Journal articles: 'SIR infection model'

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Lefèvre, Claude, and Matthieu Simon. "SIR epidemics with stages of infection." Advances in Applied Probability 48, no. 3 (September 2016): 768–91. http://dx.doi.org/10.1017/apr.2016.27.

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AbstractIn this paper we are concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population when an infective can go through several stages of infection before being removed. The transitions between stages are governed by either a Markov process or a semi-Markov process. An infective of any stage makes contacts amongst the population at the points of a Poisson process. Our main purpose is to derive the distribution of the final epidemic size and severity, as well as an approximation by branching, using simple matrix analytic methods. Some illustrations are given, including a model with treatment discussed by Gani (2006).

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WU, JIANJUN, ZIYOU GAO, and HUIJUN SUN. "SIMULATION OF TRAFFIC CONGESTION WITH SIR MODEL." Modern Physics Letters B 18, no. 30 (December 30, 2004): 1537–42. http://dx.doi.org/10.1142/s0217984904008031.

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The spread of traffic congestion is related to the rate of infection and the recovery rate. In this paper, we describe the traffic congestion spread with SIR model of a complex network. From the point of the complex network, the spread of the traffic congestion with different parameters are simulated. By simulation, we find that the behavior of the traffic system is tightly related to the average rate of infection, the average recovery rate and the topological properties of traffic network.

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Acemoglu, Daron, Victor Chernozhukov, Iván Werning, and Michael D. Whinston. "Optimal Targeted Lockdowns in a Multigroup SIR Model." American Economic Review: Insights 3, no. 4 (December 1, 2021): 487–502. http://dx.doi.org/10.1257/aeri.20200590.

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We study targeted lockdowns in a multigroup SIR model where infection, hospitalization, and fatality rates vary between groups—in particular between the “young,” the “middle-aged,” and the “old.” Our model enables a tractable quantitative analysis of optimal policy. For baseline parameter values for the COVID-19 pandemic applied to the US, we find that optimal policies differentially targeting risk/age groups significantly outperform optimal uniform policies and most of the gains can be realized by having stricter protective measures such as lockdowns on the more vulnerable, old group. Intuitively, a strict and long lockdown for the old both reduces infections and enables less strict lockdowns for the lower-risk groups. (JEL H51, I12, I18, J13, J14)

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Obolonkin, Vladimir, and Anatoly Zherelo. "Stochastic Generalization of the Epidemiological SIR Model." Nonlinear Phenomena in Complex Systems 24, no. 4 (December 10, 2021): 409–14. http://dx.doi.org/10.33581/1561-4085-2021-24-4-409-414.

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In this paper we propose stochastic modification of well-known in epidemiology SIR model. This modification allows us to simulate various scenarios of infection and can be used for the risk management.

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Dubey, Balram, Preeti Dubey, and Uma S. Dubey. "Role of media and treatment on an SIR model." Nonlinear Analysis: Modelling and Control 21, no. 2 (March 25, 2016): 185–200. http://dx.doi.org/10.15388/na.2016.2.3.

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n this paper, the impact of awareness programs as well as treatment on an SIR model has been investigated. We assume that the whole population is divided into four compartments, named as susceptible (S), infected (I), aware susceptible (Sa) and recovered (R). Analytical findings and numerical simulations of the model show that if the exposure to the awareness program is high and adequate treatment is available, then the infection can be eliminated. Analysis of the model also depicts that if treatment is not available, then infection is high even if enough awareness is present. But in absence of awareness an infection can not be eliminated inspite of adequate treatment. Effective treatment can led to a diminished level of infection. Stability analysis of the model is investigated by using stability theory of differential equations. Further, numerical simulations are carried out to validate the analytical results.

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JIN, ZHEN, MAINUL HAQUE, and QUANXING LIU. "PULSE VACCINATION IN THE PERIODIC INFECTION RATE SIR EPIDEMIC MODEL." International Journal of Biomathematics 01, no. 04 (December 2008): 409–32. http://dx.doi.org/10.1142/s1793524508000370.

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In this paper a pulse vaccination SIR model with periodic infection rate β(t) is studied. The basic reproductive number R0 is defined. The dynamical behavior of the model is analyzed. It is proved that the infection-free periodic solution is globally stable if R0 < 1. The infection-free periodic solution is unstable and the disease will uniform persistence when R0 > 1. We use standard bifurcation theory to show the existence of the positive periodic solution when R0 → 1+. Numerical simulation can give suggestion, the system has a unique positive periodic, and it is globally stable when R0 > 1.

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Liu, Ting, Yanling Bai, Mingmei Du, Yueming Gao, and Yunxi Liu. "Susceptible-Infected-Removed Mathematical Model under Deep Learning in Hospital Infection Control of Novel Coronavirus Pneumonia." Journal of Healthcare Engineering 2021 (October 27, 2021): 1–11. http://dx.doi.org/10.1155/2021/1535046.

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Objective. This research aimed to explore the application of a mathematical model based on deep learning in hospital infection control of novel coronavirus (COVID-19) pneumonia. Methods. First, the epidemic data of Beijing, China, were utilized to make a definite susceptible-infected-removed (SIR) model fitting to determine the estimated value of the COVID-19 removal intensity β, which was then used to do a determined SIR model and a stochastic SIR model fitting for the hospital. In addition, the reasonable β and γ estimates of the hospital were determined, and the spread of the epidemic in hospital was simulated, to discuss the impact of basal reproductive number changes, isolation, vaccination, and so forth on COVID-19. Results. There was a certain gap between the fitting of SIR to the remover and the actual data. The fitting of the number of infections was accurate. The growth rate of the number of infections decreased after measures, such as isolation, were taken. The effect of herd immunity was achieved after the overall immunity reached 70.9%. Conclusion. The SIR model based on deep learning and the stochastic SIR fitting model were accurate in judging the development trend of the epidemic, which can provide basis and reference for hospital epidemic infection control.

