Mathematical Analysis of a Transformed ODE from a PDE Multiscale Model of Hepatitis C Virus Infection.

Mathematical modeling has revealed the quantitative dynamics during the process of viral infection and evolved into an important tool in modern virology. Coupled with analyses of clinical and experimental data, the widely used basic model of viral dynamics described by ordinary differential equations (ODEs) has been well parameterized. In recent years, age-structured models, called "multiscale model," formulated by partial differential equations (PDEs) have also been developed and become useful tools for more detailed data analysis. However, in general, PDE models are considerably more difficult to subject to mathematical and numerical analyses. In our recently reported study, we successfully derived a mathematically identical ODE model from a PDE model, which helps to overcome the limitations of the PDE model with regard to clinical data analysis. Here, we derive the basic reproduction number from the identical ODE model and provide insight into the global stability of all possible steady states of the ODE model.

Modelling infectious diseases with relapse: a case study of HSV-2.

BACKGROUND: Herpes Simplex Virus Type 2 (HSV-2) is one of the most common sexually transmitted diseases. Although there is still no licensed vaccine for HSV-2, a theoretical investigation of the potential effects of a vaccine is considered important and has recently been conducted by several researchers. Although compartmental mathematical models were considered for each special case in the previous studies, as yet there are few global stability results. RESULTS: In this paper, we formulate a multi-group SVIRI epidemic model for HSV-2, which enables us to consider the effects of vaccination, of waning vaccine immunity, and of infection relapse. Since the number of groups is arbitrary, our model can be applied to various structures such as risk, sex, and age group structures. For our model, we define the basic reproduction number ℜ0 and prove that if ℜ0≤1, then the disease-free equilibrium is globally asymptotically stable, whereas if ℜ0>1, then the endemic equilibrium is so. Based on this global stability result, we estimate ℜ0 for HSV-2 by applying our model to the risk group structure and using US data from 2001 to 2014. Through sensitivity analysis, we find that ℜ0 is approximately in the range of 2-3. Moreover, using the estimated parameters, we discuss the optimal vaccination strategy for the eradication of HSV-2. CONCLUSIONS: Through discussion of the optimal vaccination strategy, we come to the following conclusions. (1) Improving vaccine efficacy is more effective than increasing the number of vaccines. (2) Although the transmission risk in female individuals is higher than that in male individuals, distributing the available vaccines almost equally between female and male individuals is more effective than concentrating them within the female population.

Application of the backstepping method to the prediction of increase or decrease of infected population.

BACKGROUND: In mathematical epidemiology, age-structured epidemic models have usually been formulated as the boundary-value problems of the partial differential equations. On the other hand, in engineering, the backstepping method has recently been developed and widely studied by many authors. METHODS: Using the backstepping method, we obtained a boundary feedback control which plays the role of the threshold criteria for the prediction of increase or decrease of newly infected population. Under an assumption that the period of infectiousness is same for all infected individuals (that is, the recovery rate is given by the Dirac delta function multiplied by a sufficiently large positive constant), the prediction method is simplified to the comparison of the numbers of reported cases at the current and previous time steps. RESULTS: Our prediction method was applied to the reported cases per sentinel of influenza in Japan from 2006 to 2015 and its accuracy was 0.81 (404 correct predictions to the total 500 predictions). It was higher than that of the ARIMA models with different orders of the autoregressive part, differencing and moving-average process. In addition, a proposed method for the estimation of the number of reported cases, which is consistent with our prediction method, was better than that of the best-fitted ARIMA model ARIMA(1,1,0) in the sense of mean square error. CONCLUSIONS: Our prediction method based on the backstepping method can be simplified to the comparison of the numbers of reported cases of the current and previous time steps. In spite of its simplicity, it can provide a good prediction for the spread of influenza in Japan.

Hopf bifurcation in a chronological age-structured SIR epidemic model with age-dependent infectivity.

In this paper, we examine the stability of an endemic equilibrium in a chronological age-structured SIR (susceptible, infectious, removed) epidemic model with age-dependent infectivity. Under the assumption that the transmission rate is a shifted exponential function, we perform a Hopf bifurcation analysis for the endemic equilibrium, which uniquely exists if the basic reproduction number is greater than 1. We show that if the force of infection in the endemic equilibrium is equal to the removal rate, then there always exists a critical value such that a Hopf bifurcation occurs when the bifurcation parameter reaches the critical value. Moreover, even in the case where the force of infection in the endemic equilibrium is not equal to the removal rate, we show that if the distance between them is sufficiently small, then a similar Hopf bifurcation can occur. By numerical simulation, we confirm a special case where the stability switch of the endemic equilibrium occurs more than once.

Recurrent epidemic waves in a delayed epidemic model with quarantine.

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R0 and show that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, whereas if R0>1, then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves.

Structure of epidemic models: toward further applications in economics.

