Imperfect Vaccine Research Articles

The role of asymptomatic cattle for leptospirosis dynamics in a herd with imperfect vaccination

Leptospirosis is an emerging zoonotic disease with high health and economic damage. In this study, we developed a deterministic mathematical model that describes the dynamics of leptospirosis transmission within a cattle herd, incorporating asymptomatic infected and vaccinated compartments. The study examined the transmission role of asymptomatic cattle that contaminate herds without farmers’ knowledge. We proved the well-posedness of the proposed model and found the basic reproduction number using the next-generation matrix. Analytically, we demonstrated that the disease-free equilibrium point is locally and globally asymptotically stable when R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {R}}_0$$\\end{document} is less than unity and is otherwise unstable. Graphically, we further established the local asymptotic stability of disease-free and endemic equilibria. Sensitivity analysis showed that the contact rate with asymptomatic infected cattle, βA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _A$$\\end{document}, is the most sensitive parameter in the stated model, followed by the recovery rate of asymptomatic infected cattle, σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}, and the vaccination rate of susceptible cattle, τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Numerical simulations revealed that a reduction in contact rate with asymptomatic infected cattle significantly reduced pathogen Leptospira transmission in the herd. In addition, fostering the recovery rate of asymptomatic infected cattle can significantly reduce new infections in the herd. Furthermore, augmenting the vaccination rate among susceptible cattle resulted in a notable decrease in disease prevalence within the herd. Findings of this study underscore the remarkable importance of targeted interventions, such as reducing contact rates with asymptomatic infected cattle, increasing recovery rates using proper treatments, and enhancing vaccination efforts to manage leptospirosis transmission in cattle herds.

Quantifying indirect and direct vaccination effects arising in the SIR model.

Vaccination campaigns have both direct and indirect effects that act to control an infectious disease as it spreads through a population. Indirect effects arise when vaccinated individuals block disease transmission in any infection chain they are part of, and this in turn can benefit both vaccinated and unvaccinated individuals. Indirect effects are difficult to quantify in practice but, in this article, working with the susceptible-infected-recovered (SIR) model, they are analytically calculated in important cases, through pivoting on the final size formula for epidemics. Their relationship to herd immunity is also clarified. The analysis allows us to identify the important distinction between quantifying the indirect effects of vaccination at the 'population level' versus the 'per capita' level, which often results in radically different conclusions. As an example, our analysis unpacks why the population-level indirect effect can appear significantly larger than its per capita analogue. In addition, we consider a recently proposed epidemiological non-pharmaceutical intervention (by the means of recovered individuals) used over the COVID-19 pandemic, referred to as 'shielding', and study its impact on our mathematical analysis. The shielding scheme is extended to take advantage of vaccination including imperfect vaccination.

Optimal vaccination strategies for imperfect vaccines and variable host susceptibility

I present a method to allocate a given number of vaccines to members of a population who differ in their susceptibility to the disease so that the final size of the epidemic is minimised. I consider an arbitrary distribution of protection that the vaccine confers, including the extreme cases of leaky and all-or-none vaccines. The optimal vaccination policy depends on the distribution of protection. While for low values of the basic reproduction number R0 the optimal policy prioritises the most susceptible hosts, I show that for almost any distribution the order of priority reverses and the least susceptible hosts should be vaccinated when R0 is high. The exception where this does not happen is the all-or-none vaccine. However, even a small deviation from the ideal all-or-none distribution can imply that the limited number of vaccines should be given to less susceptible hosts already at realistic values of R0.

Social vs. individual age-dependent costs of imperfect vaccination

In diseases with long-term immunity, vaccination is known to increase the average age at infection as a result of the decrease in the pathogen circulation. This implies that a vaccination campaign can have negative effects when a disease is more costly (financial or health-related costs) for higher ages. This work considers an age-structured population transmission model with imperfect vaccination. We aim to compare the social and individual costs of vaccination, assuming that disease costs are age-dependent, while the disease’s dynamic is age-independent. A model for pathogen deterministic dynamics in a population consisting of juveniles and adults, assumed to be rational agents, is introduced. The parameter region for which vaccination has a positive social impact is fully characterized and the Nash equilibrium of the vaccination game is obtained. Finally, collective strategies designed to promote voluntary vaccination, without compromising social welfare, are discussed.

