Susceptible-infected-recovered-susceptible Research Articles

Proposition of a novel SIRS epidemic model with double epidemics and coexisting epidemics

Improving epidemic models to better reflect reality has long been a prominent concern for governments and researchers. This paper presents a novel Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model for human populations, offering a comprehensive analysis. The proposed model introduces a generalized SIRS epidemics framework encompassing three propagation scenarios. The paper establishes the positivity and boundedness of the system and demonstrates the stability of its equilibrium points. Furthermore, a controlled system is introduced, accompanied by three suggested control strategies to minimize the infected population while optimizing cost. To validate the analytical findings, a numerical example is provided. The paper concludes with a summary and outlines future research directions.

Forecasting influenza epidemics in China using transmission dynamic model with absolute humidity

BackgroundAn influenza forecasting system is critical to influenza epidemic preparedness. Low temperature has long been recognized as a condition favoring influenza epidemic, yet it fails to justify the summer influenza peak in tropics/subtropics. Recent studies have suggested that absolute humidity (AH) had a U-shape relationship with influenza survival and transmission across climate zones, indicating that a unified influenza forecasting system could be established for China with various climate conditions. MethodsOur study has generated weekly influenza forecasts by season and type/subtype in northern and southern China from 2011 to 2021, using a forecasting system combining an AH-driven susceptible-infected-recovered-susceptible (SIRS) model and the ensemble adjustment Kalman filter (EAKF). Model performance was assessed by sensitivity and specificity in predicting epidemics, and by accuracies in predicting peak timing and magnitude. ResultsOur forecast system can generally well predict seasonal influenza epidemics (mean sensitivity>87.5%; mean specificity >80%). The average forecast accuracies were 82% and 60% for peak timing and magnitude at 3–6 weeks ahead for northern China, higher than those of 42% and 20% for southern China. The accuracy was generally better when the forecast was made closer to the actual peak time. DiscussionThe established AH-driven forecasting system can generally well predict the occurrence of seasonal influenza epidemics in China.

FRACTAL-FRACTIONAL SIRS MODEL FOR THE DISEASE DYNAMICS IN BOTH PREY AND PREDATOR WITH SINGULAR AND NONSINGULAR KERNELS

In this study, we consider a mathematical model for the disease dynamics in both prey and predator by considering the Susceptible–Infected–Recovered–Susceptible (SIRS) model with the prey–predator Lotka–Volterra differential equations. Carrying capacity, predation, migration, and immunity loss are also taken into account for both species. Using the law of mass action, the physical model is transformed into a nonlinear coupled system of ODEs. The classical/integer order ODE system is then generalized through the fractal-fractional differential operators of power law and Mittag-Leffler kernels. For the model under consideration, we additionally check for positivity, boundedness, the basic reproduction number, and equilibrium points. Graphical results are obtained by use of a numerical approach, and the existence and uniqueness of the solution are also established theoretically. The graphical solutions has been displayed via 2D and 3D phase plots. It has been shown that when the fractional order reduces, the amplitude of the chaotic attractor dynamics shrinks, along with the range of limit cycles and periodic trajectories while a drop in the fractal dimension parameter causes an increase in the time period of the chaotic attractor dynamics. This study not only improves the understanding of epidemic breakouts in predator–prey systems, but also highlights the efficiency of fractal-fractional calculus in ecological modeling.

Mathematical Modeling for Dengue Fever Transmission Dynamics in Kebri Dehar District, Somali Regional State, Ethiopia

Dengue is a mosquito-borne viral infection and is transmitted to humans by the bite of an infected female Aedes mosquito of the species Aedes Aegypti and Aedes Albopictus. In this study, we developed and analysed a Susceptible-Infected-Recovered-Susceptible (SIRS) mathematical model which describes the transmission dynamics of dengue in Kebri Dehar district of Somali regional state. To achieve the objective of the study, we employed different mathematical tools and available cumulative cases data from Kebri Dehar district health office. The model was developed using a system of nonlinear ordinary differential equations and we performed analysis for both local and global stability of the equilibrium points of the model and the basic reproduction number for the year 2017 to 2021 is R_0 ≈ 0.0421071. In addition the parameters of the model are estimated by using the confirmed dengue cases of Kebri Dehar district for the year 2017 to 2021. Moreover, we studied numerical simulation and sensitivity analysis of the based on reproduction number that revealed the death rate of the mosquitoes and the host transmission rate are the most influential parameters. Our results suggest that the prevention of mosquitoes from humans (measure to decreasing transmission rate from mosquito to human) and any measure the increases the mortality of mosquitoes can significantly reduce the infection of dengue fever and may reduce further spread of infection in the community.