In this paper, we review the structure of various epidemic models in mathematical epidemiology for the future applications in economics. The heterogeneity of population and the generalization of nonlinear terms play important roles in making more elaborate and realistic models. The basic, effective, control and type reproduction numbers have been used to estimate the intensity of epidemic, to evaluate the effectiveness of interventions and to design appropriate interventions. The advanced epidemic models includes the age structure, seasonality, spatial diffusion, mutation and reinfection, and the theory of reproduction numbers has been generalized to them. In particular, the existence of sustained periodic solutions has attracted much interest because they can explain the recurrent waves of epidemic. Although the theory of epidemic models has been developed in decades and the development has been accelerated through COVID-19, it is still difficult to completely answer the uncertainty problem of epidemic models. We would have to mind that there is no single model that can solve all questions and build a scientific attitude to comprehensively understand the results obtained by various researchers from different backgrounds.

Prediction of COVID-19 cases during Tokyo's Olympic and Paralympic Games.

Tokyo's Olympic and Paralympic Games set to begin in late July 2021 without spectators from abroad, but vaccine rollout has been slow in Japan compared with other developed countries. In this study, COVID-19 epidemic curve in Tokyo is developed based on weekly reported data from January 23, 2020, until April 16, 2021. The maximum daily number of infected cases in Tokyo in August 2021 would be 7,991 if the current pace of vaccinations would be used (1/1,000 per day). This daily number is greater than the highest daily cases (2,447) recorded on January 7, 2021. However, if the rollout pace could be doubled (1/500 per day), the peak daily new cases would be 4,470 in August. If it could be quadrupled (1/250 per day), the peak would be noted at 2,128 in July and the highest number in August would be 1,977. If vaccine rollout could not be enhanced, the cancellation might be an acceptable decision, since health is the most precious to local people and our Olympians.

Modeling population-wide testing of SARS-CoV-2 for containing COVID-19 pandemic in Okinawa, Japan.

To break the chains of SARS-CoV-2 transmission and contain the coronavirus disease 2019 (COVID-19) pandemic, population-wide testing has been practiced in various countries. However, scant research has addressed this topic in Japan. In this modeling exercise, we extracted the number of daily reported cases of COVID-19 in Okinawa from October 1 to November 30, 2020, and explored possible scenarios for decreasing COVID-19 incidence by combining population-wide screening and/or social distancing policy. We reveal that permanent lockdown can be theoretically replaced by mass testing but sufficient target population at an adequate frequency must be mobilized. In addition, solely imposing a circuit breaker will not bring a favorable outcome in the long run, and mass testing presents implications for minimizing a period of lockdown. Our results highlight the importance of incentivizing citizens to join the frequent testing and ensure their appropriate isolation. This study also suggests that early containment of COVID-19 will be feasible in prefectures where the mobility is low and/or can be easily controlled for its geographic characteristics. Rigorous investment in public health will be manifestly vital to contain COVID-19.

An age-structured epidemic model with boosting and waning of immune status.

In this paper, we developed an age-structured epidemic model that takes into account boosting and waning of immune status of host individuals. For many infectious diseases, the immunity of recovered individuals may be waning as time evolves, so reinfection could occur, but also their immune status could be boosted if they have contact with infective agent. According to the idea of the Aron's malaria model, we incorporate a boosting mechanism expressed by reset of recovery-age (immunity clock) into the SIRS epidemic model. We established the mathematical well-posedness of our formulation and showed that the initial invasion condition and the endemicity can be characterized by the basic reproduction number $ R_0 $. Our focus is to investigate the condition to determine the direction of bifurcation of endemic steady states bifurcated from the disease-free steady state, because it is a crucial point for disease prevention strategy whether there exist subcritical endemic steady states. Based on a recent result by Martcheva and Inaba [1], we have determined the direction of bifurcation that endemic steady states bifurcate from the disease-free steady state when the basic reproduction number passes through the unity. Finally, we have given a necessary and sufficient condition for backward bifurcation to occur.

Evaluation of the effect of the state of emergency for the first wave of COVID-19 in Japan.

In this paper, we evaluate the effect of the state of emergency for the first wave of COVID-19 in Japan, 2020 from the viewpoint of mathematical modelling. In Japan, it was announced during the period of the state of emergency from April 7 to May 25, 2020 that the 80% reduction of the contact rate is needed to control the outbreak. By numerical simulation, we show that the reduction rate seems to have reached up to 86%. Moreover, we estimate the control reproduction number R c during the period of the state of emergency as R c = 0.36 (95%CI, 0.34-0.39), and show that the effective reproduction number R e after the lifting of the state of emergency could be greater than 1. This result suggests us that the second wave of COVID-19 in Japan could possibly occur if any effective intervention will not be taken again.

Prediction of the Epidemic Peak of Coronavirus Disease in Japan, 2020.

