SIR Type Research Articles

Beyond the biting - limited impact of explicit mosquito dynamics in dengue models

Mathematical models play a crucial role in assisting public health authorities in timely disease control decision-making. For vector-borne diseases, integrating host and vector dynamics into models can be highly complex, particularly due to limited data availability, making system validation challenging. In this study, two compartmental models akin to the SIR type were developed to characterize vector-borne infectious disease dynamics. Motivated by dengue fever epidemiology, the models varied in their treatment of vector dynamics, one with implicit vector dynamics and the other explicitly modeling mosquito-host contact. Both considered temporary immunity after primary infection and disease enhancement in secondary infection, analogous to the temporary cross-immunity and the Antibody-dependent enhancement biological processes observed in dengue epidemiology. Qualitative analysis using bifurcation theory and numerical experiments revealed that the immunity period and disease enhancement outweighed the impact of explicit vector dynamics. Both models demonstrated similar bifurcation structures, indicating that explicit vector dynamics are only justified when assessing the effects of vector control methods. Otherwise, the extra equations are irrelevant, as both systems display similar dynamics scenarios. The study underscores the importance of using simple models for mathematical analysis, initiating crucial discussions among the modeling community in vector-borne diseases.

DYNAMICAL BEHAVIOR OF AN SIR EPIDEMIC MODEL WITH RATIO-DEPENDENT IMPULSIVE CONTROL AND BEDDINGTON–DEANGELIS INCIDENCE

In this paper, we propose an epidemic model of SIR type with ratio-dependent impulse control and Beddington–DeAngelis (B–D) incidence. According to the magnitude of the basic reproductive number [Formula: see text] and the relation of the endemic equilibrium [Formula: see text] and the ratio threshold [Formula: see text], dynamical analysis of the controlled system is conducted. Under the control strategy, if [Formula: see text], the solutions converge to the disease-free equilibrium. If [Formula: see text] and [Formula: see text], the impulsive system has periodic solution that is orbitally asymptotically stable, and order-[Formula: see text] periodic solution does not exist. Furthermore, if [Formula: see text] and [Formula: see text], the solution converges either to the endemic equilibrium or to a periodic solution, which is proved to be determined by the initial value. Finally, numerical simulations are performed to demonstrate the theoretical results.

Dynamics of a nonlocal SIR epidemic model with free boundaries

This paper is devoted to a nonlocal SIR type epidemic model, with free boundary denoting the expanding front. We aim to examine the full information about the dynamical behaviors of the involved disease that caused by the nonlocal diffusion and growing habitat. We obtain a classification for the spreading and vanishing cases, and show that those are determined by the new proposed threshold function R0F(t) and especially the corresponding initial data R0F(0). We prove that R0F(t)≤R0 and R0F(t)→R0 as t tends to ∞, R0 here representing the basic reproductive number of the associated ordinary differential equation. Also, the result show that the introduce of the nonlocal diffusion makes the disease becomes more easier to spread. In addition, we present some numerical simulations to confirm the theoretical results, as well as the impacts of R0 and the expanding capability.

Time-dependent non-homogeneous stochastic epidemic model of SIR type

Time-dependent non-homogeneous stochastic epidemic model of SIR type

Optimal vaccination in a SIRS epidemic model.

We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton-Jacobi-Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.

COVID-19: A Comparative Study of Contagions Peaks in Cities from Europe and the Americas

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by a group of viruses that provoke illnesses ranging from the common cold to more serious illnesses such as pneumonia. COVID-19 started in China and spread rapidly from a single city to an entire country in just 30 days and to the rest of the world in no more than 3 months. Several studies have tried to model the behavior of COVID-19 in diverse regions, based on differential equations of the SIR and stochastic SIR type, and their extensions. In this article, a statistical analysis of daily confirmed COVID-19 cases reported in eleven different cities in Europe and America is conducted. Log-linear models are proposed to model the rise or drop in the number of positive cases reported daily. A classification analysis of the estimated slopes is performed, allowing a comparison of the eleven cities at different epidemic peaks. By rescaling the curves, similar behaviors among rises and drops in different cities are found, independent of socioeconomic conditions, type of quarantine measures taken, whether more or less restrictive. The log-linear model appears to be suitable for modeling the incidence of COVID-19 both in rises and drops.

