Quantifying the basic reproduction number and underestimated fraction of Mpox cases worldwide at the onset of the outbreak.

In 2022, there was a global resurgence of mpox, with different clinical-epidemiological features compared with previous outbreaks. Sexual contact was hypothesized as the primary transmission route, and the community of men having sex with men (MSM) was disproportionately affected. Because of the stigma associated with sexually transmitted infections, the real burden of mpox could be masked. We quantified the basic reproduction number (R 0) and the underestimated fraction of mpox cases in 16 countries, from the onset of the outbreak until early September 2022, using Bayesian inference and a compartmentalized, risk-structured (high-/low-risk populations) and two-route (sexual/non-sexual transmission) mathematical model. Machine learning (ML) was harnessed to identify underestimation determinants. Estimated R 0 ranged between 1.37 (Canada) and 3.68 (Germany). The underestimation rates for the high- and low-risk populations varied between 25-93% and 65-85%, respectively. The estimated total number of mpox cases, relative to the reported cases, is highest in Colombia (3.60) and lowest in Canada (1.08). In the ML analysis, two clusters of countries could be identified, differing in terms of attitudes towards the 2SLGBTQIAP+ community and the importance of religion. Given the substantial mpox underestimation, surveillance should be enhanced, and country-specific campaigns against the stigmatization of MSM should be organized, leveraging community-based interventions.

Host behaviour driven by awareness of infection risk amplifies the chance of superspreading events.

We demonstrate that heterogeneity in the perceived risks associated with infection within host populations amplifies chances of superspreading during the crucial early stages of epidemics. Under this behavioural model, individuals less concerned about dangers from infection are more likely to be infected and attend larger sized (riskier) events, where we assume event sizes remain unchanged. For directly transmitted diseases such as COVID-19, this leads to infections being introduced at rates above the population prevalence to those events most conducive to superspreading. We develop an interpretable, computational framework for evaluating within-event risks and derive a small-scale reproduction number measuring how the infections generated at an event depend on transmission heterogeneities and numbers of introductions. This generalizes previous frameworks and quantifies how event-scale patterns and population-level characteristics relate. As event duration and size grow, our reproduction number converges to the basic reproduction number. We illustrate that even moderate levels of heterogeneity in the perceived risks of infection substantially increase the likelihood of disproportionately large clusters of infections occurring at larger events, despite fixed overall disease prevalence. We show why collecting data linking host behaviour and event attendance is essential for accurately assessing the risks posed by invading pathogens in emerging stages of outbreaks.

Source-Sink Dynamics in a Two-Patch SI Epidemic Model with Life Stages and No Recovery from Infection.

This study presents a comprehensive analysis of a two-patch, two-life stage SI model without recovery from infection, focusing on the dynamics of disease spread and host population viability in natural populations. The model, inspired by real-world ecological crises like the decline of amphibian populations due to chytridiomycosis and sea star populations due to Sea Star Wasting Disease, aims to understand the conditions under which a sink host population can present ecological rescue from a healthier, source population. Mathematical and numerical analyses reveal the critical roles of the basic reproductive numbers of the source and sink populations, the maturation rate, and the dispersal rate of juveniles in determining population outcomes. The study identifies basic reproduction numbers R 0 for each of the patches, and conditions for the basic reproduction numbers to produce a receiving patch under which its population. These findings provide insights into managing natural populations affected by disease, with implications for conservation strategies, such as the importance of maintaining reproductively viable refuge populations and considering the effects of dispersal and maturation rates on population recovery. The research underscores the complexity of host-pathogen dynamics in spatially structured environments and highlights the need for multi-faceted approaches to biodiversity conservation in the face of emerging diseases.

Global stability and optimal control of an age-structured SVEIR epidemic model with waning immunity and relapses.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number R 0 and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of R 0 is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

Prevention and control of Ebola virus transmission: mathematical modelling and data fitting.