The first case of coronavirus disease 2019 (COVID-19) in Japan was reported on 15 January 2020 and the number of reported cases has increased day by day. The purpose of this study is to give a prediction of the epidemic peak for COVID-19 in Japan by using the real-time data from 15 January to 29 February 2020. Taking into account the uncertainty due to the incomplete identification of infective population, we apply the well-known SEIR compartmental model for the prediction. By using a least-square-based method with Poisson noise, we estimate that the basic reproduction number for the epidemic in Japan is R 0 = 2 . 6 ( 95 % CI, 2 . 4 - 2 . 8 ) and the epidemic peak could possibly reach the early-middle summer. In addition, we obtain the following epidemiological insights: (1) the essential epidemic size is less likely to be affected by the rate of identification of the actual infective population; (2) the intervention has a positive effect on the delay of the epidemic peak; (3) intervention over a relatively long period is needed to effectively reduce the final epidemic size.

Possible effects of mixed prevention strategy for COVID-19 epidemic: massive testing, quarantine and social distancing.

BACKGROUND: The pandemic coronavirus disease 2019 (COVID-19) has spread and caused enormous and serious damages to many countries worldwide. One of the most typical interventions is the social distancing such as lockdown that would contribute to reduce the number of contacts among undiagnosed individuals. However, prolongation of the period of such a restrictive intervention could hugely affect the social and economic systems, and the outbreak will come back if the strong social distancing policy will end earlier due to the economic damage. Therefore, the social distancing policy should be followed by massive testing accompanied with quarantine to eradicate the infection. METHODS: In this paper, we construct a mathematical model and discuss the effect of massive testing with quarantine, which would be less likely to affect the social and economic systems, and its efficacy has been proved in South Korea, Taiwan, Vietnam and Hong Kong. RESULTS: By numerical calculation, we show that the control reproduction number is monotone decreasing and convex downward with respect to the testing rate, which implies that the improvement of the testing rate would highly contribute to reduce the epidemic size if the original testing rate is small. Moreover, we show that the recurrence of the COVID-19 epidemic in Japan could be possible after the lifting of the state of emergency if there is no massive testing and quarantine. CONCLUSIONS: If we have entered into an explosive phase of the epidemic, the massive testing could be a strong tool to prevent the disease as long as the positively reacted individuals will be effectively quarantined, no matter whether the positive reaction is pseudo or not. Since total population could be seen as a superposition of smaller communities, we could understand how testing and quarantine policy might be powerful to control the disease.

Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity.

The present work is devoted to the global stability analysis for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity. First, we characterize some subsets of attraction basins of equilibria. Next, by Lyapunov functional and fluctuation method, we obtain a series of criteria for the global stability of equilibria. Finally, we illustrate our results by applying them to a problem with Allee effect.

Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations.

We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.

Parameter estimation and prediction for coronavirus disease outbreak 2019 (COVID-19) in Algeria.

BACKGROUND: The wave of the coronavirus disease outbreak in 2019 (COVID-19) has spread all over the world. In Algeria, the first case of COVID-19 was reported on 25 February, 2020, and the number of confirmed cases of it has increased day after day. To overcome this difficult period and a catastrophic scenario, a model-based prediction of the possible epidemic peak and size of COVID-19 in Algeria is required. METHODS: We are concerned with a classical epidemic model of susceptible, exposed, infected and removed (SEIR) population dynamics. By using the method of least squares and the best fit curve that minimizes the sum of squared residuals, we estimate the epidemic parameter and the basic reproduction number ℜ 0. Moreover, we discuss the effect of intervention in a certain period by numerical simulation. RESULTS: We find that ℜ 0 = 4.1, which implies that the epidemic in Algeria could occur in a strong way. Moreover, we obtain the following epidemiological insights: the intervention has a positive effect on the time delay of the epidemic peak; the epidemic size is almost the same for a short intervention; a large epidemic can occur even if the intervention is long and sufficiently effective. CONCLUSION: Algeria must implement the strict measures as shown in this study, which could be similar to the one that China has finally adopted.

Demand and supply of invasive and noninvasive ventilators at the peak of the COVID-19 outbreak in Okinawa.

While Okinawa has been facing outbreak of the coronavirus disease 2019 (COVID-19) pandemic, healthcare collapse should be prevented by sufficient supply of ventilators for caring the rapidly growing number of critically ill patients with COVID-19. We estimated the number of invasive and noninvasive ventilators that would be required in Okinawa at the peak of the COVID-19 outbreak based on recent data of COVID-19 cases in Okinawa and data on the proportion of patients with COVID-19 in the ICU requiring ventilation. Based on our results using the current supply of all ventilators, demand for ventilators could be prepared for patients with COVID-19 who would require it and demand for noninvasive ventilators could also be prepared for those with COVID-19 who would require it. The higher supply over the demand would be achieved by flattening the epidemic curve by implementing public health interventions to delay and suppress the epidemic peak in Okinawa.

Mathematical analysis for an age-structured SIRS epidemic model.

In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number R0 to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on R0 and the critical coverage of immunization based on the reinfection threshold.

Global stability of an age-structured epidemic model with general Lyapunov functional.

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number R0 of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for R0 < 1 and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R0 > 1. Several numerical simulations are given to illustrate our results.

A note on dynamics of an age-of-infection cholera model.

A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if i(a,t)/i*(a)=0 on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.