Mathematical Analysis of the Transmission Dynamics of Skin Cancer Caused by UV Radiation

Nowadays, skin cancer is a worldwide panic. It is related to ultraviolet radiation. In this paper, we have formulated a SIRS type mathematical model to show the effects of ultraviolet radiation on skin cancer. At first, we have showed the boundedness and positivity of the model solutions to verify the model’s existence and uniqueness. The boundedness and positivity gave the solutions of our model bounded and positive, which was very important for real-world situation because in real world, population cannot be negative. Then, we have popped out all the equilibrium points of our model and verified the stability of the equilibrium points. This stability test expressed some physical situation of our model. The disease-free equilibrium point is locally asymptotically stable if R 0 < 1 and if R 0 > 1 , then it is unstable. Again, the endemic equilibrium point is stable, if R 0 > 1 and unstable if R 0 < 1 . In order to understand the dynamical behavior of the model’s equilibrium points, we examined the phase portrait. We also have observed the sensitivity of the model parameters. After this, we have investigated the different scenarios of bifurcations of the model’s parameters. At the set of Hopf bifurcation parameters when infection rate due to UV rays is less than α 1 = 0.01 , proper control may eradicate the existence of disease. From transcritical bifurcation, we can say when recovery rate greater than 1.9, then the disease of skin cancer can be eliminated and when recovery rate less than 1.9 then the disease of skin cancer cannot be eradicated. Finally, numerical analysis is done to justify our analytical findings.

Modeling secondary infections with temporary immunity and disease enhancement factor: Mechanisms for complex dynamics in simple epidemiological models

Modeling insights for epidemiological scenarios characterized by chaotic dynamics have been largely unexplored. A rigorous analysis of such systems are essential for a real predictive power and a more accurate disease control decision making. Motivated by dengue fever epidemiology, we study a basic SIR–SIR type model for the host population, capturing differences between primary and secondary infections. This model is the minimalistic version to previously suggested multi-strain models for dengue fever in which deterministic chaos was found in wider parameter regions. Without strain structure of pathogens, we consider temporary immunity after a primary infection and disease enhancement in a subsequent infection to identify to which extent these biological mechanisms can generate complex behavior in simple epidemiological models.Stability analysis of the system is performed using the classical linearization theory, and the qualitative behavior of the model is investigated with a detailed bifurcation analysis. Rich dynamical structures are identified, including the Bogdanov–Takens, cusp and Bautin bifurcations which has never been described in dengue fever epidemiology. Besides the conventional transcritical bifurcation, a backward bifurcation occurs for higher disease enhancement in secondary infections, exhibiting bi-stability when biological temporary immunity period is assumed. The backward bifurcation is formalized using the center manifold theory. While the Hopf and the global homoclinic bifurcation curves were computed numerically, analytical expressions for the transcritical and tangent bifurcations are obtained. The combination of temporary immunity and disease enhancement play a significant role in the complexity of the system dynamics, with chaotic behavior observed after including seasonal forcing.

Stochastic SIR model predicts the evolution of COVID-19 epidemics from public health and wastewater data in small and medium-sized municipalities: A one year study

The level of unpredictability of the COVID-19 pandemics poses a challenge to effectively model its dynamic evolution. In this study we incorporate the inherent stochasticity of the SARS-CoV-2 virus spread by reinterpreting the classical compartmental models of infectious diseases (SIR type) as chemical reaction systems modeled via the Chemical Master Equation and solved by Monte Carlo Methods. Our model predicts the evolution of the pandemics at the level of municipalities, incorporating for the first time (i) a variable infection rate to capture the effect of mitigation policies on the dynamic evolution of the pandemics (ii) SIR-with-jumps taking into account the possibility of multiple infections from a single infected person and (iii) data of viral load quantified by RT-qPCR from samples taken from Wastewater Treatment Plants. The model has been successfully employed for the prediction of the COVID-19 pandemics evolution in small and medium size municipalities of Galicia (Northwest of Spain).