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Jing, Wenjun, Zhen Jin, and Juping Zhang. "An SIR pairwise epidemic model with infection age and demography." Journal of Biological Dynamics 12, no. 1 (January 1, 2018): 486–508. http://dx.doi.org/10.1080/17513758.2018.1475018.

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Ichinose, Toshiaki, Danhe Tian, and Yifeng Li. "Verification of Infection Prevention Control Using a Spatial Random Walk Model." International Journal of Social Science Studies 8, no. 6 (September 29, 2020): 35. http://dx.doi.org/10.11114/ijsss.v8i6.4955.

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To stop pandemic of the 2019 novel coronavirus (COVID-19), "an 80 percent reduction of person to person contact opportunities" was proposed by the Japanese government. This guideline was based on the result of macroscopic differential equation model akin to the SIR (Susceptible-Infected-Recovered) model. For the purpose of indicating person to person’s infection mechanism intuitively, we built a new model to calculate infections between two persons who are in contact each other. This model adopted a spatial random walk model to express random movement of people in a specific 2-D geographical space. This model was applied to verify the effect of the proposed infection control procedure, "80 percent reduction". The result of the numerical simulation supported a proposed infection control procedure of "an 80 percent reduction" derived by the SIR model.

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Sagar, Surendra Kumar. "SIR-SI Mathematical Model for Zika Virus Progression Dynamics in India: A Case Study." Journal of Communicable Diseases 53, no. 02 (June 30, 2021): 100–104. http://dx.doi.org/10.24321/0019.5138.202132.

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Viral diseases are very hazardous for humanity because in the case of most viral diseases, drugs are not effective. At present, the whole world is living with the fear of COVID-19. From time to time, several viral diseases have been troubling human life. In this article, we have tried to capture the progression dynamics of Zika Virus (ZIKV) infection in the Indian scenario. A constructed model is based on compartment modelling. In the model, Susceptible-Infected-Recovered (SIR) structure is used for the human population and Susceptible-Infected (SI) structure is used for mosquito population. The value of the basic reproduction number (R0) is computed 0.36 at baseline values of parameters involved in the model. The lower value of R0 suggests that infection was unable to spread in the human population. Sensitive analysis for R0 revealed that the most accountable parameter in the spread of infection was mosquito biting rate. The modelling technique might be useful for other diseases also.

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Boston, Kelley M., Misti Ellsworth, Jocelyn Thomas, Tawanna A. McInnis-Cole, and Luis Ostrosky-Zeichner. "95. Impact of Penetrating Trauma on Surgical Site Infection Standardized Infection Ratio (SIR) for Colon Procedures." Open Forum Infectious Diseases 8, Supplement_1 (November 1, 2021): S60—S61. http://dx.doi.org/10.1093/ofid/ofab466.095.

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Abstract Background Colon surgery (COLO) is one of the focus areas for the the Centers for Medicare and Medicaid Services (CMS) Hospital Inpatient Quality Reporting (IQR) Program. Standardized criteria from the National Healthcare Surveillance Network (NSHN) are used to define surgical site infections (SSI) and to assess and weight standardized risk variables, so that all organizations can be judged to the same standard. Performance is compared though use of a standardized infection ratio (SIR), which is the observed number of infections, divided by the “predicted” number of infections, given the number and type of surgeries performed. Methods A retrospective review of medical records and NHSN documentation was conducted for 778 COLO procedures that were performed at a large academic and level 1 trauma center between January 2019 and December 2020. Initial review of the data showed that the increases in SIR were primarily concentrated in trauma patients with intestinal injury and fecal spillage. SIR for adult procedures were calculated using the NHSN Complex 30-Day SSI Data for IQR Report model, which the metric used by the CMS IQR. The CDC NHSN Statistics Calculator was used to compare SIR for procedures coded as trauma and non-trauma. As a proxy for patients with penetrating trauma, SIR for patients coded as trauma who had a surgical wound class noted as dirty was compared to SIR for patients coded as trauma with surgical wound class coded as contaminated or clean-contaminated. Results For the CMS model, there was a statistically significant difference (p = 0.0003) between SIR for trauma (SIR = 3.451) and non-trauma (SIR = 1.071) procedures. There was also a statistically significant difference (p=0.0014) between trauma procedures with dirty surgical wound class (SIR = 6.608), compared to those with wounds categorized as contaminated or clean-contaminated (SIR = 2.235). NHSN Adult Complex 30 Days SIR comparison for COLO SSI with and without trauma NHSN Adult Complex 30 Days SIR comparison for trauma COLO procedures with dirty wound class description, against COLO procedures with wound class described as clean or clean-contaminated Conclusion Risk factors currently included in the model for COLO SSI may not adequately account for the increased risk from penetrating trauma with fecal spillage. Trauma and wound class should be added to the CMS IQR risk model for SIR. Disclosures Kelley M. Boston, MPH, CIC, CPHQ, FAPIC, Infection Prevention & Management Associates (Employee, Shareholder) Luis Ostrosky-Zeichner, MD, Amplyx (Consultant)Cidara (Consultant)F2G (Consultant)Gilead (Grant/Research Support, Speaker's Bureau)Pfizer (Scientific Research Study Investigator, Speaker's Bureau)Scynexis (Grant/Research Support, Scientific Research Study Investigator)Viracor (Consultant)

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Liu, Helong, Houbao Xu, Jingyuan Yu, and Guangtian Zhu. "Stability on coupling SIR epidemic model with vaccination." Journal of Applied Mathematics 2005, no. 4 (2005): 301–19. http://dx.doi.org/10.1155/jam.2005.301.

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We develop a mathematical model for the disease which can be transmitted via vector and through blood transfusion in host population. The host population is structured by the chronological age. We assume that the instantaneous death and infection rates depend on the age. Applying semigroup theory and so forth, we investigate the existence of equilibria. We also discuss local stability of steady states.

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AlQadi, Hadeel, and Majid Bani-Yaghoub. "Incorporating global dynamics to improve the accuracy of disease models: Example of a COVID-19 SIR model." PLOS ONE 17, no. 4 (April 8, 2022): e0265815. http://dx.doi.org/10.1371/journal.pone.0265815.