Selection of models and parameter estimation for monitoring the COVID-19 epidemic in Brazil via Bayesian inference

In 2019, a new strain of coronavirus led to an outbreak of disease cases named COVID-19, evolving rapidly into a pandemic. In Brazil, delayed decision making and lack of knowledge have resulted in an alarming increase in daily transmission and deaths. In this context, researchers used mathematical models to assist in determining the parameters that act in the spread of diseases, revealing containment measures. However, numerous mathematical models exist in the literature, each with specific parameters to be specified, leading to an important question about which model best represents the pandemic behavior. In this regard, this work aims to apply the Approximate Bayesian Computation method to select the best model and simultaneously estimate the parameters to resolve the abovementioned issue. The models adopted were susceptible-infected-recovered (SIR), susceptible-exposed-infected-recovered (SEIR), susceptible-infected-recovered-susceptible (SIRS), and susceptible-exposed-infected-recovered-susceptible (SEIRS). Approximate Bayesian Computation Monte Carlo Sequencing (ABC-SMC) was used to select among four competing models to represent the number of infected individuals and to estimate the model parameters based on three periods of Brazil COVID-19 data. A forecasting test was performed to test the ABC-SMC algorithm and the selected models for two months. The result was compared with the actual number of infected that were reported. Among the teste models, it was found that the ABC-SMC algorithm had a promising performance, since the data were noisy and the models could not predict all parameters.

The Rise and Fall of Omicron BA.1 Variant as Seen in Wastewater Supports Epidemiological Model Predictions.

The COVID-19 pandemic caused by the SARS-CoV-2 virus has inflicted significant mortality and morbidity worldwide. Continuous virus mutations have led to the emergence of new variants. The Omicron BA.1 sub-lineage prevailed as the dominant variant globally at the beginning of 2022 but was subsequently replaced by BA.2 in numerous countries. Wastewater-based epidemiology (WBE) offers an efficient tool for capturing viral shedding from infected individuals, enabling early detection of potential pandemic outbreaks without relying solely on community cooperation and clinical testing resources. This study integrated RT-qPCR assays for detecting general SARS-CoV-2 and its variants levels in wastewater into a modified triple susceptible-infected-recovered-susceptible (SIRS) model. The emergence of the Omicron BA.1 variant was observed, replacing the presence of its predecessor, the Delta variant. Comparative analysis between the wastewater data and the modified SIRS model effectively described the BA.1 and subsequent BA.2 waves, with the decline of the Delta variant aligning with its diminished presence below the detection threshold in wastewater. This study demonstrates the potential of WBE as a valuable tool for future pandemics. Furthermore, by analyzing the sensitivity of different variants to model parameters, we are able to deduce real-life values of cross-variant immunity probabilities, emphasizing the asymmetry in their strength.

Pattern formation and bifurcation analysis of delay induced fractional-order epidemic spreading on networks

The spontaneous emergence of ordered structures, known as Turing patterns, in complex networks is a phenomenon that holds potential applications across diverse scientific fields, including biology, chemistry, and physics. Here, we present a novel delayed fractional-order susceptible–infected–recovered–susceptible (SIRS) reaction–diffusion model functioning on a network, which is typically used to simulate disease transmission but can also model rumor propagation in social contexts. Our theoretical analysis establishes the Turing instability resulting from delay, and we support our conclusions through numerical experiments. We identify the unique impacts of delay, average network degree, and diffusion rate on pattern formation. The primary outcomes of our study are: (i) Delays cause system instability, mainly evidenced by periodic temporal fluctuations; (ii) The average network degree produces periodic oscillatory states in uneven spatial distributions; (iii) The combined influence of diffusion rate and delay results in irregular oscillations in both time and space. However, we also find that fractional-order can suppress the formation of spatiotemporal patterns. These findings are crucial for comprehending the impact of network structure on the dynamics of fractional-order systems.

A stochastic SIRS modeling of transport-related infection with three types of noises

People’s ability to easily travel across cities is thought to play a major role in the spread of infectious diseases. This research develops and analyzes a new Susceptible-Infected-Recovered-Susceptible (SIRS) model that accounts for transport-related infection, media coverage, and three types of noise (white, telegraph, and Lévy) to examine the role of transport in disease transmission. Thus, several theoretical and numerical uses of the revised model are investigated. Using Lyapunov functions, we check whether or not the model has a positive global solution. The infection either dies out or stays put beyond that point. The researched analytical results are validated by developing and using a numerical scheme to analyze and simulate the effects of various factors on the model’s dynamics.

