An age-structured epidemic model of rotavirus with vaccination

Spreading dynamics for a time-periodic nonlocal dispersal epidemic model with delay and vaccination

It is known that vaccination plays an important strategy in eliminating infectious diseases. In this paper, we investigate the spreading dynamics for a time-periodic nonlocal dispersal epidemic model with delay and vaccination. We first establish the spreading speed of the model and an abstract framework on the existence of time-periodic traveling waves, which will help us to derive the existence of the super-critical and critical time-periodic traveling waves. Then we show that the spreading speed coincides with the minimal waves speed of time-periodic traveling waves. Further, we consider the effects of delay, periodicity, nonlocality and vaccination on the spreading speed. In the absence of delay, we find a large class of the time-periodic systems that have the same spreading speed. When delay is introduced, some numerical simulations reveal that the spreading speed initially exhibits oscillatory behavior and ultimately converges to a constant as time-period increases. Moreover, we observe that both delay and efficacy of vaccination decrease the spreading speed; both diffusion rate and nonlocality of infectious individuals increase the spreading speed; while the diffusion rates, nonlocalities of susceptible and vaccinated individuals do not affect the spreading speed. In particular, it is worth mentioning that the spreading speed is highly sensitivity to the efficacy of vaccination than the rate of vaccination.

Imperfect vaccine can yield multiple Nash equilibria in vaccination games

As infectious diseases continue to threaten communities across the globe, people are faced with a choice to vaccinate, or not. Many factors influence this decision, such as the cost of the disease, the chance of contracting the disease, the population vaccination coverage, and the efficacy of the vaccine. While the vaccination games in which individuals decide whether to vaccinate or not based on their own interests are gaining in popularity in recent years, the vaccine imperfection has been an overlooked aspect so far. In this paper we investigate the effects of an imperfect vaccine on the outcomes of a vaccination game. We use a simple SIR compartmental model for the underlying model of disease transmission. We model the vaccine imperfection by adding vaccination at birth and maintain a possibility for the vaccinated individual to become infected. We derive explicit conditions for the existence of different Nash equilibria, the solutions of the vaccination game. The outcomes of the game depend on the complex interplay between disease transmission dynamics (the basic reproduction number), the relative cost of the infection, and the vaccine efficacy. We show that for diseases with relatively low basic reproduction numbers (smaller than about 2.62), there is a little difference between outcomes for perfect or imperfect vaccines and thus the simpler models assuming perfect vaccines are good enough. However, when the basic reproduction number is above 2.62, then, unlike in the case of a perfect vaccine, there can be multiple equilibria. Moreover, unless there is a mandatory vaccination policy in place that would push the vaccination coverage above the value of unstable Nash equilibrium, the population could eventually slip to the "do not vaccinate" state. Thus, for diseases that have relatively high basic reproduction numbers, the potential for the vaccine not being perfect should be explicitly considered in the models.

An age-dependent immuno-epidemiological model with distributed recovery and death rates

The work is devoted to a new immuno-epidemiological model with distributed recovery and death rates considered as functions of time after the infection onset. Disease transmission rate depends on the intra-subject viral load determined from the immunological submodel. The age-dependent model includes the viral load, recovery and death rates as functions of age considered as a continuous variable. Equations for susceptible, infected, recovered and dead compartments are expressed in terms of the number of newly infected cases. The analysis of the model includes the proof of the existence and uniqueness of solution. Furthermore, it is shown how the model can be reduced to age-dependent SIR or delay model under certain assumptions on recovery and death distributions. Basic reproduction number and final size of epidemic are determined for the reduced models. The model is validated with a COVID-19 case data. Modelling results show that proportion of young age groups can influence the epidemic progression since disease transmission rate for them is higher than for other age groups.