The Ebola virus disease (EVD) has been endemic since 1976, and the case fatality rate is extremely high. EVD is spread by infected animals, symptomatic individuals, dead bodies, and contaminated environment. In this paper, we formulate an EVD model with four transmission modes and a time delay describing the incubation period. Through dynamical analysis, we verify the importance of blocking the infection source of infected animals. We get the basic reproduction number without considering the infection source of infected animals. And, it is proven that the model has a globally attractive disease-free equilibrium when the basic reproduction number is less than unity; the disease eventually becomes endemic when the basic reproduction number is greater than unity. Taking the EVD epidemic in Sierra Leone in 2014-2016 as an example, we complete the data fitting by combining the effect of the media to obtain the unknown parameters, the basic reproduction number and its time-varying reproduction number. It is shown by parameter sensitivity analysis that the contact rate and the removal rate of infected group have the greatest influence on the prevalence of the disease. And, the disease-controlling thresholds of these two parameters are obtained. In addition, according to the existing vaccination strategy, only the inoculation ratio in high-risk areas is greater than 0.4, the effective reproduction number can be less than unity. And, the earlier the vaccination time, the greater the inoculation ratio, and the faster the disease can be controlled.

Analysis of the impact of COVID-19 variants and vaccination on the time-varying reproduction number: statistical methods.

Introduction: The COVID-19 pandemic has profoundly impacted global health systems, requiring the monitoring of infection waves and strategies to control transmission. Estimating the time-varying reproduction number is crucial for understanding the epidemic and guiding interventions. Methods: Probability distributions of serial interval are estimated for Pre-Delta and Delta periods. We conducted a comparative analysis of time-varying reproduction numbers, taking into account population immunity and variant differences. We incorporated the regional heterogeneity and age distribution of the population, as well as the evolving variants and vaccination rates over time. COVID-19 transmission dynamics were analyzed with variants and vaccination. Results: The reproduction number is computed with and without considering variant-based immunity. In addition, values of reproduction number significantly differed by variants, emphasizing immunity's importance. Enhanced vaccination efforts and stringent control measures were effective in reducing the transmission of the Delta variant. Conversely, Pre-Delta variant appeared less influenced by immunity levels, due to lower vaccination rates. Furthermore, during the Pre-Delta period, there was a significant difference between the region-specific and the non-region-specific reproduction numbers, with particularly distinct pattern differences observed in Gangwon, Gyeongbuk, and Jeju in Korea. Discussion: This research elucidates the dynamics of COVID-19 transmission concerning the dominance of the Delta variant, the efficacy of vaccinations, and the influence of immunity levels. It highlights the necessity for targeted interventions and extensive vaccination coverage. This study makes a significant contribution to the understanding of disease transmission mechanisms and informs public health strategies.

Studying the impacts of variant evolution for a generalized age-group transmission model.

The differences of SARS-CoV-2 variants brought the changes of transmission characteristics and clinical manifestations during the prevalence of COVID-19. In order to explore the evolution mechanisms of SARS-CoV-2 variants and the impacts of variant evolution, the classic SIR (Susceptible-Infected-Recovered) compartment model was modified to a generalized SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) compartment model with age-group and varying variants in this study. By using of the SVEIR model and least squares method, the optimal fittings against the surveillance data from Fujian Provincial Center for Disease Control and Prevention were performed for the five epidemics of Fujian Province. The main epidemiological characteristics such as basic reproduction number, effective reproduction number, sensitivity analysis, and cross-variant scenario investigations were extensively investigated during dynamic zero-COVID policy. The study results showed that the infectivities of the variants became fast from wild strain to the Delta variant, further to the Omicron variant. Meanwhile, the cross-variant investigations showed that the average incubation periods were shortened, and that the infection scales quickly enhanced. Further, the risk estimations with the new variants were performed without implements of the non-pharmaceutical interventions, based on the dominant variants XBB.1.9.1 and EG.5. The results of the risk estimations suggested that non-pharmaceutical interventions were necessary on the Chinese mainland for controlling severe infections and deaths, and also that the regular variant monitors were still workable against the aggressive variant evolution and the emergency of new transmission risks in the future.