The use of an imperfect vaccination and awareness campaign in the control of antibiotic resistant gonorrhoea infection: A mathematical modelling perspective

The multi drug-resistant Neisseria gonorrhoeae has been classified by the World Health Organisation (WHO) as a high-priority global public health problem. This underlines the need for better understanding of the transmission dynamics and proposing an optimal intervention strategy to control the disease. In this article, a deterministic mathematical model for the transmission dynamics of gonorrhoea as an antibiotic resistant disease in a population with an imperfect vaccination is proposed and analysed. The model incorporates the classes of vaccinated individuals and individuals equipped with self protection interventions to reduce antibiotic resistant cases. The threshold parameter R0, the basic reproduction number, for the analysis of the model is calculated. In the given setting, the model exhibits a backward bifurcation for R0<1. However, if the efficacy of the vaccine is 100% without a waning effect, the model is shown to be without a backward bifurcation and the disease-free equilibrium is globally asymptotically stable whenever R0<1. The global sensitivity analysis of the model to variations in parameter values is also performed to determine the most influential parameters on the disease transmission. Moreover, the optimal control analysis of the full model is presented and the optimal intervention strategies are proposed. The proposed intervention strategies are shown to be able to control the disease within a relatively shorter period of time. Finally, numerical experiments are presented to support the theoretical analysis of the model.

A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19

In this paper a mathematical model that investigates how vaccination affects the dynamics of COVID-19 was considered. More particularly the model takes into account the waning rate of immunity after vaccination as well as administration of booster vaccine. Posititivity and boundedness of solutions of the model were proved. The disease free equilibrium of the model was determined and by using the next generation matrix method both the basic and effective reproduction numbers of the model were determined. Further, from the effective reproduction number, the minimum critical value of individuals to be vaccinated for containment of the diseases was determined. It was found that the value is less for a perfect vaccine compared to an imperfect vaccine. Numerical simulation of the model was done to determine how the parameters of interest in the study (waning rate of immunity, vaccination rate, administration of booster vaccine and efficacy of the vaccine) affect the effective reproduction number. The results show that increasing the rates of vaccination, administering booster vaccine will decrease the effective reproduction number while an increase in waning rate of immunity increases the effective reproduction number. The disease persist in the population due to the declining of immunity after vaccination which increases the effective reproduction number.

Dynamic vaccination strategies in dual-strain epidemics: A multi-agent-based game-theoretic approach on scale-free hybrid networks

This paper proposes a multi-agent-based game-theoretic framework to explore the dynamics of decision-making among agents in a situation involving a dual-strain epidemic, where the emergence of the second strain is directly related to the first strain. It considers the impact of imperfect vaccination on dual strains within the SIRV framework, in a hybrid network that combines scale-free and grid-based structures. This research presents three models based on the level of decision-making of agents: Individual Decision Making (IDM), Collective Decision Making (CDM), and Stochastic Decision Making (SDM) model. In the IDM model, agents can continuously update their strategies based on the infection and vaccination statuses of their immediate neighbors. The CDM model offers a macro-level approach where agents’ willingness to vaccinate varies uniformly, while the SDM model maintains a constant level of willingness for vaccination across agents. This study demonstrates that agents using the IDM model tend to initiate vaccination earlier in scenarios of high infection rates, effectively mitigating the spread of the disease in subsequent epidemic seasons. However, when both infection and vaccination rates heavily influence decision-making, there is a noticeable decrease in vaccination uptake, thereby aggravating the spread of infection. Additionally, the authors found that in scenarios of high infection, an over reliance on vaccination without adequate initial uptake can jeopardize disease control efforts. Another important finding of this study is that despite the CDM and SDM models showing a quicker emergence of the second strain and reaching equilibrium points faster than the IDM model, the overall infection rate remains lower in the IDM model. This research underscores the complexity of managing dual-strain epidemics through vaccination strategies and highlights the significance of micro-level decision-making in epidemic control.