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Mathematical models of infectious diseases exhibit robust dynamics, such as stable endemic, disease-free equilibriums or convergence of the solutions to periodic epidemic waves. The present work shows that the accuracy of such dynamics can be significantly improved by including global effects of host movements in disease models. To demonstrate improved accuracy, we extended a standard Susceptible-Infected-Recovered (SIR) model by incorporating the global dynamics of the COVID-19 pandemic. The extended SIR model assumes three possibilities for susceptible individuals traveling outside of their community: • They can return to the community without any exposure to the infection. • They can be exposed and develop symptoms after returning to the community. • They can be tested positively during the trip and remain quarantined until fully recovered. To examine the predictive accuracy of the extended SIR model, we studied the prevalence of the COVID-19 infection in six randomly selected cities and states in the United States: Kansas City, Saint Louis, San Francisco, Missouri, Illinois, and Arizona. The extended SIR model was parameterized using a two-step model-fitting algorithm. The extended SIR model significantly outperformed the standard SIR model and revealed oscillatory behaviors with an increasing trend of infected individuals. In conclusion, the analytics and predictive accuracy of disease models can be significantly improved by incorporating the global dynamics of the infection.

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Okabe, Yutaka, and Akira Shudo. "Microscopic Numerical Simulations of Epidemic Models on Networks." Mathematics 9, no. 9 (April 22, 2021): 932. http://dx.doi.org/10.3390/math9090932.

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Mathematical models of the spread of epidemic diseases are studied, paying special attention to networks. We treat the Susceptible-Infected-Recovered (SIR) model and the Susceptible-Exposed-Infectious-Recovered (SEIR) model described by differential equations. We perform microscopic numerical simulations for corresponding epidemic models on networks. Comparing a random network and a scale-free network for the spread of the infection, we emphasize the role of hubs in a scale-free network. We also present a simple derivation of the exact solution of the SIR model.

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Postnikov, Eugene B., and Igor M. Sokolov. "Continuum description of a contact infection spread in a SIR model." Mathematical Biosciences 208, no. 1 (July 2007): 205–15. http://dx.doi.org/10.1016/j.mbs.2006.10.004.

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ANGULO, OSCAR, FABIO MILNER, and LAURENTIU SEGA. "A SIR EPIDEMIC MODEL STRUCTURED BY IMMUNOLOGICAL VARIABLES." Journal of Biological Systems 21, no. 04 (December 2013): 1340013. http://dx.doi.org/10.1142/s0218339013400135.

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Standard mathematical models for analyzing the spread of a disease are usually either epidemiological or immunological. The former are mostly ordinary differential equation (ODE)-based models that use classes like susceptibles, recovered, infectives, latently infected, and others to describe the evolution of an epidemic in a population. Some of them also use structure variables, such as size or age. The latter describe the evolution of the immune system/pathogen in the infected host — evolution that usually results in death, recovery or chronic infection. There is valuable insight to be gained from combining these two types of models, as that may lead to a better understanding of the severity of an epidemic. In this article, we propose a new type of model that combines the two by using variables of immunological nature as structure variables for epidemiological models. We prove the well-posedness of the proposed model under some restrictions and conclude with a look at a practical application of the model.

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Bayraktar, Erhan, Asaf Cohen, and April Nellis. "A Macroeconomic SIR Model for COVID-19." Mathematics 9, no. 16 (August 10, 2021): 1901. http://dx.doi.org/10.3390/math9161901.

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The COVID-19 pandemic and subsequent lockdowns highlight the close and delicate relationship between a country’s public health and economic health. Models that combine macroeconomic factors with traditional epidemic dynamics to calculate the impacts of a disease outbreak are therefore extremely useful for policymakers seeking to evaluate the best course of action in such a crisis. We developed a macroeconomic SIR model that considers herd immunity, behavior-dependent transmission rates, remote workers, and the indirect externalities of lockdowns. It is formulated as an exit time control problem where a social planner is able to prescribe separate levels of the lockdown low-risk and high-risk portions of the adult population. The model predicts that by considering the possibility of reaching herd immunity, high-risk individuals are able to leave lockdown sooner than in models where herd immunity is not considered. Additionally, a behavior-dependent transmission rate (which represents increased personal caution in response to increased infection levels) can lower both output loss and total mortality. Overall, the model-determined optimal lockdown strategy, combined with individual actions to slow virus transmission, is able to reduce total mortality to one-third of the model-predicted no-lockdown level of mortality.

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Baez Sanchez, A. D., and N. Bobko. "On Equilibria Stability in an Epidemiological SIR Model with Recovery-dependent Infection Rate." TEMA (São Carlos) 21, no. 3 (November 27, 2020): 409. http://dx.doi.org/10.5540/tema.2020.021.03.409.

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We consider an epidemiological SIR model with an infection rate depending on the recovered population. We establish sufficient conditions for existence, uniqueness, and stability (local and global) of endemic equilibria and consider also the stability of the disease-free equilibrium. We show that, in contrast with classical SIR models, a system with a recovery-dependent infection rate can have multiple endemic stable equilibria (multistability) and multiple stable and unstable saddle points of equilibria. We establish conditions for the occurrence of these phenomena and illustrate the results with some examples.

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Arasli, Batuhan, and Sennur Ulukus. "Dynamic Infection Spread Model Based Group Testing." Algorithms 16, no. 1 (January 2, 2023): 25. http://dx.doi.org/10.3390/a16010025.