A mathematical model of COVID-19 with multiple variants of the virus under optimal control in Ghana.

In this paper, we suggest a mathematical model of COVID-19 with multiple variants of the virus under optimal control. Mathematical modeling has been used to gain deeper insights into the transmission of COVID-19, and various prevention and control strategies have been implemented to mitigate its spread. Our model is a SEIR-based model for multi-strains of COVID-19 with 7 compartments. We also consider the circulatory structure to account for the termination of immunity for COVID-19. The model is established in terms of the positivity and boundedness of the solution and the existence of equilibrium points, and the local stability of the solution. As a result of fitting data of COVID-19 in Ghana to the model, the basic reproduction number of the original virus and Delta variant was estimated to be 1.9396, and the basic reproduction number of the Omicron variant was estimated to be 3.4905, which is 1.8 times larger than that. We observe that even small differences in the incubation and recovery periods of two strains with the same initial transmission rate resulted in large differences in the number of infected individuals. In the case of COVID-19, infections caused by the Omicron variant occur 1.5 to 10 times more than those caused by the original virus. In terms of the optimal control strategy, we formulate three control strategies focusing on social distancing, vaccination, and testing-treatment. We have developed an optimal control model for the three strategies outlined above for the multi-strain model using the Pontryagin's Maximum Principle. Through numerical simulations, we analyze three optimal control strategies for each strain and also consider combinations of the two control strategies. As a result of the simulation, all control strategies are effective in reducing disease spread, in particular, vaccination strategies are more effective than the other two control strategies. In addition the combination of the two strategies also reduces the number of infected individuals by 1/10 compared to implementing one strategy, even when mild levels are implemented. Finally, we show that if the testing-treatment strategy is not properly implemented, the number of asymptomatic and unidentified infections may surge. These results could help guide the level of government intervention and prevention strategy formulation.

Emergence of the reproduction matrix in epidemic forecasting.

During the recent COVID-19 pandemic, the instantaneous reproduction number, R(t), has surged as a widely used measure to target public health interventions aiming at curbing the infection rate. In analogy with the basic reproduction number that arises from the linear stability analysis, R(t) is typically interpreted as a threshold parameter that separates exponential growth (R(t) > 1) from exponential decay (R(t) < 1). In real epidemics, however, the finite number of susceptibles, the stratification of the population (e.g. by age or vaccination state), and heterogeneous mixing lead to more complex epidemic courses. In the context of the multidimensional renewal equation, we generalize the scalar R(t) to a reproduction matrix, [Formula: see text], which details the epidemic state of the stratified population, and offers a concise epidemic forecasting scheme. First, the reproduction matrix is computed from the available incidence data (subject to some a priori assumptions), then it is projected into the future by a transfer functional to predict the epidemic course. We demonstrate that this simple scheme allows realistic and accurate epidemic trajectories both in synthetic test cases and with reported incidence data from the COVID-19 pandemic. Accounting for the full heterogeneity and nonlinearity of the infection process, the reproduction matrix improves the prediction of the infection peak. In contrast, the scalar reproduction number overestimates the possibility of sustaining the initial infection rate and leads to an overshoot in the incidence peak. Besides its simplicity, the devised forecasting scheme offers rich flexibility to be generalized to time-dependent mitigation measures, contact rate, infectivity and vaccine protection.

Stability analysis of a nonlinear malaria transmission epidemic model using an effective numerical scheme.

