability (GAS) of the DFE. In many cases, we can prove that a necessary and sufficient condition for the global asymptotic stability of the DFE is R(0)< or =1, where R(0) is the basic reproduction number [O. Diekmann, J.A. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000]. To illustrate our results, we apply our techniques to examples taken from the literature. In these examples we improve the results already obtained for the GAS of the DFE. We show that our algorithm is relevant for high dimensional epidemiological models.     

SVIR epidemic models with vaccination strategies

Vaccination is important for the elimination of infectious diseases. To finish a vaccination process, doses usually should be taken several times and there must be some fixed time intervals between two doses. The vaccinees (susceptible individuals who have started the vaccination process) are different from both susceptible and recovered individuals. Considering the time for them to obtain immunity and the possibility for them to be infected before this, two SVIR models are established to describe continuous vaccination strategy and pulse vaccination strategy (PVS), respectively. It is shown that both systems exhibit strict threshold dynamics which depend on the basic reproduction number. If this number is below unity, the disease can be eradicated. And if it is above unity, the disease is endemic in the sense of global asymptotical stability of a positive equilibrium for continuous vaccination strategy and disease permanence for PVS. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before this is neglected, this condition disappears and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.     

A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R0

We present a novel SEIR (susceptible-exposure-infective-recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R(0) in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R(0) for most known diseases, that are calculated by our formulas, coincide very well with the values of R(0) estimated by the usual method of fitting the models to observed data.     

An SIRVS epidemic model with pulse vaccination strategy

The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value.     

Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity

We derive and analyse a deterministic model for the transmission of malaria disease with mass action form of infection. Firstly, we calculate the basic reproduction number, R(0), and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of malaria, R(0)<1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory we derive the conditions under which it is optimal to eradicate the disease and examine the impact of a possible combined vaccination and treatment strategy on the disease transmission. When eradication is impossible, we derive the necessary conditions for optimal control of the disease using Pontryagin's Maximum Principle. The results obtained from the numerical simulations of the model show that a possible vaccination combined with effective treatment regime would reduce the spread of the disease appreciably.     

Circulating Vaccine Derived Polio Viruses and their Impact on Global Polio Eradication

Poliomyelitis vaccination via live Oral Polio Vaccine (OPV) suffers from the inherent problem of reversion: the vaccine may, upon replication in the human gut, mutate back to virulence and transmissibility resulting in circulating vaccine derived polio viruses (cVDPVs). We formulate a general mathematical model to assess the impact of cVDPVs on prospects for polio eradication. We find that for OPV coverage levels below a certain threshold, cVDPVs have a small impact in comparison to the expected endemic level of the disease in the absence of reversion. Above this threshold, the model predicts a small but significant endemic level of the disease, even where standard models predict eradication. In light of this, we consider and analyze three alternative eradication strategies involving a transition from continuous OPV vaccination to either continuous Inactivated Polio Vaccine (IPV), pulsed OPV vaccination, or a one-time IPV pulse vaccination. Stochastic modeling shows continuous IPV vaccination is effective at achieving eradication for moderate coverage levels, while pulsed OPV is effective if higher coverage levels are maintained. The one-time pulse IPV method may also be a viable strategy, especially in terms of the number of vaccinations required and time to eradication, provided that a sufficiently large pulse is practically feasible. More investigation is needed regarding the frequency of revertant virus infection resulting directly from vaccination, the ability of IPV to induce gut immunity, and the potential role of spatial transmission dynamics in eradication efforts. B.G. Wagner’s research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Doctoral Scholarship. D.J.D. Earn’s research is supported by the Canadian Institutes of Health Research (CIHR), Natural Sciences and Engineering Research Council of Canada (NSERC) and the J.S. McDonnell Foundation.     

Multiple attractors in the response to a vaccination program

Though it is well known that multiple attractors may co-exist in the SEIR (susceptible/exposed/infective/recovered) epidemic model with vital dynamics and seasonally forced oscillations in transmission, the epidemiological significance of multiple attractors has been a subject of debate. I show that the co-existence of attractors is relevant in using the model to study the dynamics of the introduction of a vaccination program into a stable epidemic cycle. Responses to the program may include more than one attractor. The exact timing of the introduction of the program relative to the original epidemic cycle is critical in determining which attractor appears in the response. Analysis of this simple model suggests that the role of multiple attractors in the response to vaccination should be examined in more realistic epidemiological models.     