Analysis of a fractional-order model for dengue transmission dynamics with quarantine and vaccination measures

A comprehensive mathematical model is proposed to study two strains of dengue virus with saturated incidence rates and quarantine measures. Imperfect dengue vaccination is also assumed in the model. Existence, uniqueness and stability of the proposed model are proved using the results from fixed point and degree theory. Additionally, well constructed Lyapunov function candidates are also applied to prove the global stability of infection-free equilibria. It is also demonstrated that the model is generalized Ulam-Hyers stable under some appropriate conditions. The model is fitted to the real data of dengue epidemic taken from the city of Espirito Santo in Brazil. For the approximate solution of the model, a non-standard finite difference(NSFD) approach is applied. Sensitivity analysis is also carried out to show the influence of different parameters involved in the model. The behaviour of the NSFD is also assessed under different denominator functions and it is observed that the choice of the denominator function could influence the solution trajectories. Different scenario analysis are also assessed when the reproduction number is below or above one. Furthermore, simulations are also presented to assess the epidemiological impact of dengue vaccination and quarantine measures for infected individuals.

Fractional mathematical model for the transmission dynamics and control of Lassa fever

This paper aims to investigate various epidemiological aspects of Lassa fever viral infection using a fractional-order mathematical model so as to assess the impacts of treatment and vaccination on the spread of Lassa fever transmission dynamics. Initially, the model employs integer-order nonlinear differential equations, incorporating imperfect vaccination and treatment as control measures for the human population. Subsequently, the model is redefined using a fractional order derivative with power law to enhance understanding of disease dynamics. The paper establishes conditions ensuring the existence and uniqueness of the model’s solution in the fractional case, alongside presenting stability analysis of the endemic equilibrium using the Lyapunov function approach. Numerical simulations are performed using the fractional Adams–Bashforth–Moulton method, shedding light on the influence of model parameters and fractional order values on Lassa fever dynamics and control. Additional numerical simulations conducted utilizing surface and contour plots revealed that elevating the contact rates and diminishing the efficacy of vaccines would escalate the prevalence of Lassa fever among the human populace. It was also observed that optimizing treatment and enhancing vaccination strategies would ultimately mitigate the prevalence of Lassa fever within the human population.

A Metapopulation Model for Cholera with Variable Media Efficacy and Imperfect Vaccine

In this paper, a metapopulation model has been developed and analysed to describe the transmission dynamics of cholera between two communities linked by migration, in the presence of an imperfect vaccine and a varying media awareness impact. Stability analysis shows that the disease-free equilibrium is both locally and globally asymptotically stable when the vaccine reproduction number is less than unity. The endemic equilibria have also been shown to be locally asymptotically stable when the vaccine reproduction number is greater than unity. The simulation results show that with an imperfect vaccine and efficient media awareness, cholera transmission is reduced. The transmission rates have also been shown to be nonidentical in the two communities. It is therefore advisable, that health practitioners embrace the use of both vaccination and media awareness when designing and implementing community-specific cholera intervention strategies.

Impact of immune evasion, waning and boosting on dynamics of population mixing between a vaccinated majority and unvaccinated minority.