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Group testing idea is an efficient approach to detect prevalence of an infection in the test samples taken from a group of individuals. It is based on the idea of pooling the test samples and performing tests to the mixed samples. This approach results in possible reduction in the required number of tests to identify infections. Classical group testing works consider static settings where the infection statuses of the individuals do not change throughout the testing process. In our paper, we study a dynamic infection spread model, inspired by the discrete time SIR model, where infections are spread via non-isolated infected individuals, while infection keeps spreading over time, a limited capacity testing is performed at each time instance as well. In contrast to the classical, static group testing problem, the objective in our setup is not to find the minimum number of required tests to identify the infection status of every individual in the population, but to control the infection spread by detecting and isolating the infections over time by using the given, limited number of tests. In order to analyze the performance of the proposed algorithms, we focus on the average-case analysis of the number of individuals that remain non-infected throughout the process of controlling the infection. We propose two dynamic algorithms that both use given limited number of tests to identify and isolate the infections over time, while the infection spreads, while the first algorithm is a dynamic randomized individual testing algorithm, in the second algorithm we employ the group testing approach similar to the original work of Dorfman. By considering weak versions of our algorithms, we obtain lower bounds for the performance of our algorithms. Finally, we implement our algorithms and run simulations to gather numerical results and compare our algorithms and theoretical approximation results under different sets of system parameters.

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Shah, Nita H., Ankush H. Suthar, Ekta N. Jayswal, and Ankit Sikarwar. "Fractional SIR-Model for Estimating Transmission Dynamics of COVID-19 in India." J 4, no. 2 (April 30, 2021): 86–100. http://dx.doi.org/10.3390/j4020008.

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In this article, a time-dependent susceptible-infected-recovered (SIR) model is constructed to investigate the transmission rate of COVID-19 in various regions of India. The model included the fundamental parameters on which the transmission rate of the infection is dependent, like the population density, contact rate, recovery rate, and intensity of the infection in the respective region. Looking at the great diversity in different geographic locations in India, we determined to calculate the basic reproduction number for all Indian districts based on the COVID-19 data till 7 July 2020. By preparing district-wise spatial distribution maps with the help of ArcGIS 10.2, the model was employed to show the effect of complete lockdown on the transmission rate of the COVID-19 infection in Indian districts. Moreover, with the model’s transformation to the fractional ordered dynamical system, we found that the nature of the proposed SIR model is different for the different order of the systems. The sensitivity analysis of the basic reproduction number is done graphically which forecasts the change in the transmission rate of COVID-19 infection with change in different parameters. In the numerical simulation section, oscillations and variations in the model compartments are shown for two different situations, with and without lockdown.

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Geng, Xiaolong, Gabriel G. Katul, Firas Gerges, Elie Bou-Zeid, Hani Nassif, and Michel C. Boufadel. "A kernel-modulated SIR model for Covid-19 contagious spread from county to continent." Proceedings of the National Academy of Sciences 118, no. 21 (May 6, 2021): e2023321118. http://dx.doi.org/10.1073/pnas.2023321118.

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The tempo-spatial patterns of Covid-19 infections are a result of nested personal, societal, and political decisions that involve complicated epidemiological dynamics across overlapping spatial scales. High infection “hotspots” interspersed within regions where infections remained sporadic were ubiquitous early in the outbreak, but the spatial signature of the infection evolved to affect most regions equally, albeit with distinct temporal patterns. The sparseness of Covid-19 infections in the United States was analyzed at scales spanning from 10 to 2,600 km (county to continental scale). Spatial evolution of Covid-19 cases in the United States followed multifractal scaling. A rapid increase in the spatial correlation was identified early in the outbreak (March to April). Then, the increase continued at a slower rate and approached the spatial correlation of human population. Instead of adopting agent-based models that require tracking of individuals, a kernel-modulated approach is developed to characterize the dynamic spreading of disease in a multifractal distributed susceptible population. Multiphase Covid-19 epidemics were reasonably reproduced by the proposed kernel-modulated susceptible–infectious–recovered (SIR) model. The work explained the fact that while the reproduction number was reduced due to nonpharmaceutical interventions (e.g., masks, social distancing, etc.), subsequent multiple epidemic waves still occurred; this was due to an increase in susceptible population flow following a relaxation of travel restrictions and corollary stay-at-home orders. This study provides an original interpretation of Covid-19 spread together with a pragmatic approach that can be imminently used to capture the spatial intermittency at all epidemiologically relevant scales while preserving the “disordered” spatial pattern of infectious cases.

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Chekroun, Abdennasser, and Toshikazu Kuniya. "An infection age-space structured SIR epidemic model with Neumann boundary condition." Applicable Analysis 99, no. 11 (November 30, 2018): 1972–85. http://dx.doi.org/10.1080/00036811.2018.1551997.

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Akimenko, Vitalii. "An age-structured SIR epidemic model with fixed incubation period of infection." Computers & Mathematics with Applications 73, no. 7 (April 2017): 1485–504. http://dx.doi.org/10.1016/j.camwa.2017.01.022.

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Xia, Chengyi, Li Wang, Shiwen Sun, and Juan Wang. "An SIR model with infection delay and propagation vector in complex networks." Nonlinear Dynamics 69, no. 3 (January 6, 2012): 927–34. http://dx.doi.org/10.1007/s11071-011-0313-y.

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Chekroun, Abdennasser, and Toshikazu Kuniya. "An infection age-space-structured SIR epidemic model with Dirichlet boundary condition." Mathematical Modelling of Natural Phenomena 14, no. 5 (2019): 505. http://dx.doi.org/10.1051/mmnp/2019048.

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In this paper, we are concerned with the global asymptotic behavior of an SIR epidemic model with infection age-space structure. Under the homogeneous Dirichlet boundary condition, we first reformulate the model into the coupled reaction-diffusion and difference system by using the method of characteristics. We then obtain the spatially heterogeneous disease-free steady state and define the basic reproduction number ℛ0 by the spectral radius of the next generation operator. We then show the existence and uniqueness of the global classical solution by constructing suitable upper and lower solutions. As a threshold result, we establish that the disease-free steady state is globally attractive if ℛ0 < 1, whereas the system is uniformly weakly persistent in norm if ℛ0 > 1. Finally, numerical simulations are exhibited to illustrate our theoretical results together with how to compute ℛ0.

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Laila, Arina Nur, Usman Pagalay, and Heni Widayani. "Analisis Model Stokastik Penularan Virus Hepatitis B." Jurnal Riset Mahasiswa Matematika 2, no. 1 (November 1, 2022): 1–9. http://dx.doi.org/10.18860/jrmm.v2i1.14467.