Malaria is a fever condition that results from Plasmodium parasites, which are transferred to humans by the attacks of infected female Anopheles mosquitos. The deterministic compartmental model was examined using stability theory of differential equations. The reproduction number was obtained to be asymptotically stable conditions for the disease-free, and the endemic equilibria were determined. More so, the qualitatively evaluated model incorporates time-dependent variable controls which was aimed at reducing the proliferation of malaria disease. The reproduction number R o was determined to be an asymptotically stable condition for disease free and endemic equilibria. In this paper, we used various schemes such as Runge-Kutta order 4 (RK-4) and non-standard finite difference (NSFD). All of the schemes produce different results, but the most appropriate scheme is NSFD. This is true for all step sizes. Various criteria are used in the NSFD scheme to assess the local and global stability of disease-free and endemic equilibrium points. The Routh-Hurwitz condition is used to validate the local stability and Lyapunov stability theorem is used to prove the global asymptotic stability. Global asymptotic stability is proven for the disease-free equilibrium when R 0 ≤ 1 . The endemic equilibrium is investigated for stability when R 0 ≥ 1 . All of the aforementioned schemes and their effects are also numerically demonstrated. The comparative analysis demonstrates that NSFD is superior in every way for the analysis of deterministic epidemic models. The theoretical effects and numerical simulations provided in this text may be used to predict the spread of infectious diseases.

Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment.

BACKGROUND AND OBJECTIVE: Hepatitis virus infections are affecting millions of people worldwide, causing death, disability, and considerable expenditure. Chronic infection with hepatitis C virus (HCV) can cause severe public health problems because of their high prevalence and poor long-term clinical outcomes. Thus a fractional-order epidemic model of the hepatitis C virus involving partial immunity under the influence of memory effect to know the transmission patterns and prevalence of HCV infection is studied. Investigating the transmission dynamics of HCV makes the issue more interesting. The HCV epidemic model and worldwide dynamics are examined in this study. Calculate the basic reproduction number for the HCV model using the next-generation matrix technique. We determine the model's global dynamics using reproduction numbers, the Lyapunov functional approach, and the Routh-Hurwitz criterion. The model's reproduction number shows how the disease progresses. METHODS: A fractional differential equation model of HCV infection has been created. Maximum relevant parameters, such as fractional power, HCV transmission rate, reproduction number, etc., influencing the dynamic process, have been incorporated. The model's numerical solutions are obtained using the fractional Adams method. Finally, numerical simulations support the theoretical conclusions, showing the great agreement between the two. RESULTS: In the fractional-order HCV infection model, the memory effect, which is not seen in the classical model, was shown on graphs so that disease dynamics and vector compartments could be seen. We found that the fractional-order HCV infection model has more stages of freedom than regular derivatives. Fractional-order derivations, which may be the best and most reliable, explained bodily approaches better than classical order. CONCLUSION: The current study modeled and analyzed a fractional-order HCV infection model. The current approach results in a much better understanding of HCV transmission in a population, which leads to important insights into its spread and control, such as better treatment dosage for different age groups, identifying the best control measure, improving health, prolonging life, reducing the risk of HCV transmission, and effectively increasing the quality of life of HCV patients. The creation of a fractional-order HCV infection model, which provides a better understanding of HCV transmission dynamics and leads to significant insights for better treatment dosages, identification of optimal control measures, and ultimately improvement of the quality of life for HCV patients, is the study's major outcome.

Analysis of the COVID-19 model with self-protection and isolation measures affected by the environment.

Since the global outbreak of COVID-19, the virus has continuously mutated and can survive in the air for long periods of time. This paper establishes and analyzes a model of COVID-19 with self-protection and quarantine measures affected by viruses in the environment to investigate the influence of viruses in the environment on the spread of the outbreak, as well as to develop a rational prevention and control measure to control the spread of the outbreak. The basic reproduction number was calculated and Lyapunov functions were constructed to discuss the stability of the model equilibrium points. The disease-free equilibrium point was proven to be globally asymptotically stable when $ R_0 < 1 $, and the endemic equilibrium point was globally asymptotically stable when $ R_0 > 1 $. The model was fitted using data from COVID-19 cases in Chongqing between November 1 to November 25, 2022. Based on the numerical analysis, the following conclusion was obtained: clearing the virus in the environment and strengthening the isolation measures for infected people can control the epidemic to a certain extent, but enhancing the self-protection of individuals can be more effective in reducing the risk of being infected and controlling the transmission of the epidemic, which is more conducive to the practical application.