Crossover Dynamics of Rotavirus Disease under Fractional Piecewise Derivative with Vaccination Effects: Simulations with Real Data from Thailand, West Africa, and the US

Many diseases are caused by viruses of different symmetrical shapes. Rotavirus particles are approximately 75 nm in diameter. They have icosahedral symmetry and particles that possess two concentric protein shells, or capsids. In this research, using a piecewise derivative framework with singular and non-singular kernels, we investigate the evolution of rotavirus with regard to the effect of vaccination. For the considered model, the existence of a solution of the piecewise rotavirus model is investigated via fixed-point results. The Adam–Bashforth numerical method along with the Newton polynomial is implemented to deduce the numerical solution of the considered model. Various versions of the stability of the solution of the piecewise rotavirus model are presented using the Ulam–Hyres concept and nonlinear analysis. We use MATLAB to perform the numerical simulation for a few fractional orders to study the crossover dynamics and evolution and effect of vaccination on rotavirus disease. To check the validity of the proposed approach, we compared our simulated results with real data from various countries.

Structural identifiability analysis of age-structured PDE epidemic models

Computational and mathematical models rely heavily on estimated parameter values for model development. Identifiability analysis determines how well the parameters of a model can be estimated from experimental data. Identifiability analysis is crucial for interpreting and determining confidence in model parameter values and to provide biologically relevant predictions. Structural identifiability analysis, in which one assumes data to be noiseless and arbitrarily fine-grained, has been extensively studied in the context of ordinary differential equation (ODE) models, but has not yet been widely explored for age-structured partial differential equation (PDE) models. These models present additional difficulties due to increased number of variables and partial derivatives as well as the presence of boundary conditions. In this work, we establish a pipeline for structural identifiability analysis of age-structured PDE models using a differential algebra framework and derive identifiability results for specific age-structured models. We use epidemic models to demonstrate this framework because of their wide-spread use in many different diseases and for the corresponding parallel work previously done for ODEs. In our application of the identifiability analysis pipeline, we focus on a Susceptible-Exposed-Infected model for which we compare identifiability results for a PDE and corresponding ODE system and explore effects of age-dependent parameters on identifiability. We also show how practical identifiability analysis can be applied in this example.

COVID-19 pandemic control using restrictions and vaccination

This work deals with the impact of the vaccination in combination with a restriction parameter that represents non-pharmaceutical interventions measures applied to the compartmental SEIR model in order to control the COVID-19 epidemic. This restriction parameter is used as a control parameter, and the univariate autoregressive integrated moving average (ARIMA) is used to forecast the time series of vaccination of all individuals of a specific country. Having in hand the time series of the population fully vaccinated (real data + forecast), the Levenberg-Marquardt algorithm is used to fit an analytic function that models this evolution over time. Here, it is used two time series of real data that refer to a slow vaccination obtained from India and Brazil, and two faster vaccination as observed in Israel and the United States of America. Together with vaccination, two different control approaches are presented in this paper, which enable reduces the infected people successfully: namely, the feedback and nonfeedback control methods. Numerical results predict that vaccination can reduce the peaks of infections and the duration of the pandemic, however, a better result is achieved when the vaccination is combined with any restrictions or prevention policy.

Stationary Distribution and Extinction in a Stochastic SIQR Epidemic Model Incorporating Media Coverage and Markovian Switching

This paper is concerned with the dynamic characteristics of the SIQR model with media coverage and regime switching. Firstly, the existence of the unique positive solution of the proposed system is investigated. Secondly, by constructing a suitable random Lyapunov function, some sufficient conditions for the existence of a stationary distribution is obtained. Meanwhile, the conditions for extinction is also given. Finally, some numerical simulation examples are carried out to demonstrate the effectiveness of theoretical results.