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The spread of hepatitis B virus (HBV) infection has been widely studied using the deterministic SIR model, in which individuals who recover from acute infection have temporary immunity to the virus. However, this deterministic model uses a constant rate of viral infection over time. This is not in accordance with the fact that the infection rate is a random parameter that depends on time. This study discusses the analysis of the stochastic model of hepatitis B virus transmission. The purpose of this study is to construct the SIR stochastic model by dividing the infection rate into two, namely the rate of acute and chronic infection following the Wiener process. The model is then searched for an analytical solution referring to the Ito formula. The analytical solution and the Wiener process are described by substituting parameter values in the form of acute and chronic infection rates (β+α), cure rate (γ), and initial values (S(0) and I(0)) to obtain the mean value (μ). and the standard deviation (σ) of dS(t) and dI(t). The results of the simulation show that the number of infected individuals (I(t)) will decrease rapidly if (γ) is greater but will increase rapidly if (β+α) and (I(0)) are greater.

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Laila, Arina Nur, Usman Pagalay, and Heni Widayani. "Analisis Model Stokastik Penularan Virus Hepatitis B." Jurnal Riset Mahasiswa Matematika 2, no. 1 (November 1, 2022): 298–306. http://dx.doi.org/10.18860/jrmm.v1i7.14467.

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The spread of hepatitis B virus (HBV) infection has been widely studied using the deterministic SIR model, in which individuals who recover from acute infection have temporary immunity to the virus. However, this deterministic model uses a constant rate of viral infection over time. This is not in accordance with the fact that the infection rate is a random parameter that depends on time. This study discusses the analysis of the stochastic model of hepatitis B virus transmission. The purpose of this study is to construct the SIR stochastic model by dividing the infection rate into two, namely the rate of acute and chronic infection following the Wiener process. The model is then searched for an analytical solution referring to the Ito formula. The analytical solution and the Wiener process are described by substituting parameter values in the form of acute and chronic infection rates (β+α), cure rate (γ), and initial values (S(0) and I(0)) to obtain the mean value (μ). and the standard deviation (σ) of dS(t) and dI(t). The results of the simulation show that the number of infected individuals (I(t)) will decrease rapidly if (γ) is greater but will increase rapidly if (β+α) and (I(0)) are greater.

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Datta, Samik, James C. Bull, Giles E. Budge, and Matt J. Keeling. "Modelling the spread of American foulbrood in honeybees." Journal of The Royal Society Interface 10, no. 88 (November 6, 2013): 20130650. http://dx.doi.org/10.1098/rsif.2013.0650.

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We investigate the spread of American foulbrood (AFB), a disease caused by the bacterium Paenibacillus larvae , that affects bees and can be extremely damaging to beehives. Our dataset comes from an inspection period carried out during an AFB epidemic of honeybee colonies on the island of Jersey during the summer of 2010. The data include the number of hives of honeybees, location and owner of honeybee apiaries across the island. We use a spatial SIR model with an underlying owner network to simulate the epidemic and characterize the epidemic using a Markov chain Monte Carlo (MCMC) scheme to determine model parameters and infection times (including undetected ‘occult’ infections). Likely methods of infection spread can be inferred from the analysis, with both distance- and owner-based transmissions being found to contribute to the spread of AFB. The results of the MCMC are corroborated by simulating the epidemic using a stochastic SIR model, resulting in aggregate levels of infection that are comparable to the data. We use this stochastic SIR model to simulate the impact of different control strategies on controlling the epidemic. It is found that earlier inspections result in smaller epidemics and a higher likelihood of AFB extinction.

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MCCULLOCH, K., M. G. ROBERTS, and C. R. LAING. "EXACT ANALYTICAL EXPRESSIONS FOR THE FINAL EPIDEMIC SIZE OF AN SIR MODEL ON SMALL NETWORKS." ANZIAM Journal 57, no. 4 (April 2016): 429–44. http://dx.doi.org/10.1017/s1446181116000043.

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We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of$N$nodes, there are$3^{N}$configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

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Sifriyani, Sifriyani, and Dedi Rosadi. "SUSCEPTIBLE INFECTED RECOVERED (SIR) MODEL FOR ESTIMATING COVID-19 REPRODUCTION NUMBER IN EAST KALIMANTAN AND SAMARINDA." MEDIA STATISTIKA 13, no. 2 (December 28, 2020): 170–81. http://dx.doi.org/10.14710/medstat.13.2.170-181.

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Modeling and analysis of Covid-19 data, especially on the modeling the spread and the prediction of the total number of cases for Indonesian data, has been conducted by several researchers. However, to the best of our knowledge, it has not been studied specifically for East Kalimantan Province data. The study of the data on the level of provincial and District/City level could help the government in making policies. In this study, we estimate the Covid-19 reproduction number, calculate the rate of recovery, the rate of infection, and the rate of death of East Kalimantan Province and Samarinda City. We also provide a prediction of the peak of the infection cases and forecast the total incidence of Covid-19 cases until the end of 2020. The model used in this research is the Susceptible Infected Recovered (SIR) model and the data used in the study was obtained from the East Kalimantan Public Health Office.

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Peng, Zhenghong, Siya Ao, Lingbo Liu, Shuming Bao, Tao Hu, Hao Wu, and Ru Wang. "Estimating Unreported COVID-19 Cases with a Time-Varying SIR Regression Model." International Journal of Environmental Research and Public Health 18, no. 3 (January 26, 2021): 1090. http://dx.doi.org/10.3390/ijerph18031090.