A deterministic transmission model for analytics-driven optimization of COVID-19 post-pandemic vaccination and quarantine strategies.

This study developed a deterministic transmission model for the coronavirus disease of 2019 (COVID-19), considering various factors such as vaccination, awareness, quarantine, and treatment resource limitations for infected individuals in quarantine facilities. The proposed model comprised five compartments: susceptible, vaccinated, quarantined, infected, and recovery. It also considered awareness and limited resources by using a saturated function. Dynamic analyses, including equilibrium points, control reproduction numbers, and bifurcation analyses, were conducted in this research, employing analytics to derive insights. Our results indicated the possibility of an endemic equilibrium even if the reproduction number for control was less than one. Using incidence data from West Java, Indonesia, we estimated our model parameter values to calibrate them with the real situation in the field. Elasticity analysis highlighted the crucial role of contact restrictions in reducing the spread of COVID-19, especially when combined with community awareness. This emphasized the analytics-driven nature of our approach. We transformed our model into an optimal control framework due to budget constraints. Leveraging Pontriagin's maximum principle, we meticulously formulated and solved our optimal control problem using the forward-backward sweep method. Our experiments underscored the pivotal role of vaccination in infection containment. Vaccination effectively reduces the risk of infection among vaccinated individuals, leading to a lower overall infection rate. However, combining vaccination and quarantine measures yields even more promising results than vaccination alone. A second crucial finding emphasized the need for early intervention during outbreaks rather than delayed responses. Early interventions significantly reduce the number of preventable infections, underscoring their importance.

Modeling different infectious phases of hepatitis B with generalized saturated incidence: An analysis and control.

Hepatitis B is one of the global health issues caused by the hepatitis B virus (HBV), producing 1.1 million deaths yearly. The acute and chronic phases of HBV are significant because worldwide, approximately 250 million people are infected by chronic hepatitis B. The chronic stage is a long-term, persistent infection that can cause liver damage and increase the risk of liver cancer. In the case of multiple phases of infection, a generalized saturated incidence rate model is more reasonable than a simply saturated incidence because it captures the complex dynamics of the different infection phases. In contrast, a simple saturated incidence rate model assumes a fixed shape for the incidence rate curve, which may not accurately reflect the dynamics of multiple infection phases. Considering HBV and its various phases, we constructed a model to present the dynamics and control strategies using the generalized saturated incidence. First, we proved that the model is well-posed. We then found the reproduction quantity and model equilibria to discuss the time dynamics of the model and investigate the conditions for stabilities. We also examined a control mechanism by introducing various controls to the model with the aim to increase the population of those recovered and minimize the infected people. We performed numerical experiments to check the biological significance and control implementation.

Approximating reproduction numbers: a general numerical method for age-structured models.

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments.

Tuberculosis has affected human beings for thousands of years, and until today, tuberculosis still ranks third among 29 infectious diseases in China. However, most of the existing mathematical models consider a single factor, which is not conducive to the study of tuberculosis transmission dynamics. Therefore, this study considers the combined effects of vaccination, treatment, and contaminated environments on tuberculosis, and builds a new model with seven compartments of $ SVEITRW $ based on China's tuberculosis data. The study shows that when the basic reproduction number $ R_{0} $ is less than 1, the disease will eventually disappear, but when $ R_{0} $ is greater than 1, the disease may persist. In the numerical analysis part, we use Markov-chain Monte-Carlo method to obtain the optimal parameters of the model. Through the next generation matrix theory, we calculate that the $ R_{0} $ value of tuberculosis in China is $ 2.1102 $, that is, if not controlled, tuberculosis in China will not disappear over time. At the same time, through partial rank correlation coefficients, we find the most sensitive parameter to the basic reproduction number $ R_{0} $. On this basis, we combine the actual prevalence of tuberculosis in China, apply Pontryagin's maximum principle, and perform cost-effectiveness analysis to obtain the conditions required for optimal control. The analysis shows that four control strategies could effectively reduce the prevalence of TB, and simultaneously controlling $ u_{2}, u_{3}, u_{4} $ is the most cost-effective control strategy.