Optimal Control of Information Diffusion in Temporal Networks

Effective control of information diffusion is quite a significant and universal task in information science and control engineering. The diffusion process and network evolution often share a common time scale. However, previous studies mainly focus on fixed topological structures, ignoring the temporal property of real networks. To fill this gap, we comprehensively investigate the optimal control of information diffusion in temporal networks. First, activity-driven networks (ADNs) are introduced to represent realistic temporal networks. Meanwhile, the node-based susceptible-infected-recovered-susceptible (SIRS) model is employed to describe the diffusion dynamics from a microcosmic perspective. Second, we propose two synergistic optimal control approaches to suppress negative diffusion and enhance positive diffusion, respectively. Then, the optimal control problems of minimizing cumulative system cost and maximizing total system revenue are formulated. Subsequently, the optimal control signals are obtained using the optimal control theory and Pontryagin’s minimum principle. Third, we exploit an implementation framework with a multi-layer hybrid feedback control scheme for large-scale temporal networks. This framework utilizes centralized and distributed techniques to implement different control strategies. Finally, experimental results on various real-world datasets validate the efficiency of the proposed synergistic optimal control approaches and implementation framework.

Intervention against information diffusion in static and temporal coupling networks

Information diffusion in complex networks has become quite an active research topic. As an important part of this field, intervention against information diffusion processes is attracting ever-increasing attention from network and control engineers. In particular, it is urgent to design intervention schemes for the coevolutionary dynamics between information diffusion processes and coupled networks. For this purpose, we comprehensively study the problem of information diffusion intervention over static and temporal coupling networks. First, individual interactions are described by a modified activity-driven network (ADN) model. Then, we establish a novel node-based susceptible–infected–recovered–susceptible (SIRS) model to characterize the information diffusion dynamics. On these bases, three synergetic intervention strategies are formulated. Second, we derive the critical threshold of the controlled-SIRS system via stability analysis. Accordingly, we exploit a spectral optimization scheme to minimize the outbreak risk or the required budget. Third, we develop an optimal control scheme of dynamically allocating resources to minimize both system loss and intervention expense, in which the optimal intervention inputs are obtained through optimal control theory and a forward–backward sweep algorithm. Finally, extensive simulation results validate the accuracy of theoretical derivation and the performance of our proposed intervention schemes.

Comparison of theoretical approaches for epidemic processes with waning immunity in complex networks.

The role of waning immunity in basic epidemic models on networks has been undervalued while being noticeably fundamental for real epidemic outbreaks. One central question is which mean-field approach is more accurate in describing the epidemic dynamics. We tackled this problem considering the susceptible-infected-recovered-susceptible (SIRS) epidemic model on networks. Two pairwise mean-field theories, one based on recurrent dynamical message-passing (rDMP) theory and the other on the pair quenched mean-field (PQMF) theory, are compared with extensive stochastic simulations on large networks of different levels of heterogeneity. For waning immunity times longer than or comparable with the recovering time, rDMP outperforms PQMF theory on power-law networks with degree distribution P(k)∼k^{-γ}. In particular, for γ>3, the epidemic threshold observed in simulations is finite, in qualitative agreement with rDMP, while PQMF leads to an asymptotically null threshold. The critical epidemic prevalence for γ>3 is localized in a finite set of vertices in the case of the PQMF theory. In contrast, the localization happens in a subextensive fraction of the network in rDMP theory. Simulations, however, indicate that localization patterns of the actual epidemic lay between the two mean-field theories, and improved theoretical approaches are necessary to understanding the SIRS dynamics.

Dynamical behavior of SIRS model incorporating government action and public response in presence of deterministic and fluctuating environments

In the present article, global characteristics of a generalized SIRS (susceptible–infected–recovered–susceptible) epidemic model have been investigated incorporating government policy, public response and social behavioral reaction. The effects of environmental fluctuations and time-dependent control strategies on the disease dynamics have also been analyzed. In the case of deterministic model, it is shown that the disease invades in this system when the basic reproduction number (R0) is greater than 1, whereas the dynamics of the stochastic model can be controlled by its associated basic reproduction number R̃s. Specifically, this work emphasizes the importance of nonlinear dynamic analysis of epidemic modeling, as well as the significant impact of social and government actions on disease dynamics. Numerical figure depicts that the governmental action plays a crucial role to control an epidemic situation, and the system turns out to be disease-free sooner if the government takes action at an early stage during a disease outbreak. Furthermore, one of the most key developments is that random fluctuations can prevent disease outbreaks, which can lead to the development of useful control techniques to restrict disease dynamics. The governmental actions and the clinical treatment are considered to be the effective control pair in this model, and it can be observed that the simultaneous implementation of the control strategies significantly reduces the disease burden.