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Background: Potential unreported infection might impair and mislead policymaking for COVID-19, and the contemporary spread of COVID-19 varies in different counties of the United States. It is necessary to estimate the cases that might be underestimated based on county-level data, to take better countermeasures against COVID-19. We suggested taking time-varying Susceptible-Infected-Recovered (SIR) models with unreported infection rates (UIR) to estimate factual COVID-19 cases in the United States. Methods: Both the SIR model integrated with unreported infection rates (SIRu) of fixed-time effect and SIRu with time-varying parameters (tvSIRu) were applied to estimate and compare the values of transmission rate (TR), UIR, and infection fatality rate (IFR) based on US county-level COVID-19 data. Results: Based on the US county-level COVID-19 data from 22 January (T1) to 20 August (T212) in 2020, SIRu was first tested and verified by Ordinary Least Squares (OLS) regression. Further regression of SIRu at the county-level showed that the average values of TR, UIR, and IFR were 0.034%, 19.5%, and 0.51% respectively. The ranges of TR, UIR, and IFR for all states ranged from 0.007–0.157 (mean = 0.048), 7.31–185.6 (mean = 38.89), and 0.04–2.22% (mean = 0.22%). Among the time-varying TR equations, the power function showed better fitness, which indicated a decline in TR decreasing from 227.58 (T1) to 0.022 (T212). The general equation of tvSIRu showed that both the UIR and IFR were gradually increasing, wherein, the estimated value of UIR was 9.1 (95%CI 5.7–14.0) and IFR was 0.70% (95%CI 0.52–0.95%) at T212. Interpretation: Despite the declining trend in TR and IFR, the UIR of COVID-19 in the United States is still on the rise, which, it was assumed would decrease with sufficient tests or improved countersues. The US medical system might be largely affected by severe cases amidst a rapid spread of COVID-19.

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Zhao, Wencai, Tongqian Zhang, Zhengbo Chang, Xinzhu Meng, and Yulin Liu. "Dynamical Analysis of SIR Epidemic Models with Distributed Delay." Journal of Applied Mathematics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/154387.

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SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction numberRis got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.

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Chen, Yuming, Shaofen Zou, and Junyuan Yang. "Global analysis of an SIR epidemic model with infection age and saturated incidence." Nonlinear Analysis: Real World Applications 30 (August 2016): 16–31. http://dx.doi.org/10.1016/j.nonrwa.2015.11.001.

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Gilligan, Christopher A., Simon Gubbins, and Sarah A. Simons. "Analysis and fitting of an SIR model with host response to infection load for a plant disease." Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 352, no. 1351 (March 29, 1997): 353–64. http://dx.doi.org/10.1098/rstb.1997.0026.

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We reformulate a model for botanical epidemics into an SIR form for susceptible ( S ), infected ( I ) and removed ( R ) plant organs, in order to examine the effects of different models for the effect of host responses to the load of infection on the production of susceptible tissue. The new formulation also allows for a decline in host susceptibility with age. The model is analysed and tested for the stem canker disease of potatoes, caused by the soil–borne fungus, Rhizoctonia solani . Using a combination of model fitting to field data and analysis of model behaviour, we show that a function for host response to the amount (load) of parasite infection is critical in the description of the temporal dynamics of susceptible and infected stems in epidemics of R. solani . Several different types of host response to infection are compared including two that allow for stimulation of the plant to produce more susceptible tissue at low levels of disease and inhibition at higher levels. We show that when the force of infection decays with time, due to increasing resistance of the host, the equilibrium density of susceptible stems depends on the parameters and initial conditions. The models differ in sensitivity to small changes in disease transmission with some showing marked qualitative changes leading to a flush of susceptible stems at low levels of disease transmission. We conclude that there is no evidence to reject an SIR model with a simpler linear term for the effect of infection load on the production of healthy tissue, even though biological considerations suggest greater complexity in the relationship between disease and growth. We show that reduction in initial inoculum density, and hence in the force of infection, is effective in controlling disase when the simple model applies.

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Zhang, Juping, Zhen Jin, Yakui Xue, and Youwen Li. "The Dynamics of the Pulse Birth in an SIR Epidemic Model with Standard Incidence." Discrete Dynamics in Nature and Society 2009 (2009): 1–18. http://dx.doi.org/10.1155/2009/490437.

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An SIR epidemic model with pulse birth and standard incidence is presented. The dynamics of the epidemic model is analyzed. The basic reproductive numberR∗is defined. It is proved that the infection-free periodic solution is global asymptotically stable ifR∗<1. The infection-free periodic solution is unstable and the disease is uniform persistent ifR∗>1. Our theoretical results are confirmed by numerical simulations.

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Song, Haitao, Weihua Jiang, and Shengqiang Liu. "Global dynamics of two heterogeneous SIR models with nonlinear incidence and delays." International Journal of Biomathematics 09, no. 03 (February 25, 2016): 1650046. http://dx.doi.org/10.1142/s1793524516500467.

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To investigate the effect of heterogeneity on the global dynamics of two SIR epidemic models with general nonlinear incidence rate and infection delays, we formulate a multi-group model corresponding to the heterogeneity in the host population and a multi-stage model corresponding to heterogeneous stages of infection. Under biologically motivated considerations, we establish that the global dynamics for each of the two models is determined completely by the corresponding basic reproduction number: if the basic reproduction number is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable and the disease dies out in all groups or stages; if the basic reproduction number is larger than one, then the disease will persist in all groups or stages, and there is a unique endemic equilibrium which is globally asymptotically stable. Then we conclude that the heterogeneity does not change the global dynamics of the SIR model when the incidence rate is a general nonlinear function. Our results extend a class of previous results and can be applied to the other epidemiological models. The proofs of the main results use Lyapunov functional and graph-theoretic approach.

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Gao, Shujing, Zhidong Teng, Juan J. Nieto, and Angela Torres. "Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay." Journal of Biomedicine and Biotechnology 2007 (2007): 1–10. http://dx.doi.org/10.1155/2007/64870.

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Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease.

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Stanovov, Vladimir, Stanko Grabljevec, Shakhnaz Akhmedova, Eugene Semenkin, Radovan Stojanović, Črtomir Rozman, and Andrej Škraba. "Identification of COVID-19 spread mechanisms based on first-wave data, simulation models, and evolutionary algorithms." PLOS ONE 17, no. 12 (December 28, 2022): e0279427. http://dx.doi.org/10.1371/journal.pone.0279427.