Optimizing reactive responses to outbreaks of immunizing infections: balancing case management and vaccination

For vaccine-preventable infections, immunization generally needs to be supplemented by palliative care of individuals missed by the vaccination. Costs and availability of vaccine doses and palliative care vary by disease and by region. In many situations, resources for delivery of palliative care are independent of resources required for vaccination; however we also need to consider the conservative scenario where there is some trade-off between efforts, which is of potential relevance for resource-poor settings. We formulate an SEIR model that includes those two control strategies - vaccination and palliative care. We consider their relative merit and optimal allocation in the context of a highly efficacious vaccine, and under the assumption that palliative care may reduce transmission. We investigate the utility of a range of mixed or pure strategies that can be implemented after an epidemic has started, and look for rule-of-thumb principles of how best to reduce the burden of disease during an acute outbreak over a spectrum of vaccine-preventable infections. Intuitively, we expect the best strategy to initially focus on vaccination, and enhanced palliative care after the infection has peaked, but a number of plausible realistic constraints for control result in important qualifications on the intervention strategy. The time in the epidemic when one should switch strategy depends sensitively on the relative cost of vaccine to palliative care, the available budget, and [Formula: see text]. Crucially, outbreak response vaccination may be more effective in managing low-[Formula: see text] diseases, while high [Formula: see text] scenarios enhance the importance of routine vaccination and case management.     

Pulse vaccination strategy in the SIR epidemic model

Theoretical results show that the measles ‘pulse’ vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.     

Assessing the best time interval between doses in a two-dose vaccination regimen to reduce the number of deaths in an ongoing epidemic of SARS-CoV-2

The SARS-CoV-2 pandemic is a major concern all over the world and, as vaccines became available at the end of 2020, optimal vaccination strategies were subjected to intense investigation. Considering their critical role in reducing disease burden, the increasing demand outpacing production, and that most currently approved vaccines follow a two-dose regimen, the cost-effectiveness of delaying the second dose to increment the coverage of the population receiving the first dose is often debated. Finding the best solution is complex due to the trade-off between vaccinating more people with lower level of protection and guaranteeing higher protection to a fewer number of individuals. Here we present a novel extended age-structured SEIR mathematical model that includes a two-dose vaccination schedule with a between-doses delay modelled through delay differential equations and linear optimization of vaccination rates. By maintaining the minimum stock of vaccines under a given production rate, we evaluate the dose interval that minimizes the number of deaths. We found that the best strategy depends on an interplay between the vaccine production rate and the relative efficacy of the first dose. In the scenario of low first-dose efficacy, it is always better to vaccinate the second dose as soon as possible, while for high first-dose efficacy, the best strategy of time window depends on the production rate and also on second-dose efficacy provided by each type of vaccine. We also found that the rate of spread of the infection does not affect significantly the thresholds of the best window, but is an important factor in the absolute number of total deaths. These conclusions point to the need to carefully take into account both vaccine characteristics and roll-out speed to optimize the outcome of vaccination strategies.     

Sortase A Induces Th17-Mediated and Antibody-Independent Immunity to Heterologous Serotypes of Group A Streptococci

Group A streptococci (GAS) are associated with a variety of mucosal and invasive human infections. Recurrent infections by highly heterologous serotypes indicate that cross-serotype immunity is critical for prevention of GAS infections; however, mechanisms underlying serotype-independent protection are poorly understood. Here we report that intranasal vaccination of mice with Sortase A (SrtA), a conserved cell wall bound protein, reduced colonization of nasal-associated lymphoid tissue (NALT) by heterologous serotypes of GAS. Vaccination significantly increased CD4⁺ IL-17A⁺ cells in NALT and depletion of IL-17A by neutralizing antibody prevented GAS clearance from NALT which was dependent on immunization with SrtA. Vaccination also induced high levels of SrtA-specific antibodies; however, immunized, B cell-deficient mice cleared streptococcal challenges as efficiently as wild type mice, indicating that the cross-serotype protection is Th17-biased and antibody-independent. Furthermore, efficient GAS clearance from NALT was associated with a rapid neutrophil influx into NALT of immunized mice. These results suggest that serotype independent immune protection against GAS mucosal infection can be achieved by intranasal vaccination with SrtA and enhanced neutrophil function is critical for anti-GAS defense and might be a target for prevention of GAS infections.     

Genetically attenuated P36p-deficient Plasmodium berghei sporozoites confer long-lasting and partial cross-species protection

Immunisation with live, radiation-attenuated sporozoites (RAS) or genetically attenuated sporozoites (GAS) of rodent plasmodial parasites protects against subsequent challenge infections. We recently showed that immunisation with Plasmodium berghei GAS that lack the microneme protein P36p protects mice for a period of up to 4 months. Here, we show that the period of full protection induced by p36p(-)-sporozoites lasts 12 and 18 months in C57Bl6 and BALB/c mice, respectively. Full protection is also achieved with three doses of only 1000 p36p(-) (but not RAS) sporozoites. Subcutaneous, intradermal or intramuscular routes of administration also lead to partial protection. In addition, immunisation with either P. berghei RAS- or, to a lesser extent, p36p(-)-sporozoites inhibits parasite intrahepatic development in mice challenged with Plasmodium yoelii sporozoites. Since naturally acquired malaria infections or subunit-based vaccines only induce short-term immune responses, the protection conferred by immunisation with p36p(-)-sporozoites described here further emphasises the potential of GAS as a vaccination strategy for malaria.     

