Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy

Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin incidence and Holling type II treatment rates

This article presents a time-delayed SIR epidemiological model that has been quantitatively examined. The model incorporates a logistic growth function for the susceptible population, a Crowley-Martin type incidence, and Holling type II treatment rates. We investigated two separate time delays. The first delay refers to the rate at which new infections occur, allowing us to evaluate the impact of the latent period. The second delay relates to the rate of treatment for those who have contracted the infection, which allows us to examine the consequences of postponed access to therapy. The investigation of the steady-state behavior of the model emphasizes two equilibria, namely, the infection-free equilibrium and the endemic equilibrium. The determination of critical values involves the use of the fundamental reproduction number, denotedR 0 , which serves as a predictive measure to determine the potential elimination of a disease within a specific population. Using the fundamental reproduction number, it can be shown that infection-free equilibrium exhibits local asymptotic stability when the value ofR 0 is less than 1. In contrast, whenR 0 exceeds 1, the infection-free equilibrium becomes unstable in the context of the time-delayed system. Furthermore, an analysis of the steady-state dynamics of the endemic equilibrium indicates the appearance of oscillations and periodic solutions with the Hopf bifurcation for all feasible combinations of two-time delays as the bifurcated parameter. In sensitivity analysis, a sensitivity index is utilized to evaluate the relative modification in the fundamental reproduction number caused by each parameter. In summary, numerical simulations are employed to offer empirical evidence for the theoretical findings.

Analyzing the dynamics of COVID-19 transmission in select regions of the Philippines: A modeling approach to assess the impact of various tiers of community quarantines

The COVID-19 pandemic has significantly impacted communities worldwide, and effective management strategies are critical to reduce transmission rates and minimize the impact of the disease. In this study, we modeled and analyzed the COVID-19 transmission dynamics and derived relevant epidemiological values for three regions of the Philippines, namely, the National Capital Region (NCR), Davao City, and Baguio City, under different community quarantine implementations. The unique features and differences of these regions-of-interest were accounted for in simulating the disease spread and in estimating key epidemiological parameters fitted to the reported COVID-19 cases. Results support the robustness of the model formulated and provides insights into the effect of the government's implemented intervention protocols. With a forecasting feature, this modeling framework is beneficial for science-based decision support, policy making, and assessment for recent and future pandemics wherever regions-of-interest.

Beyond six feet: The collective behavior of social distancing

In a severe epidemic such as the COVID-19 pandemic, social distancing can be a vital tool to stop the spread of the disease and save lives. However, social distancing may induce profound negative social or economic impacts as well. How to optimize social distancing is a serious social, political, as well as public health issue yet to be resolved. This work investigates social distancing with a focus on how every individual reacts to an epidemic, what role he/she plays in social distancing, and how every individual's decision contributes to the action of the population and vice versa. Social distancing is thus modeled as a population game, where every individual makes decision on how to participate in a set of social activities, some with higher frequencies while others lower or completely avoided, to minimize his/her social contacts with least possible social or economic costs. An optimal distancing strategy is then obtained when the game reaches an equilibrium. The game is simulated with various realistic restraints including (i) when the population is distributed over a social network, and the decision of each individual is made through the interactions with his/her social neighbors; (ii) when the individuals in different social groups such as children vs. adults or the vaccinated vs. unprotected have different distancing preferences; (iii) when leadership plays a role in decision making, with a certain number of leaders making decisions while the rest of the population just follow. The simulation results show how the distancing game is played out in each of these scenarios, reveal the conflicting yet cooperative nature of social distancing, and shed lights on a self-organizing, bottom-up perspective of distancing practices.

The public health impact of COVID-19 variants of concern on the effectiveness of contact tracing in Vermont, United States

Case investigation and contact tracing (CICT) are public health measures that aim to break the chain of pathogen transmission. Changes in viral characteristics of COVID-19 variants have likely affected the effectiveness of CICT programs. We estimated and compared the cases averted in Vermont when the original COVID-19 strain circulated (Nov. 25, 2020-Jan. 19, 2021) with two periods when the Delta strain dominated (Aug. 1-Sept. 25, 2021, and Sept. 26-Nov. 20, 2021). When the original strain circulated, we estimated that CICT prevented 7180 cases (55% reduction in disease burden), compared to 1437 (15% reduction) and 9970 cases (40% reduction) when the Delta strain circulated. Despite the Delta variant being more infectious and having a shorter latency period, CICT remained an effective tool to slow spread of COVID-19; while these viral characteristics did diminish CICT effectiveness, non-viral characteristics had a much greater impact on CICT effectiveness.