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Background The COVID-19 epidemic has shown that efficient prediction models are required, and the well-known SI, SIR, and SEIR models are not always capable of capturing the real dynamics. Modified models with novel structures could help identify unknown mechanisms of COVID-19 spread. Objective Our objective is to provide additional insights into the COVID-19 spread mechanisms based on different models’ parameterization which was performed using evolutionary algorithms and the first-wave data. Methods Data from the Our World in Data COVID-19 database was analysed, and several models—SI, SIR, SEIR, SEIUR, and Bass diffusion—and their variations were considered for the first wave of the COVID-19 pandemic. The models’ parameters were tuned with differential evolution optimization method L-SHADE to find the best fit. The algorithm for the automatic identification of the first wave was developed, and the differential evolution was applied to model parameterization. The reproduction rates (R0) for the first wave were calculated for 61 countries based on the best fits. Results The performed experiments showed that the Bass diffusion model-based modification could be superior compared to SI, SIR, SEIR and SEIUR due to the component responsible for spread from an external factor, which is not directly dependent on contact with infected individuals. The developed modified models containing this component were shown to perform better when fitting to the first-wave cumulative infections curve. In particular, the modified SEIR model was better fitted to the real-world data than the classical SEIR in 43 cases out of 61, based on Mann–Whitney U tests; the Bass diffusion model was better than SI for 57 countries. This showed the limitation of the classical models and indicated ways to improve them. Conclusions By using the modified models, the mechanism of infection spread, which is not directly dependent on contacts, was identified, which significantly influences the dynamics of the spread of COVID-19.

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Hoque, Md Enamul. "Estimation of the number of affected people due to the Covid-19 pandemic using susceptible, infected and recover model." International Journal of Modern Physics C 31, no. 08 (July 30, 2020): 2050111. http://dx.doi.org/10.1142/s0129183120501119.

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The Susceptible, Infected and Recover (SIR) model is a very simple model to estimate the dynamics of an epidemic. In the current pandemic due to Covid-19, the SIR model has been used to estimate the dynamics of infection for various infected countries. Numerical solutions are used to obtain the value of parameters for the SIR model. The maximum and minimum basic reproduction number (14.5 and 2.3) are predicted to be in Turkey and China, respectively.

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Yang, Junyuan, and Yuming Chen. "Effect of infection age on an SIR epidemic model with demography on complex networks." Physica A: Statistical Mechanics and its Applications 479 (August 2017): 527–41. http://dx.doi.org/10.1016/j.physa.2017.03.006.

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Liu, Zhuanzhuan, Zhongwei Shen, Hao Wang, and Zhen Jin. "Analysis of a Local Diffusive SIR Model with Seasonality and Nonlocal Incidence of Infection." SIAM Journal on Applied Mathematics 79, no. 6 (January 2019): 2218–41. http://dx.doi.org/10.1137/18m1231493.

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Röst, G., Z. Vizi, and I. Z. Kiss. "Pairwise approximation for SIR -type network epidemics with non-Markovian recovery." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2210 (February 2018): 20170695. http://dx.doi.org/10.1098/rspa.2017.0695.

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We present the generalized mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma- and uniformly distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalized pairwise model lies in approximating the time evolution of the epidemic.

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Avram, Florin, Rim Adenane, and David I. Ketcheson. "A Review of Matrix SIR Arino Epidemic Models." Mathematics 9, no. 13 (June 28, 2021): 1513. http://dx.doi.org/10.3390/math9131513.

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Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”,which may be useful to obtain approximate control policies.

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Record, Nicholas R., and Andrew Pershing. "A note on the effects of epidemic forecasts on epidemic dynamics." PeerJ 8 (August 7, 2020): e9649. http://dx.doi.org/10.7717/peerj.9649.

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The purpose of a forecast, in making an estimate about the future, is to give people information to act on. In the case of a coupled human system, a change in human behavior caused by the forecast can alter the course of events that were the subject of the forecast. In this context, the forecast is an integral part of the coupled human system, with two-way feedback between forecast output and human behavior. However, forecasting programs generally do not examine how the forecast might affect the system in question. This study examines how such a coupled system works using a model of viral infection—the susceptible-infected-removed (SIR) model—when the model is used in a forecasting context. Human behavior is modified by making the contact rate responsive to other dynamics, including forecasts, of the SIR system. This modification creates two-way feedback between the forecast and the infection dynamics. Results show that a faster rate of response by a population to system dynamics or forecasts leads to a significant decline in peak infections. Responding to a forecast leads to a lower infection peak than responding to current infection levels. Inaccurate forecasts can lead to either higher or lower peak infections depending on whether the forecast under-or over-estimates the peak. The direction of inaccuracy in a forecast determines whether the outcome is better or worse for the population. While work is still needed to constrain model functional forms, forecast feedback can be an important component of epidemic dynamics that should be considered in response planning.

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Flayyih, Hadeer S. "Stability Analysis of Fractional SIR Model Related to Delay in State and Control Variables." BASRA JOURNAL OF SCIENCE 39, no. 2 (April 1, 2021): 204–20. http://dx.doi.org/10.29072/basjs.202123.

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The study of a nonlinear mathematical fractional SIR (Susceptible - Infected - Recovered) epidemiological model related to the delay in state and control variables in terms of time is the focus of this paper. The existence of a bounded solution for the fractional SIR epidemic model has been demonstrated, and it is unique. A new set of infection-free equilibrium points has been discovered, and their local stability has been investigated. In addition, using the next-generation matrix method, the basic reproductive number Ro was calculated

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CAO, WENJUN, and ZHEN JIN. "THE DYNAMICS OF THE CONSTANT AND PULSE BIRTH IN AN SIR EPIDEMIC MODEL WITH CONSTANT RECRUITMENT." Journal of Biological Systems 15, no. 02 (June 2007): 203–18. http://dx.doi.org/10.1142/s0218339007002118.

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In this paper, an SIR epidemic model with constant recruitment is considered. The dynamic behavior of this disease model with constant and pulse birth are analyzed. With constant birth, the infection-free equilibrium is locally and globally stable when the basic reproductive number R0 < 1. However, with pulse birth the system converges to a stable period solution with the number of infectious individuals equal to zero. Furthermore, the local and global stability of the periodic infection-free solution is obtained if the basic reproductive number [Formula: see text]. Numerical simulation shows that the periodic infection-free solution is unstable and the disease will persist when [Formula: see text]. The effectiveness of the constant and pulse birth to eliminating the disease are compared.