We previously demonstrated that when vaccines prevent infection, the dynamics of mixing between vaccinated and unvaccinated sub-populations is such that use of imperfect vaccines markedly decreases risk for vaccinated people, and for the population overall. Risks to vaccinated people accrue disproportionately from contact with unvaccinated people. In the context of the emergence of Omicron SARS-CoV-2 and evolving understanding of SARS-CoV-2 epidemiology, we updated our analysis to evaluate whether our earlier conclusions remained valid. We modified a previously published Susceptible-Infectious-Recovered (SIR) compartmental model of SARS-CoV-2 with two connected sub-populations: vaccinated and unvaccinated, with non-random mixing between groups. Our expanded model incorporates diminished vaccine efficacy for preventing infection with the emergence of Omicron SARS-CoV-2 variants, waning immunity, the impact of prior immune experience on infectivity, "hybrid" effects of infection in previously vaccinated individuals, and booster vaccination. We evaluated the dynamics of an epidemic within each subgroup and in the overall population over a 10-year time horizon. Even with vaccine efficacy as low as 20%, and in the presence of waning immunity, the incidence of COVID-19 in the vaccinated subpopulation was lower than that among the unvaccinated population across the full 10-year time horizon. The cumulative risk of infection was 3-4 fold higher among unvaccinated people than among vaccinated people, and unvaccinated people contributed to infection risk among vaccinated individuals at twice the rate that would have been expected based on the frequency of contacts. These findings were robust across a range of assumptions around the rate of waning immunity, the impact of "hybrid immunity", frequency of boosting, and the impact of prior infection on infectivity in unvaccinated people. Although the emergence of the Omicron variants of SARS-CoV-2 has diminished the protective effects of vaccination against infection with SARS-CoV-2, updating our earlier model to incorporate loss of immunity, diminished vaccine efficacy and a longer time horizon, does not qualitatively change our earlier conclusions. Vaccination against SARS-CoV-2 continues to diminish the risk of infection among vaccinated people and in the population as a whole. By contrast, the risk of infection among vaccinated people accrues disproportionately from contact with unvaccinated people.

Imperfect Vaccination Evolutionary Game Incorporating Individual Social Difference and Subjective Perception

Previous studies have focused on the assumption that individual attitudes toward epidemics, epidemic-related information, and vaccination are identical with much less effort attempting to consider psychological factors and individual heterogeneity. In this article, we employ a two-layer susceptible–infected–recovered–vaccinated/unaware–aware–unaware (SIRV-UAU) coupled network to depict the interplay between epidemic spreading and information diffusion. We propose an imperfect vaccination evolutionary game model integrating prospect theory (PT) to explore the effect of individual subjective perception and social difference on epidemic spreading and vaccination equilibrium. We find that the individual social difference in the epidemic spreading layer has a more significant impact on the epidemic threshold than that in the information diffusion layer. Individual psychological perception and vaccination behavior decision-making determine the vaccination equilibrium: under the PT, vaccination equilibrium increases with the decrease of the rationality coefficient when the vaccination cost is small. Furthermore, we analyze the effect of social differences on individual vaccination behavior decision-making and find that vaccination equilibrium increases with the increase of social reinforcement strength and primary protective probability. Finally, we discuss the effect of infection cost on vaccination equilibrium when the infection cost of vaccinated individuals is less than that of unvaccinated individuals and observe that a small infection cost helps improve the vaccination equilibrium.

Mutations make pandemics worse or better: modeling SARS-CoV-2 variants and imperfect vaccination.

COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, , which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when , that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate leads to oscillations. When increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation.

Dynamics of two-strain epidemic model with imperfect vaccination on complex networks

Dynamics of two-strain epidemic model with imperfect vaccination on complex networks

Dynamics of a mathematical model of virus spreading incorporating the effect of a vaccine

The COVID-19 pandemic led to widespread interest in epidemiological models. In this context the role of vaccination in influencing the spreading of the disease is of particular interest. There has also been a lot of debate on the role of non-pharmaceutical interventions such as the disinfection of surfaces. We investigate a mathematical model for the spread of a disease which includes both imperfect vaccination and infection due to virus in the environment. The latter is studied with the help of two phenomenological models for the force of infection. In one of these models we find that backward bifurcations take place so that for some parameter values an endemic steady state exists although the basic reproduction ratio R0 is less than one. We also prove that in that case there can exist more than one endemic steady state. In the other model all generic transcritical bifurcations are forward bifurcations so that these effects cannot occur. Thus we see that the occurrence of backward bifurcations, which can be important for disease control strategies, is dependent on the details of the function describing the force of infection. By means of simulations the predictions of this model are compared with data for COVID-19 from Turkey. A sensitivity analysis is also carried out.