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, R 0 , 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 whenR 0 < 1 , 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 ν 1 leads to oscillations. When ν 1 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.

Dynamically adjustable SVEIR(MH) model of multiwave epidemics: Estimating the effects of public health measures against COVID-19

The COVID-19 pandemic was characterized by multiple subsequent, overlapping outbreaks, as well as extremely rapid changes in viral genomes. The information about local epidemics spread and the epidemic control measures was shared on a daily basis (number of cases and deaths) via centralized repositories. The vaccines were developed within the first year of the pandemic. New modes of monitoring and sharing of epidemic data were implemented using Internet resources. We modified the basic SEIR compartmental model to include public health measures, multiwave scenarios, and the variation of viral infectivity and transmissibility reflected by the basic reproduction number R0 of emerging viral variants. SVEIR(MH) model considers the capacity of the medical system, lockdowns, vaccination, and changes in viral reproduction rate on the epidemic spread. The developed model uses daily infection reports for assessing the epidemic dynamics, and daily changes of mobility data from mobile phone networks to assess the lockdown effectiveness. This model was deployed to six European regions Baden-Württemberg (Germany), Belgium, Czechia, Lombardy (Italy), Sweden, and Switzerland for the first 2 years of the pandemic. The correlation coefficients between observed and reported infection data showed good concordance for both years of the pandemic (ρ = 0.84-0.94 for the raw data and ρ = 0.91-0.98 for smoothed 7-day averages). The results show stability across the regions and the different epidemic waves. Optimal control of epidemic waves can be achieved by dynamically adjusting epidemic control measures in real-time. SVEIR(MH) model can simulate different scenarios and inform adjustments to the public health policies to achieve the target outcomes. Because this model is highly representative of actual epidemic situations, it can be used to assess both the public health and socioeconomic effects of the public health measures within the first 7 days of the outbreak.

SIRC epidemic model with cross-immunity and multiple time delays

Multi-strain diseases lead to the development of some degree of cross-immunity among people. In the present paper, we propose a multi-delayed SIRC epidemic model with incubation and immunity time delays. Here we aim to examine and investigate the effects of incubation delay [Formula: see text] and the impact of vaccine which provides partial/cross-immunity with immunity delay parameter ([Formula: see text]) on the disease dynamics. Also, we study the impact of the strength of cross-immunity [Formula: see text] on the disease prevalence. The positivity and boundedness of the solutions of the epidemic model have been established. Two different types of equilibrium points (disease-free and endemic) have been deduced. Expression for basic reproduction number has been derived. The stability conditions and Hopf-bifurcation about both the equilibrium points in the absence and presence of both delays have been discussed. The Lyapunov stability conditions about the endemic equilibrium point have been established. Numerical simulations have been performed to support our analytical results. We quantitatively demonstrate how oscillations and Hopf-bifurcation allow time delays to alter the dynamics of the system. The combined impacts of both the delays on disease prevalence has been studied. Through parameter sensitivity analysis, we observe that the infected population decreases with an increase in vaccination rate and the system starts to stabilize early with the increase in cross-immunity rate. Global sensitivity analysis for the basic reproduction number has been performed using Latin hypercube sampling and partial rank correlation coefficients techniques. The combined effect of vaccination rate with transmission rate and vaccination rate with re-infection probability (i.e. strength of cross-immunity) on [Formula: see text] have been discussed. Our research underlines the need to take cross-immunity and time delays into account in the epidemic model in order to better understand disease dynamics.