Stability analysis of within-host parasite models with delays

We provide a global analysis of systems of within-host parasitic infections. The systems studied have parallel classes of different length of latently infected target cells. These systems can also be thought as systems arising from within-host parasitic systems with distributed continuous delays. We compute the basic reproduction ratio R0 for the systems under consideration. If R0< or =1 the parasite is cleared, if R0>1 and if a sufficient condition is satisfied we conclude to the global asymptotic stability (GAS) of the endemic equilibrium. For some generic class of models this condition reduces to R0>1. These results make possible to revisit some parasitic models including intracellular delays and to study their global stability.     

Epidemiological effects of seasonal oscillations in birth rates

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the "infected period" (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.     

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.     

Modelling the effect of a booster vaccination on disease epidemiology

Despite the effectiveness of vaccines in dramatically decreasing the number of new infectious cases and severity of illnesses, imperfect vaccines may not completely prevent infection. This is because the immunity afforded by these vaccines is not complete and may wane with time, leading to resurgence and epidemic outbreaks notwithstanding high levels of primary vaccination. To prevent an endemic spread of disease, and achieve eradication, several countries have introduced booster vaccination programs. The question of whether this strategy could eventually provide the conditions for global eradication is addressed here by developing a seasonally-forced mathematical model. The analysis of the model provides the threshold condition for disease control in terms of four major parameters: coverage of the primary vaccine; efficacy of the vaccine; waning rate; and the rate of booster administration. The results show that if the vaccine provides only temporary immunity, then the infection typically cannot be eradicated by a single vaccination episode. Furthermore, having a booster program does not necessarily guarantee the control of a disease, though the level of epidemicity may be reduced. In addition, these findings strongly suggest that the high coverage of primary vaccination remains crucial to the success of a booster strategy. Simulations using estimated parameters for measles illustrate model predictions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). One of the authors (P.R.) acknowledges the support of the Ellison Medical Foundation.     

Toxin–antitoxin-stabilized reporter plasmids for biophotonic imaging of Group A streptococcus

Bioluminescence is a rapid and cost-efficient optical imaging technology that allows the detection of bacteria in real-time during disease development. Here, we report a novel strategy to generate a wide range of bioluminescent group A streptococcus (GAS) strains by using a toxin–antitoxin-stabilized plasmid. The bacterial luciferin–luciferase operon (lux) or the firefly luciferase gene (ffluc) was introduced into GAS via a stabilized plasmid. The FFluc reporter gave significantly stronger bioluminescent signals than the Lux reporter, and was generally more stable. Plasmid-based luciferase reporters could easily be introduced into a variety of GAS strains and the signals correlated linearly with viable cell counts. Co-expression of the streptococcal ω–ε–ζ toxin–antitoxin operon provided segregational stability in the absence of antibiotics for at least 17 passages in vitro and up to 7 days in a mouse infection model. In addition, genome-integrated reporter constructs were also generated by site-specific recombination, but were found to be technically more challenging. The quick and efficient generation of various M-type GAS strains expressing plasmid-based luciferase reporters with comparable and quantifiable bioluminescence signals allows for comparative analysis of different GAS strains in vitro and in vivo.     

小滴玻璃化法脱除蝴蝶兰种苗建兰花叶病毒的研究

以感染建兰花叶病毒(Cymbidium mosaic virus,CymMV)的蝴蝶兰(Phalaenopsis aphrodite)品种‘满天红’为试材,通过筛选蔗糖预培养浓度、预培养时间、PVS2(Plant vitrification solution 2,PVS2)处理时间三个关键因素,建立蝴蝶兰茎尖小滴玻璃化超低温脱毒体系,将再生的茎尖诱导类原球茎,再分化成苗,经RT-PCR检测CymMV的脱除情况,阴性结果的再生植株进行增殖和诱导生根。结果显示:最佳预培养为:BM+0.6 mol·L-1蔗糖处理1~2 d,超低温茎尖的成活率为70%~76.7%,再生率为53.3%~56.7%;PVS2最佳处理时间为60~90 min,超低温茎尖的成活率为73.3%~76.7%,再生率为50.0%~56.7%。再生植株经RT-PCR检测,CymMV的脱除率为50%。该研究为兰科植物脱除CymMV提供了理论和技术基础。     

Control of emerging infectious diseases using responsive imperfect vaccination and isolation

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.     

Modelling Cultivar Mixtures Using SEIR Compartmental Models

A SEIR (susceptible, exposed, infectious, removed) compartmental model is constructed to represent disease progress in a two cultivar mixture. The concept of the spore pool is the means by which inoculum exchange between the constituent cultivars is represented. Infection frequencies for each cultivar are permitted to vary with that cultivar's susceptible fraction according to a power-law relationship. For each cultivar, new additions to the susceptible class balance deaths from the removed class so that the total leaf area of all four SEIR classes remains constant. It is shown that an equilibrium with non-zero diseased classes exists in a certain parameter regime. A numerical stability analysis is performed using the model equations linearised about this equilibrium. The effects of changing induced resistance parameters within the model are demonstrated graphically. It is also demonstrated that the proportion of susceptibles as a function of mixture composition has an optimum in the regime where nontrivial equilibria exist, a feature of practical interest provided that equilibrium is reached within a growing season.