Pulse vaccination of an epidemic model with two parallel infectious stages and time delays

An epidemic model with two parallel infectious stages and time delays and pulse vaccination is proposed. We introduce four thresholds and further obtain the conditions that the disease will be extinct or not. Corollaries show that under condition that θ > max { θ ∗ 1 , θ ∗ 2 } the disease will fade out, and if θ < min { θ 1 ∗ , θ 2 ∗ } , the disease will be endemic. Our results indicate that a larger pulse vaccination rate will lead to the eradication of a disease. Furthermore, two thresholds ℜ ∗ 1 and ℜ ∗ 2 show that the diversity of the contagiousness affects the basic properties of these models. In addition, numerical results indicate that the probability for an infected individual to enter different infective compartments greatly affects two infective compartments.

The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model

In this paper, we introduce stochasticity into an SIR epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number [Formula: see text]. If [Formula: see text], the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If [Formula: see text], there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.

Demographic buffering: titrating the effects of birth rate and imperfect immunity on epidemic dynamics

Host demography can alter the dynamics of infectious disease. In the case of perfectly immunizing infections, observations of strong sensitivity to demographic variation have been mechanistically explained through analysis of the susceptible–infected–recovered (SIR) model that assumes lifelong immunity following recovery from infection. When imperfect immunity is incorporated into this framework via the susceptible–infected–recovered–susceptible (SIRS) model, with individuals regaining full susceptibility following recovery, we show that rapid loss of immunity is predicted to buffer populations against the effects of demographic change. However, this buffering is contrary to the dependence on demography recently observed for partially immunizing infections such as rotavirus and respiratory syncytial virus. We show that this discrepancy arises from a key simplification embedded in the SIR(S) framework, namely that the potential for differential immune responses to repeat exposures is ignored. We explore the minimum additional immunological information that must be included to reflect the range of observed dependencies on demography. We show that including partial protection and lower transmission following primary infection is sufficient to capture more realistic reduced levels of buffering, in addition to changes in epidemic timing, across a range of partially and fully immunizing infections. Furthermore, our results identify key variables in this relationship, including R₀.

Qualitative Effects of Monovalent Vaccination Against Rotavirus: A Comparison of North America and South America

Rotavirus is the most common cause of severe gastroenteritis in young children worldwide. The introduction of vaccination programs has led to a significant reduction in number of hospitalizations due to rotavirus in North and South American countries. Little work has been done, however, to examine the differential impact of vaccination as a function of strain distribution and strain-specific vaccine efficacy. We developed a two-strain epidemiological model of rotavirus transmission, and used it to examine the effects of a monovalent vaccine (Rotarix) on the qualitative behaviors of infection levels in a population. For contrast, we parameterized our model with strain distribution data from North America and from South America. In all cases, the introduction of the vaccine led to significant decreases in the prevalence of primary infection due to both strains for a decade or more, after which the overall prevalence recovers to near pre-vaccination levels. The prevalence of G1P[8] is significantly higher in North America (73 % of all rotavirus infections) compared to that in South America (34 %). Our model predicts that the introduction of Rotarix might result in major strain replacement in regions such as North America where the prevalence of G1P[8] is relatively high, due to higher efficacy of Rotarix against infection caused by G1P[8], while regions with lower prevalence of G1P[8], such as South America, are not susceptible to major strain replacement.

Mathematical modelling and prediction in infectious disease epidemiology

We discuss to what extent disease transmission models provide reliable predictions. The concept of prediction is delineated as it is understood by modellers, and illustrated by some classic and recent examples. A precondition for a model to provide valid predictions is that the assumptions underlying it correspond to the reality, but such correspondence is always limited—all models are simplifications of reality. A central tenet of the modelling enterprise is what we may call the ‘robustness thesis’: a model whose assumptions approximately correspond to reality will make predictions that are approximately valid. To examine which of the predictions made by a model are trustworthy, it is essential to examine the outcomes of different models. Thus, if a highly simplified model makes a prediction, and if the same or a very similar prediction is made by a more elaborate model that includes some mechanisms or details that the first model did not, then we gain some confidence that the prediction is robust. An important benefit derived from mathematical modelling activity is that it demands transparency and accuracy regarding our assumptions, thus enabling us to test our understanding of the disease epidemiology by comparing model results and observed patterns. Models can also assist in decision-making by making projections regarding important issues such as intervention-induced changes in the spread of disease.