Optimal Control of SIRS Model with Limited Medical Resources and Reinfection Problems

COVID-19 is a global public health problem that causes severe acute respiratory syndrome (SARS). It is also extremely contagious with rapidly increasing death rates. In this paper, we propose an optimal control model with SIRS (Susceptible–Infected–Recovered- Susceptible) kinetics to examine the effects of several intervention measures (e.g., vaccination and treatment) under the limited medical resources scenarios. This model is also employed to investigate the possibility of reinfection because of the fading of immunity problem. As a case study, the modeling framework is parametrised using COVID-19 daily confirmed and recovered cases in Malaysia. The parameters have been approximated by relying on the model's best fit to actual data published by the Malaysian Ministry of Health (MOH). Our numerical simulation results show that the inclusion of optimal control components with vaccination and treatment strategies would dramatically reduce the number of active cases even in the presence of reinfection forces. Regardless of the relative weightage (or costs) of vaccination and treatment, as well as the possibility of reinfection, it is critical to plan effective COVID-19 control measures by vaccinating as many people as possible (and as early as possible). Overall, these insights help explore the importance of intervention measures and the allocation of medical resources to control the severity of this pandemic.

Modelling control strategies against classical swine fever: Influence of traders and markets using static and temporal networks in Ecuador

Pig farming in Ecuador represents an important economic and cultural sector, challenged by classical swine fever (CSF). Recently, the National Veterinary Service (NVS), has dedicated its efforts to control the disease by implementing pig identification, mandatory vaccination against CSF and movement control. Our objective was to characterise pig premises according to risk criteria, to model the effect of movement restriction strategies and to consider the temporal evolution of the network. Social network analysis (SNA), SIRS (susceptible, infected, recovered, susceptible) network modelling and temporal analysis were used. The network contained 751,003 shipments and 6 million pigs from 2017 to 2019. Participating premises consisted of 144,118 backyard farms, 138 industrial farms, 21,337 traders and 51 markets. The 10 most influential markets, in the Andean highlands, received between 500 and 4600 pigs each week. The 10 most influential traders made about 3 shipments with 17 pigs per week. Simulations without control strategy resulted in an average CSF prevalence of 14.4 %; targeted movement restriction reduced the prevalence to 7.2 %, while with random movement restriction it was 13 %. Targeting the top 10 national traders and markets and one of the high-risk premises in every parish was one of the best strategies with the surveillance infrastructure available, highlighting its major influence and epidemiological importance in the network. When comparing the static network with its temporal counterpart, causal fidelity (c = 0.62) showed a 38 % overestimation in the number of transmission paths, also traversing the network required 4.39 steps, lasting approximately 233 days. In conclusion, NVS surveillance strategies could be more efficient by targeting the most at-risk premises, and in particular, taking into account the temporal information would make the risk assessment much more precise. This information could contribute to implement risk-based surveillance reducing the time to eradicate CSF and other infectious animal diseases.

Bifurcation Analysis of a Fractional-Order Simplicial SIRS System Induced by Double Delays

In this paper, a fractional-order susceptible-infected-recovered-susceptible (SIRS) model is studied, focusing on delay effects and high-order interactions. Two types of time delays are considered to describe latent period and healing cycle, respectively. From the ecological point of view, we found that the increasing delays caused by either the latent period or the healing cycle lead to the periodic outbreak of disease. The finding provided us with an important implication to preventing periodic outbreaks of disease by reducing the time delay, like accelerating the healing process with effective medication and medical intervention. Specifically, taking the time delays as bifurcation parameters, the stability of endemic equilibria and the existence of Hopf bifurcation are studied by analyzing the characteristic equation of the SIRS model. From a general point of view, based on the establishment of a fractional-order SIRS model, we found that the order of the fractional order is critical for describing the dynamic behavior of the model. Typically, the decrease of the order appears to bring about the disappearance of the periodic phenomenon (i.e. the periodic oscillation) of the originally stable system.

Stability of an SIRS Epidemic Model with Saturated Incidence Rate and Saturated Treatment Function

In this paper the global dynamics of susceptible-infected-recovered-susceptible (SIRS) epidemic model with saturated incidence rate and saturated treatment function is studied. Firstly, the basic reproduction number $R_0$ is calculated and the existence of the disease-free and positive equilibria is showed. In addition, local stability of the equilibria is investigated. Then, sufficient conditions are achieved for global stability of disease-free and endemic equilibria. Finally, the numerical examples are presented to validate the theoretical results.