Dynamic behaviors of a cholera model with nonlinear incidences, multiple transmission pathways, and imperfect vaccine

Dynamic behaviors of a cholera model with nonlinear incidences, multiple transmission pathways, and imperfect vaccine

Analyzing a class of stochastic SIRS models under imperfect vaccination

This work is devoted to a class of stochastic infectious disease models, namely, stochastic SIRS models under imperfect vaccination. Because of the inclusion of the vaccination, the underlying models are more difficult to analyze than that of the previously considered in the literature. Our main effort is devoted to overcoming the difficulties and deriving a complete characterization of longtime behavior of systems. Numerical examples are provided to present computational evidence. These examples also provide insight that the discrete event process (the random switching) can reverse persistence to extinction, and vice versa.

On the exact and population bi-dimensional reproduction numbers in a stochastic SVIR model with imperfect vaccine

We aim to quantify the spread of a direct contact infectious disease that confers permanent immunity after recovery, within a non-isolated finite and homogeneous population. Prior to the onset of the infection and to prevent the spread of this disease, a proportion of individuals was vaccinated. But the administered vaccine is imperfect and can fail, which implies that some vaccinated individuals get the infection when being in contact with infectious individuals. We study the evolution of the epidemic process over time in terms of a continuous-time Markov chain, which represents a general SIR model with an additional compartment for vaccinated individuals. In our stochastic framework, we study two bi-dimensional variables recording infection events, produced by a single infectious individual or by the whole infected group, taking into account if the newly infected individual was previously vaccinated or not. Theoretical schemes and recursive algorithms are derived in order to compute joint probability mass functions and factorial moments for these random variables. We illustrate the applicability of our techniques by means of a set of numerical experiments.

Global stability analysis of two strains epidemic model with imperfect vaccination and immunity waning in a complex network

For some epidemic diseases, there exist several strains of the pathogen, which can pose a challenge in controlling the disease and result in complicated dynamics, such as dengue fever and HIV. However, we lack the deep understanding of the dynamic features of interacting multiple strains. In order to reduce the final size of infections and achieve the herd immunity, vaccination is the most effective way of controlling an outbreak. To this end, we put forward the two-strain epidemic model in which vaccination is not perfect and immunity wanes in a complex network. It can be revealed that the disease free equilibrium point of two strains is global asymptotically stable when max{R1,R1}<1, but the coexistence equilibrium point remains and is stable when min{R1,R1}>1. Then, to verify the global stability of each strain dominance equilibrium point, the critical values are further derived. Finally, numerical simulations have been performed to visualize the results of the theoretical analysis. The current results are beneficial to comprehending the dynamics behaviors of multi-strain epidemics.

Effect of unequal vaccination coverage and migration on long-term pathogen evolution in a metapopulation.

Worldwide inequalities in vaccine availability are expected to affect the spread and spatial distribution of infectious diseases. It is unclear, however, how spatial variation in vaccination coverage can affect the long-term evolution of pathogens. Here we use an analytical model and numerical simulations to analyse the influence of different imperfect vaccines on the potential evolution of pathogen virulence in a two-population model where vaccination coverage varies between populations. We focus on four vaccines, with different modes of action on the life cycle of a pathogen infecting two host populations coupled by migration. We show that, for vaccines that reduce infection risk or transmissibility, spatial heterogeneity has little effect on pathogen prevalence and host mortality, and no effect on the evolution of pathogen virulence. In contrast, vaccines that reduce pathogen virulence can select for more virulent pathogens and may lead to the coexistence of different pathogen strains, depending on the degree of spatial heterogeneity in the metapopulation. This heterogeneity is driven by two parameters: pathogen migration and the difference in the vaccination rate between the two populations. We show that vaccines that only reduce pathogen virulence select mainly for a single pathogen strategy in the long term, while vaccines that reduce both transmission and virulence can favor the coexistence of two pathogen genotypes. We discuss the implications and potential extensions of our analysis.