Assessing potential insights of an imperfect testing strategy: Parameter estimation and practical identifiability using early COVID-19 data in India

A deterministic model with testing of infected individuals has been proposed to investigate the potential consequences of the impact of testing strategy. The model exhibits global dynamics concerning the disease-free and a unique endemic equilibrium depending on the basic reproduction number when the recruitment of infected individuals is zero; otherwise, the model does not have a disease-free equilibrium, and disease never dies out in the community. Model parameters have been estimated using the maximum likelihood method with respect to the data of early COVID-19 outbreak in India. The practical identifiability analysis shows that the model parameters are estimated uniquely. The consequences of the testing rate for the weekly new cases of early COVID-19 data in India tell that if the testing rate is increased by 20% and 30% from its baseline value, the weekly new cases at the peak are decreased by 37.63% and 52.90%; and it also delayed the peak time by four and fourteen weeks, respectively. Similar findings are obtained for the testing efficacy that if it is increased by 12.67% from its baseline value, the weekly new cases at the peak are decreased by 59.05% and delayed the peak by 15 weeks. Therefore, a higher testing rate and efficacy reduce the disease burden by tumbling the new cases, representing a real scenario. It is also obtained that the testing rate and efficacy reduce the epidemic's severity by increasing the final size of the susceptible population. The testing rate is found more significant if testing efficacy is high. Global sensitivity analysis using partial rank correlation coefficients (PRCCs) and Latin hypercube sampling (LHS) determine the key parameters that must be targeted to worsen/contain the epidemic.

Multi-regional COVID-19 epidemic forecast in Sweden

The novel coronavirus disease 2019 (COVID-19) is a contagious disease with high transmissibility to spread worldwide, reported to present a certain burden on worldwide public health. This study aimed to determine epidemic occurrence probability at any reasonable time horizon in any region of interest by applying modern novel statistical methods directly to raw clinical data. This paper describes a novel bio-system reliability approach, particularly suitable for multi-regional health and stationary environmental systems, observed over a sufficient period of time, resulting in a reliable long-term forecast of the highly pathogenic virus outbreak probability. For this study, COVID-19 daily recorded patient numbers in most affected Sweden regions were chosen. This work aims to benchmark state-of-the-art methods, making it possible to extract necessary information from dynamically observed patient numbers while considering relevant territorial mapping. The method proposed in this paper opens up the possibility of accurately predicting epidemic outbreak probability for multi-regional biological systems. Based on their clinical survey data, the suggested methodology can be used in various public health applications. Key findings are: A novel spatiotemporal health system reliability method has been developed and applied to COVID-19 epidemic data.Accurate multi-regional epidemic occurrence prediction is made.Epidemic threshold confidence bands given.

Optimal control strategies to combat COVID-19 transmission: A mathematical model with incubation time delay

The coronavirus disease 2019, started spreading around December 2019, still persists in the population all across the globe. Though different countries have been able to cope with the disease to some extent and vaccination for the same has been developed, it cannot be ignored that the disease is still not on the verge of completely eradicating, which in turn creates a need for having deeper insights of the disease in order to understand it well and hence be able to work towards its eradication. Meanwhile, using mitigation strategies like non-pharmaceutical interventions can help in controlling the disease. In this work, our aim is to study the dynamics of COVID-19 using compartmental approach by applying various analytical methods. We obtain formula for important tools like R0 and establish the stability of disease-free equilibrium point for R0<1. Further, based on R0, we discuss the stability and existence of the endemic equilibrium point. We incorporate various control strategies possible and using optimal control theory, study their expected positive impacts on the spread of the disease. Later, using a biologically feasible set of parameters, we numerically analyse the model. We even study the trend of the outbreak in China, for over 120 days, where the active cases rise up to a peak and then the curve flattens.

Assessing the Dynamic Outcomes of Containment Strategies against COVID-19 under Different Public Health Governance Structures: A Comparison between Pakistan and Bangladesh

COVID-19 scenarios were run using an epidemiological mathematical model (system dynamics model) and counterfactual analysis to simulate the impacts of different control and containment measures on cumulative infections and deaths in Bangladesh and Pakistan. The simulations were based on national-level data concerning vaccination level, hospital capacity, and other factors, from the World Health Organization, the World Bank, and the Our World in Data web portal. These data were added to cumulative infections and death data from government agencies covering the period from 18 March 2020 to 28 February 2022. Baseline curves for Pakistan and Bangladesh were obtained using piecewise fitting with a consideration of different events against the reported data and allowing for less than 5% random errors in cumulative infections and deaths. The results indicate that Bangladesh could have achieved more reductions in each key outcome measure by shifting its initial lockdown at least five days backward, while Pakistan would have needed to extend its lockdown to achieve comparable improvements. Bangladesh's second lockdown appears to have been better timed than Pakistan's. There were potential benefits from starting the third lockdown two weeks earlier for Bangladesh and from combining this with the fourth lockdown or canceling the fourth lockdown altogether. Adding a two-week lockdown at the beginning of the upward slope of the second wave could have led to a more than 40 percent reduction in cumulative infections and a 35 percent reduction in cumulative deaths for both countries. However, Bangladesh's reductions were more sensitive to the duration of the lockdown. Pakistan's response was more constrained by medical resources, while Bangladesh's outcomes were more sensitive to both vaccination timing and capacities. More benefits were lost when combining multiple scenarios for Bangladesh compared to the same combinations in Pakistan. Clearly, cumulative infections and deaths could have been highly impacted by adjusting the control and containment measures in both national settings. However, COVID-19 outcomes were more sensitive to adjustment interventions for the Bangladesh context. Disaggregated analyses, using a wider range of factors, may reveal several sub-national dynamics. Nonetheless, the current research demonstrates the relevance of lockdown timing adjustments and discrete adjustments to several other control and containment measures.