Increased urbanization reduced the effectiveness of school closures on seasonal influenza epidemics in China

BackgroundSchool closure is a common mitigation strategy during severe influenza epidemics and pandemics. However, the effectiveness of this strategy remains controversial. In this study, we aimed to explore the effectiveness of school closure on seasonal influenza epidemics in provincial-level administrative divisions (PLADs) with varying urbanization rates in China.MethodsThis study analyzed influenza surveillance data between 2010 and 2019 provided by the Chinese National Influenza Center. Taking into consideration the climate, this study included a region with 3 adjacent PLADs in Northern China and another region with 4 adjacent PLADs in Southern China. The effect of school closure on influenza transmission was evaluated by the reduction of the effective reproductive number of seasonal influenza during school winter breaks compared with that before school winter breaks. An age-structured Susceptible-Infected-Recovered-Susceptible (SIRS) model was built to model influenza transmission in different levels of urbanization. Parameters were determined using the surveillance data via robust Bayesian method.ResultsBetween 2010 and 2019, in the less urbanized provinces: Hebei, Zhejiang, Jiangsu and Anhui, during school winter breaks, the effective reproductive number of seasonal influenza epidemics reduced 14.6% [95% confidential interval (CI): 6.2–22.9%], 9.6% (95% CI: 2.5–16.6%), 7.3% (95% CI: 0.1–14.4%) and 8.2% (95% CI: 1.1–15.3%) respectively. However, in the highly urbanized cities: Beijing, Tianjin and Shanghai, it reduced only 5.2% (95% CI: -0.7–11.2%), 4.1% (95% CI: -0.9–9.1%) and 3.9% (95% CI: -1.6–9.4%) respectively. In China, urbanization is associated with decreased proportion of children and increased social contact. According to the SIRS model, both factors could reduce the impact of school closure on seasonal influenza epidemics, and the proportion of children in the population is thought to be the dominant influencing factor.ConclusionsEffectiveness of school closure on the epidemics varies with the age structure in the population and social contact patterns. School closure should be recommended in the low urbanized regions in China in the influenza seasons.Graphical abstract

On the local and global stability of an sirs epidemic model with logistic growth and information intervention

In this study, we investigate an susceptible-infected-recovered-susceptible (SIRS) epidemic model with logistic growth and information intervention. Firstly, the basic reproduction number $R_0$ is defined and the main results are given in terms of local stability. Then, sufficient conditions for the global stability of endemic equilibrium are obtained. Finally, some numerical simulations are given to validate our theoretical conclusions.

Transmissibility of epidemic diseases caused by delay with local proportional fractional derivative

Epidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of change and constant solutions of each compartment are studied with incorporation of a special case of proportional fractional derivative (PFD). In addition, existence and uniqueness of the system are also comprehensively elaborated. Moreover, novel dynamic assessment of the system is carried out in context with the fractional order index. Succinctly, the manuscript accomplishes cyclic epidemiological behavior of the infectious disease due to the delay in treatment of the infected individuals.

Endemic information-contagion outbreaks in complex networks with potential spreaders based recurrent-state transmission dynamics

Epidemic models have long been used to study information diffusion in complex artificial, biological and social networks. Here, we propose a novel recurrent-state epidemic model for information diffusion in networked systems by introducing a potential-spreaders compartment into the susceptible–infected–recovered–susceptible (SIRS) model. Our model assumes that not all susceptible nodes are equally likely to become efficient information spreaders, and that information can be repeatedly disseminated in cycles, even after its temporary decay. We observed that the introduced potential spreaders compartment in our model enables a more convenient state-transition process and an accurate description of information diffusion based on discrete time. Specifically, we found that our analytic results are in good agreement with numerical simulations on both artificial systems and eight different types of real-world biological, medical, and social networks. Unlike susceptible or recovered nodes, we further observed that potential spreader nodes in our model can serve as a relatively good predictor of the peaks of contagion outbreaks. Monitoring the number of potential spreaders could thus be beneficial for predicting both infection transmission and information propagation in complex networks and for characterizing their temporal dynamical patterns. Our model could be applied to a wide variety of spreading phenomena with recurrent-state endemic dynamics, such as seasonal influenza, re-tweeting behavior, or re-emergence of rumors and fashion trends in online social networks.