IMPACT OF AGE-DEPENDENT RELAPSE AND IMMUNITY ON MALARIA DYNAMICS

An age-structured mathematical model for malaria is presented. The model explicitly includes the human and mosquito populations, structured by chronological age of humans. The infected human population is divided into symptomatic infectious, asymptomatic infectious and asymptomatic chronic infected individuals. The original partial differential equation (PDE) model is reduced to an ordinary differential equation (ODE) model with multiple age groups coupled by aging. The basic reproduction number [Formula: see text] is derived for the PDE model and the age group model in the case of general n age groups. We assume that infectiousness of chronic infected individuals gets triggered by bites of even susceptible mosquitoes. Our analysis points out that this assumption contributes greatly to the [Formula: see text] expression and therefore needs to be further studied and understood. Numerical simulations for n = 2 age groups and a sensitivity/uncertainty analysis are presented. Results suggest that it is important not only to consider asymptomatic infectious individuals as a hidden cause for malaria transmission, but also asymptomatic chronic infections (>60%), which often get neglected due to undetectable parasite loads. These individuals represent an important reservoir for future human infectiousness. By considering age-dependent immunity types, the model helps generate insight into effective control measures, by targeting age groups in an optimal way.

From the guest editors

Carlos Castilo-Chavez is a Regents Professor, a Joaquin Bustoz Jr. Professor of Mathematical Biology, and a Distinguished Sustainability Scientist at Arizona State University. His research program is at the interface of the mathematical and natural and social sciences with emphasis on (i) the role of dynamic social landscapes on disease dispersal; (ii) the role of environmental and social structures on the dynamics of addiction and disease evolution, and (iii) Dynamics of complex systems at the interphase of ecology, epidemiology and the social sciences. Castillo-Chavez has co-authored over two hundred publications (see goggle scholar citations) that include journal articles and edited research volumes. Specifically, he co-authored a textbook in Mathematical Biology in 2001 (second edition in 2012); a volume (with Harvey Thomas Banks) on the use of mathematical models in homeland security published in SIAM's Frontiers in Applied Mathematics Series (2003); and co-edited volumes in the Series Contemporary Mathematics entitled '' Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges'' (American Mathematical Society, 2006) and Mathematical and Statistical Estimation Approaches in Epidemiology (Springer-Verlag, 2009) highlighting his interests in the applications of mathematics in emerging and re-emerging diseases. Castillo-Chavez is a member of the Santa Fe Institute's external faculty, adjunct professor at Cornell University, and contributor, as a member of the Steering Committee of the '' Committee for the Review of the Evaluation Data on the Effectiveness of NSF-Supported and Commercially Generated Mathematics Curriculum Materials,'' to a 2004 NRC report. The CBMS workshop '' Mathematical Epidemiology with Applications'' lectures delivered by C. Castillo-Chavez and F. Brauer in 2011 have been published by SIAM in 2013.

Distinguishing vaccine efficacy and effectiveness

BACKGROUND

Mathematical models of disease transmission and vaccination typically assume that protective vaccine efficacy (i.e. the relative reduction in the transmission rate among vaccinated individuals) is equivalent to direct effectiveness of vaccine. This assumption has not been evaluated.

METHODS

We used dynamic epidemiological models of influenza and measles vaccines to evaluate the common measures of vaccine effectiveness in terms of both the protection of individuals and disease control within populations. We determined how vaccine-mediated reductions in attack rates translate into vaccine efficacy as well as into the common population measures of 'direct', 'indirect', 'total', and 'overall' effects of vaccination with examples of compartmental models of influenza and measles vaccination.