Optimal Incentives to Mitigate Epidemics: A Stackelberg Mean Field Game Approach

Motivated by the models of epidemic control in large populations, we consider a Stackelberg mean field game model between a principal and a mean field of agents whose states evolve in a finite state space. The agents play a noncooperative game in which they control their rates of transition between states to minimize an individual cost. The principal influences the nature of the resulting Nash equilibrium through incentives to optimize its own objective. We analyze this game using a probabilistic approach. We then propose an application to an epidemic model of SIR type in which the agents control the intensities of their interactions, and the principal is a regulator acting with nonpharmaceutical interventions. To compute the solutions, we propose an innovative numerical approach based on Monte Carlo simulations and machine learning tools for stochastic optimization. We conclude with numerical experiments illustrating the impact of the agents' and the regulator's optimal decisions in two specific models: a basic SIR model with semiexplicit solutions and a more complex model with a larger state space.

Epidemiological theory of virus variants

We propose a physics-inspired mathematical model underlying the temporal evolution of competing virus variants that relies on the existence of (quasi) fixed points capturing the large time scale invariance of the dynamics. To motivate our result we first modify the time-honoured compartmental models of the SIR type to account for the existence of competing variants and then show how their evolution can be naturally re-phrased in terms of flow equations ending at quasi fixed points. As the natural next step we employ (near) scale invariance to organise the time evolution of the competing variants within the effective description of the epidemic Renormalisation Group framework. We test the resulting theory against the time evolution of COVID-19 virus variants that validate the theory empirically.

Complex dynamics of a fractional-order SIR system in the context of COVID-19.

This paper proposes and analyses a new fractional-order SIR type epidemic model with a saturated treatment function. The detailed dynamics of the corresponding system, including the equilibrium points and their existence and uniqueness, uniform-boundedness, and stability of the solutions are studied. The threshold parameter, basic reproduction number of the system which determines the disease dynamics is derived, and the condition of occurrence of backward bifurcation is also determined. Some numerical works are conducted to validate our analytical results for the commensurate fractional-order system. Hopf bifurcations for the fractional-order system are studied by taking the order of the fractional differential as a bifurcation parameter.

Epidemic local final size in a metapopulation network as indicator of geographical priority for control strategies in SIR type diseases

The main limitation on designing epidemic control strategies lies in their economic and social costs. Thus, a practical and efficient approach takes into consideration these factors. Most epidemics evolve in a structured population, being the geographical structure the most evident. In this situation, having a criteria for identifying the most effective locations where control measures can optimize available resources is desirable. In this paper, a regional index based on the final epidemic size predicted by a metapopulation model is proposed. An efficient algorithm to calculate explicit index values was developed, and different control strategies that used the recommended index were compared with others that do not take the index information into account. We found that the proposed index represents an easy and fast criterion to guide simple control strategies. This type of index offers a new powerful approach where the information encoded in a deterministic mathematical model can be summarized to guide realistic and practical control strategies for disease spreading and epidemics.

Spreading Properties for SIR Models on Homogeneous Trees.

We consider an epidemic model of SIR type set on a homogeneous tree and investigate the spreading properties of the epidemic as a function of the degree of the tree, the intrinsic basic reproduction number and the strength of the interactions between the populations of infected individuals at each node. When the degree is one, the homogeneous tree is nothing but the standard lattice on the integers and our model reduces to a SIR model with discrete diffusion for which the spreading properties are very similar to the continuous case. On the other hand, when the degree is larger than two, we observe some new features in the spreading properties. Most notably, there exists a critical value of the strength of interactions above which spreading of the epidemic in the tree is no longer possible.