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Volpert, Vitaly, Malay Banerjee, and Sergei Petrovskii. "On a quarantine model of coronavirus infection and data analysis." Mathematical Modelling of Natural Phenomena 15 (2020): 24. http://dx.doi.org/10.1051/mmnp/2020006.

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Attempts to curb the spread of coronavirus by introducing strict quarantine measures apparently have different effect in different countries: while the number of new cases has reportedly decreased in China and South Korea, it still exhibit significant growth in Italy and other countries across Europe. In this brief note, we endeavour to assess the efficiency of quarantine measures by means of mathematical modelling. Instead of the classical SIR model, we introduce a new model of infection progression under the assumption that all infected individual are isolated after the incubation period in such a way that they cannot infect other people. Disease progression in this model is determined by the basic reproduction number R0 (the number of newly infected individuals during the incubation period), which is different compared to that for the standard SIR model. If R0 > 1, then the number of latently infected individuals exponentially grows. However, if R0 < 1 (e.g. due to quarantine measures and contact restrictions imposed by public authorities), then the number of infected decays exponentially. We then consider the available data on the disease development in different countries to show that there are three possible patterns: growth dynamics, growth-decays dynamics, and patchy dynamics (growth-decay-growth). Analysis of the data in China and Korea shows that the peak of infection (maximum of daily cases) is reached about 10 days after the restricting measures are introduced. During this period of time, the growth rate of the total number of infected was gradually decreasing. However, the growth rate remains exponential in Italy. Arguably, it suggests that the introduced quarantine is not sufficient and stricter measures are needed.

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Abu Hassan, Suzanawati, Yeong Kin Teoh, Diana Sirmayunie Mohd Nasir, and Nur Shamira Sharil. "Simulation of COVID-19 Trend in Selangor via SIR Model of Infectious Disease." Journal of Computing Research and Innovation 7, no. 2 (September 30, 2022): 294–303. http://dx.doi.org/10.24191/jcrinn.v7i2.322.

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Coronavirus Disease 2019 (COVID-19) was initially reported in December 2019 in Wuhan City, China, as a result of a respiratory pandemic. Since then, the infection has spread rapidly and uncontrollably around the globe, prompting the World Health Organization (WHO) to declare it a pandemic. The study's overall objective is to imitate the COVID-19 infectious trend in Selangor. The SIR model is used to forecast infection and the course of COVID-19 diffusion and estimate the fraction of the population infected. As a result, the Susceptible, Infectious, and Recovered (SIR) model was used to accomplish the study's aims. From March 23, 2020, to June 30, 2020, 100 days of COVID-19 data were extracted from a database on the Malaysian Ministry of Health's website. The RStudio software was used to analyse data on infectious trends in this study. The SIR model is used to predict the basic reproduction ratio, , based on actual and simulated infectious trends for comparison. The value of the basic reproduction ratio for simulating the infectious trend is 2.0, and the basic reproduction ratio for modelling the infectious trend with the entire population of Selangor is 1.15429. According to the findings of this study, the reproduction ratio would affect the number of infected individuals by reducing the number of recovered individuals. The effectiveness of lockdown in preventing COVID-19 disease in Selangor was demonstrated by a significant reduction in the basic reproduction ratio, .

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Hamieh, Mohamad, Ali ElMoussaoui, Hassan Ayoub, Hicham Aboudaya, and Zeinab Hamie. "IMPACT OF SOCIAL BEHAVIOR ON THE DYNAMIC SPREAD SARS-COV-2 IN LEBANON ACCORDING TO THE SIR MODEL." Euroasia Journal of Mathematics, Engineering, Natural & Medical Sciences 9, no. 20 (March 25, 2022): 169–88. http://dx.doi.org/10.38065/euroasiaorg.943.

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Analyzing the dynamics of Sars-Cov-2 spread in the Lebanese society is what this article mainly aspires and points to, where the study was predicated on a compartmental model, namely SIR, the widely known model in epidemiology. SIR. (Susceptible-Infected-Recovered) materializes a basic conceptional structure for theoretically investigating the virus spread and its dynamics within a community, through focusing on the interaction and communication between infected and recovered people. Consequently, providing the necessary attempts to overcome the epidemic, and diminishes its expansion to rescue lives. In which, limiting contact absolutely reduces the possibility of transmitting or contracting an infection. This investigation on a representative sample of the Lebanese population highlights the various drivers and dynamics of this proliferation. These drivers or factors clarify the behavior of the population (wearing a mask, washing their hands) in experiencing the epidemic crisis and their abuse for measures (safety distance, closures) adopted by the authorities to combat the epidemic. So, it turns out that the careless and incautious attitude of the Lebanese population, besides the unsatisfactory control to fulfill the government rules against the dynamics of virus spread was shown by the modeling of Sars-Cov-2 dynamics through the SIR model.

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Ball, Frank, and Owen D. Lyne. "Stochastic multi-type SIR epidemics among a population partitioned into households." Advances in Applied Probability 33, no. 1 (March 2001): 99–123. http://dx.doi.org/10.1017/s000186780001065x.

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We consider a stochastic model for the spread of an SIR (susceptible → infective → removed) epidemic among a closed, finite population that contains several types of individuals and is partitioned into households. The infection rate between two individuals depends on the types of the transmitting and receiving individuals and also on whether the infection is local (i.e., within a household) or global (i.e., between households). The exact distribution of the final outcome of the epidemic is outlined. A branching process approximation for the early stages of the epidemic is described and made fully rigorous, by considering a sequence of epidemics in which the number of households tends to infinity and using a coupling argument. This leads to a threshold theorem for the epidemic model. A central limit theorem for the final outcome of epidemics which take off is derived, by exploiting an embedding representation.

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Journal articles on the topic 'SIR infection model'

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