RESULTS

We found that the typical parameterization of vaccine efficacy using direct effectiveness of vaccine can lead to the underestimation of the impact of vaccine. Such underestimation occurs when the vaccine is assumed to offer partial protection to every vaccinated person, and becomes worse when the level of vaccine coverage is low. Nevertheless, estimates of 'total', 'indirect' and 'overall' effectiveness increase with vaccination coverage in the population. Furthermore, we show how the measures of vaccine efficacy and vaccine effectiveness can be correctly calculated.

CONCLUSIONS

Typical parameterization of vaccine efficacy in mathematical models may underestimate the actual protective effect of the vaccine, resulting in discordance between the actual effects of vaccination at the population level and predictions made by models. This work shows how models can be correctly parameterized from clinical trial data.

Direct and indirect effects of rotavirus vaccination: comparing predictions from transmission dynamic models

Early observations from countries that have introduced rotavirus vaccination suggest that there may be indirect protection for unvaccinated individuals, but it is unclear whether these benefits will extend to the long term. Transmission dynamic models have attempted to quantify the indirect protection that might be expected from rotavirus vaccination in developed countries, but results have varied. To better understand the magnitude and sources of variability in model projections, we undertook a comparative analysis of transmission dynamic models for rotavirus. We fit five models to reported rotavirus gastroenteritis (RVGE) data from England and Wales, and evaluated outcomes for short- and long-term vaccination effects. All of our models reproduced the important features of rotavirus epidemics in England and Wales. Models predicted that during the initial year after vaccine introduction, incidence of severe RVGE would be reduced 1.8–2.9 times more than expected from the direct effects of the vaccine alone (28–50% at 90% coverage), but over a 5-year period following vaccine introduction severe RVGE would be reduced only by 1.1–1.7 times more than expected from the direct effects (54–90% at 90% coverage). Projections for the long-term reduction of severe RVGE ranged from a 55% reduction at full coverage to elimination with at least 80% coverage. Our models predicted short-term reductions in the incidence of RVGE that exceeded estimates of the direct effects, consistent with observations from the United States and other countries. Some of the models predicted that the short-term indirect benefits may be offset by a partial shifting of the burden of RVGE to older unvaccinated individuals. Nonetheless, even when such a shift occurs, the overall reduction in severe RVGE is considerable. Discrepancies among model predictions reflect uncertainties about age variation in the risk and reporting of RVGE, and the duration of natural and vaccine-induced immunity, highlighting important questions for future research.

TWO DIFFERENT VACCINATION STRATEGIES IN AN SIR EPIDEMIC MODEL WITH SATURATED INFECTIOUS FORCE

Two different vaccination and treatment strategies in the SIR epidemic model with saturation infectious force are analyzed. With the continuous vaccination and treatment, it is obtained that the disease free equilibrium and endemic equilibrium are globally asymptotically stable by using Lassall theorem and Pioncare–Bendixon trichotomy. Moreover, with pulse vaccination and treatment at different time, the dynamics of the epidemic model is globally investigated by using Floquet theory and comparison theorem of impulsive differential equation and analytic method. We obtain the conditions of global asymptotical stability of the infection-free periodic solution and permanence of the model. Finally, we compare the two different vaccination and treatment strategies, and obtain that the elimination of disease is independent of treatment in the case of the pulse vaccination.

Influence of birth rates and transmission rates on the global seasonality of rotavirus incidence

Rotavirus is a major cause of mortality in developing countries, and yet the dynamics of rotavirus in such settings are poorly understood. Rotavirus is typically less seasonal in the tropics, although recent observational studies have challenged the universality of this pattern. While numerous studies have examined the association between environmental factors and rotavirus incidence, here we explore the role of intrinsic factors. By fitting a mathematical model of rotavirus transmission dynamics to published age distributions of cases from 15 countries, we obtain estimates of local transmission rates. Model-predicted patterns of seasonal incidence based solely on differences in birth rates and transmission rates are significantly correlated with those observed (Spearman's ρ = 0.65, p < 0.05). We then examine seasonal patterns of rotavirus predicted across a range of different birth rates and transmission rates and explore how vaccination may impact these patterns. Our results suggest that the relative lack of rotavirus seasonality observed in many tropical countries may be due to the high birth rates and transmission rates typical of developing countries rather than being driven primarily by environmental conditions. While vaccination is expected to decrease the overall burden of disease, it may increase the degree of seasonal variation in the incidence of rotavirus in some settings.