Effect of density dependence on coinfection dynamics

In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R_0approx 1. We show even more, that for the values R_0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

The impact of hospital resources and environmental perturbations to the dynamics of SIRS model

Incorporating the environmental perturbations and available resources of the public health system, we construct both deterministic and stochastic models of SIRS type. The deterministic model exhibits very rich dynamics, such as Hopf bifurcation and backward bifurcation which leads to the co-existence of the stable disease-free state and a stable endemic equilibrium. For the stochastic model, we show that under mild extra conditions, if the basic reproduction number is less than one, then the disease will be eradicated almost surely, and if the basic reproduction number is greater than one, the stochastic model will admit a unique ergodic stationary distribution, which implies that the disease persists almost surely. Part of our numerical simulations indicate that: (i) The introduction of environmental perturbations may drift the endemic equilibrium to the disease-free equilibrium, or vice versa; (ii) Increasing available resources is necessary in order to mitigate the infections.

A SIR-type model describing the successive waves of COVID-19.

It is well-known that the classical SIR model is unable to make accurate predictions on the course of illnesses such as COVID-19. In this paper, we show that the official data released by the authorities of several countries (Italy, Spain and The USA) regarding the expansion of COVID-19 are compatible with a non-autonomous SIR type model with vital dynamics and non-constant population, calibrated according to exponentially decaying infection and death rates. Using this calibration we construct a model whose outcomes for most relevant epidemiological paramenters, such as the number of active cases, cumulative deaths, daily new deaths and daily new cases (among others) fit available real data about the first and successive waves of COVID-19. In addition to this, we also provide predictions on the evolution of this pandemic in Italy and the USA in several plausible scenarios.

Epidemic Conditions with Temporary Link Deactivation on a Network SIR Disease Model

The spread of an infectious disease depends on intrinsic properties of the disease as well as the connectivity and actions of the population. This study investigates the dynamics of an SIR type model which accounts for human tendency to avoid infection while also maintaining preexisting, interpersonal relationships. Specifically, we use a network model in which individuals probabilistically deactivate connections to infected individuals and later reconnect to the same individuals upon recovery. To analyze this network model, a mean field approximation consisting of a system of fourteen ordinary differential equations for the number of nodes and edges is developed. This system of equations is closed using a moment closure approximation for the number of triple links. By analyzing the differential equations, it is shown that, in addition to force of infection and recovery rate, the probability of deactivating edges and the average node degree of the underlying network determine if an epidemic occurs.

Mathematical modeling and optimal control strategies of Buruli ulcer in possum mammals

<abstract> Buruli is a neglected tropical disease that can now be found in many countries including developed countries like Australia. It is a skin disorder that usually occurs in the arms and legs. The disease has been identified in a number of mammals, particularly possum. Adequate eradication and control programs are needed to minimize infection and its spread to less developed countries before it becomes an epidemic. In this work, a SIR type possum epidemic model is proposed. The properties of the model are thoroughly studied and obtained its stability results. We determine the stability of the model at its fixed points and show that the model is locally and globally asymptomatically stable. The stability of the disease-free case is shown for $ \mathcal{R}_0 &lt; 1 $ and the endemic case is examined for $ \mathcal{R}_0 &gt; 1$. We further extend the model using the control variables and obtain an optimal control system and using the optimal control theory to characterize the necessary condition for controlling the spread of Buruli ulcer (BU). The model results are plotted to determine the best strategies for disease elimination. Numerical simulation has shown that the useful strategy consists in implementing all suggested controls. </abstract>

Rumor spreading dynamics with an online reservoir and its asymptotic stability

&lt;p style='text-indent:20px;'&gt;The spread of rumors is a phenomenon that has heavily impacted society for a long time. Recently, there has been a huge change in rumor dynamics, through the advent of the Internet. Today, online communication has become as common as using a phone. At present, getting information from the Internet does not require much effort or time. In this paper, the impact of the Internet on rumor spreading will be considered through a simple SIR type ordinary differential equation. Rumors spreading through the Internet are similar to the spread of infectious diseases through water and air. From these observations, we study a model with the additional principle that spreaders lose interest and stop spreading, based on the SIWR model. We derive the basic reproduction number for this model and demonstrate the existence and global stability of rumor-free and endemic equilibriums.&lt;/p&gt;