An SIS epidemic model with individual variation.

We study an extension of the stochastic SIS (Susceptible-Infectious-Susceptible) model in continuous time that accounts for variation amongst individuals. By examining its limiting behaviour as the population size grows we are able to exhibit conditions for the infection to become endemic.

A novel within-host model of HIV and nutrition.

In this paper we develop a four compartment within-host model of nutrition and HIV. We show that the model has two equilibria: an infection-free equilibrium and infection equilibrium. The infection free equilibrium is locally asymptotically stable when the basic reproduction number $ \mathcal{R}_0 < 1 $, and unstable when $ \mathcal{R}_0 > 1 $. The infection equilibrium is locally asymptotically stable if $ \mathcal{R}_0 > 1 $ and an additional condition holds. We show that the within-host model of HIV and nutrition is structured to reveal its parameters from the observations of viral load, CD4 cell count and total protein data. We then estimate the model parameters for these 3 data sets. We have also studied the practical identifiability of the model parameters by performing Monte Carlo simulations, and found that the rate of clearance of the virus by immunoglobulins is practically unidentifiable, and that the rest of the model parameters are only weakly identifiable given the experimental data. Furthermore, we have studied how the data frequency impacts the practical identifiability of model parameters.

Threshold dynamics of a switching diffusion SIR model with logistic growth and healthcare resources.

In this article, we have constructed a stochastic SIR model with healthcare resources and logistic growth, aiming to explore the effect of random environment and healthcare resources on disease transmission dynamics. We have showed that under mild extra conditions, there exists a critical parameter, i.e., the basic reproduction number $ R_0/ $, which completely determines the dynamics of disease: when $ R_0/ < 1 $, the disease is eradicated; while when $ R_0/ > 1 $, the disease is persistent. To validate our theoretical findings, we conducted some numerical simulations using actual parameter values of COVID-19. Both our theoretical and simulation results indicated that (1) the white noise can significantly affect the dynamics of a disease, and importantly, it can shift the stability of the disease-free equilibrium; (2) infectious disease resurgence may be caused by random switching of the environment; and (3) it is vital to maintain adequate healthcare resources to control the spread of disease.

The impact of temperature, humidity and closing school on the mumps epidemic: a case study in the mainland of China.

BACKGROUND: To control resurging infectious diseases like mumps, it is necessary to resort to effective control and preventive measures. These measures include increasing vaccine coverage, providing the community with advice on how to reduce exposure, and closing schools. To justify such intervention, it is important to understand how well each of these measures helps to limit transmission. METHODS: In this paper, we propose a simple SEILR (susceptible-exposed-symptomatically infectious-asymptomatically infectious-recovered) model by using a novel transmission rate function to incorporate temperature, humidity, and closing school factors. This new transmission rate function allows us to verify the impact of each factor either separately or combined. Using reported mumps cases from 2004 to 2018 in the mainland of China, we perform data fitting and parameter estimation to evaluate the basic reproduction number  R 0 . As a wide range of one-dose measles, mumps, and rubella (MMR) vaccine programs in China started only in 2008, we use different vaccination proportions for the first Stage I period (from 2004 to 2008) and the second Stage II period (from 2009 to 2018). This allows us to verify the importance of higher vaccine coverage with a possible second dose of MMR vaccine. RESULTS: We find that the basic reproduction number  R 0  is generally between 1 and 3. We then use the Akaike Information Criteria to assess the extent to which each of the three factors contributed to the spread of mumps. The findings suggest that the impact of all three factors is substantial, with temperature having the most significant impact, followed by school opening and closing, and finally humidity. CONCLUSION: We conclude that the strategy of increasing vaccine coverage, changing micro-climate (temperature and humidity), and closing schools can greatly reduce mumps transmission.