Dynamic model of rotavirus transmission and the impact of rotavirus vaccination in Kyrgyzstan

New rotavirus vaccines show promise to reduce the burden of severe diarrhea among children in developing countries. We present an age-specific dynamic rotavirus model to assess the effect of rotavirus vaccination in Kyrgyzstan, a country in Central Asia that is eligible for funds from the GAVI Alliance. A routine rotavirus vaccination program at 95% coverage and 54% effectiveness against severe infection is estimated to lead to a 56% reduction in rotavirus-associated deaths and a 50% reduction in hospital admissions, while outpatient visits and homecare episodes would decrease by 52% compared to baseline levels after 5 years of intervention. A 10% reduction in vaccine efficacy due to incomplete 3-dose regimen is estimated to increase the numbers of severe cases by 6-8%. Herd immunity was found to account for 1% or less of averted cases of severe gastroenteritis, while an extra 7-8% of all rotavirus infections would be avoided due to reduced transmission.

CONCLUSION

Rotavirus vaccines would reduce the burden of rotavirus disease substantially, but the results are sensitive to delay in age-appropriate vaccination.

Unexpectedly high burden of rotavirus gastroenteritis in very young infants

BACKGROUND

The highest incidence of rotavirus gastroenteritis has generally been reported in children 6-24 months of age. Young infants are thought to be partially protected by maternal antibodies acquired transplacentally or via breast milk. The purpose of our study was to assess the age distribution of children with confirmed community-acquired rotavirus gastroenteritis presenting to an urban referral hospital.

METHODS

Children presenting to The Children's Hospital of Philadelphia with acute gastroenteritis have been monitored for the presence of rotavirus antigen in the stool by ELISA (followed by genotyping if ELISA-positive) since the 1994-95 epidemic season.

RESULTS

Over the last 12 rotavirus seasons prior to the introduction of the pentavalent rotavirus vaccine in 2006, stool specimens from 1646 patients tested positive for community-acquired rotavirus infection. Gender or age was not recorded in 6 and 5 cases, respectively. Overall, 58% of the cases occurred in boys. G1 was the predominant VP7 serotype, accounting for 72% of cases. The median (IQR) age was 11 (5-21) months. A total of 790 (48%) cases occurred in children outside the commonly quoted peak age range, with 27% in infants <6 months of age and 21% in children >24 months of age. A total of 220 (13%) cases occurred during the first 3 months of life, and the highest number of episodes per month of age [97 (6%)] was observed during the second month of life.

CONCLUSIONS

The incidence of community-acquired rotavirus gastroenteritis monitored over 12 seasons in the prevaccine era at a major university hospital was nearly constant for each month of age during the first year of life, revealing an unexpectedly high incidence of symptomatic rotavirus disease in infants <3 months old. A sizeable fraction of cases occurred in children too young to have been vaccinated according to current recommendations.

A mathematical model of the indirect effects of rotavirus vaccination

Rotavirus (RV) infections progressively confer natural immunity against subsequent infection. Similarly to natural infection, vaccination with a live attenuated vaccine potentially reduces RV transmission and induces herd protection. A mathematical transmission model was developed to project the impact of a vaccination programme on the incidence of RV infection and disease for five countries in the European Union. With vaccination coverage rates of 70%, 90% and 95% the model predicted that, in addition to the direct effect of vaccination, herd protection induced a reduction in RV-related gastroenteritis (GE) incidence of 25%, 22% and 20%, respectively, for RV-GE of any severity, and of 19%, 15%, and 13%, respectively, for moderate-to-severe RV-GE, 5 years after implementation of a vaccination programme.