Assessing the dynamic impacts of non-pharmaceutical and pharmaceutical intervention measures on the containment results against COVID-19 in Ethiopia

The rapid spread of COVID-19 in Ethiopia was attributed to joint effects of multiple factors such as low adherence to face mask-wearing, failure to comply with social distancing measures, many people attending religious worship activities and holiday events, extensive protests, country election rallies during the pandemic, and the war between the federal government and Tigray Region. This study built a system dynamics model to capture COVID-19 characteristics, major social events, stringencies of containment measures, and vaccination dynamics. This system dynamics model served as a framework for understanding the issues and gaps in the containment measures against COVID-19 in the past period (16 scenarios) and the spread dynamics of the infectious disease over the next year under a combination of different interventions (264 scenarios). In the counterfactual analysis, we found that keeping high mask-wearing adherence since the outbreak of COVID-19 in Ethiopia could have significantly reduced the infection under the condition of low vaccination level or unavailability of the vaccine supply. Reducing or canceling major social events could achieve a better outcome than imposing constraints on people's routine life activities. The trend analysis found that increasing mask-wearing adherence and enforcing more stringent social distancing were two major measures that can significantly reduce possible infections. Higher mask-wearing adherence had more significant impacts than enforcing social distancing measures in our settings. As the vaccination rate increases, reduced efficacy could cause more infections than shortened immunological periods. Offsetting effects of multiple interventions (strengthening one or more interventions while loosening others) could be applied when the levels or stringencies of one or more interventions need to be adjusted for catering to particular needs (e.g., less stringent social distancing measures to reboot the economy or cushion insufficient resources in some areas).

Modeling the impacts of awareness and limited medical resources on the epidemic size of a multi-group SIR epidemic model

The pharmaceutical interventions of emerging infectious diseases are constrained by the available medical resources such as drugs, vaccines, hospital beds, isolation places and the efficiency of the treatment. The awareness of the population also plays an important role in reducing contacts and consequently, reducing the disease transmission rate. In this paper, we propose a multi-group Susceptible, Infected and Recovered (SIR) epidemic model incorporating the awareness of population and the saturated treatment function that describes the effects of the availability of medical resources for treatment. We assume that the treatment of the infected individuals of a group is affected by the medical resources for the treatment of each group. We calculate the basic reproduction number [Formula: see text] in the term of the awareness parameter using the next generation approach. We determine the local and global stabilities of equilibrium (disease free equilibrium and endemic equilibrium) in terms of [Formula: see text] and the availability of medical resources for treatment. We obtain that backward bifurcation occurs at [Formula: see text] along with the existence of multiple endemic equilibria when [Formula: see text] Further, we consider the special case with a single group epidemic system and ensure the existence of multiple endemic equilibria. We showed a necessary condition on the parameter related to the availability of medical resources when backward bifurcation occurs. This situation indicates that reducing the basic reproduction number below unity is not sufficient to remove the disease when the medical resources for treatment are scarce. We used numerical simulations to support and counterpart our theoretical results and discussed the impacts of the awareness of susceptible population and availability of medical resources for treatment in each group, on the epidemic size of each group. Our findings suggest that in the case of limited medical resources, the high treatment rate and awareness of the population are very helpful to control the disease (to reduce the prevalence of infection) and the eradication of disease also depends on initial population sizes. More importantly, it is also obtained that sufficient medical resources for every group are required to eradicate the disease from an entire population.

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and L e ´ vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold λ by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.