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A) Density-independent dynamics of the discrete time SIR model. Parameter values were: p s = p i = p r = 0.8, ˇ = 0.3, and = 0.3. Fecundity rate (m) was assumed to be 1.0 for all disease states. Initial population sizes were 100, 10, and 0 for susceptible, infective, and recovered states, respectively. (B-D) Depict the dynamics of SIR model with density-dependentˇanddependentˇ dependentˇand fecundity rates. Density of infective individuals was assumed to influencě such thatˇ(thatˇ(N) = 1 − exp(−kI), where k quantifies the strength of density-dependence and I is the density of infective individuals. Ricker function was used to incorporate density-dependence in fecundity rates, which was assumed to be influenced by total population size (N(t) = S(t) + I(t) + R(t)): m(N) = m × exp(−cN), where c quantifies the strength of density-dependence. Values of density-dependence parameters were (B) k = 0.001, c = 0.005, (C) k = 0.01, c = 0.05, and (D) k = 0.01, c = 0.001.
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Mathematical models of infectious diseases can provide important insight into our understanding of epidemiological processes, the course of infection within a host, the transmission dynamics in a host population, and formulation or implementation of infection control programs. We present a framework for modeling the dynamics of infectious diseases...
Citations
... Treatment involves antibiotics and diphtheria antitoxin, while prevention relies on timely vaccination Lamichhane and Radhakrishnan (2022), Britannica (2023). Mathematical models aid in understanding transmission dynamics and shaping public health strategies Oli et al. (2006), Sweileh (2022). Various authors have explored diphtheria dynamics, noting the role of contaminated environments and the necessity of booster vaccines Islam (2018), Husain (2019), Izzati et al. (2020), Kanchanarat et al. (2022), Ghani et al. 2022, Akhi et al. (2023, Amalia (2022), Rahmi & Pratama (2023), Djaafara (2020). ...
Diphtheria, a bacterial infection caused by Corynebacterium diphtheriae, remains a significant public health concern worldwide. In this study, we employ mathematical modeling to analyze the spread and control of diphtheria, focusing on the efficacy of Diphtheria Antitoxin in mitigating the disease's impact. Through the development of compartmental models, system of differential equations governing the dynamics was formulated. Due to the complexity and non-linearity of the dynamics, a numerical solutions that utilizes Runge-Kutta Fehlberg order 4 and 5 method. The dynamics of diphtheria transmission and the potential impact of DAT administration on disease outcomes was investigate. Our findings highlight the critical role of Antitoxin efficiency in reducing disease burden, preventing severe cases, and containing epidemic spread. By exploring various scenarios and parameter sensitivities, we provide insights into optimal control strategies and intervention measures to combat diphtheria outbreaks effectively. This research contributes to a better understanding of diphtheria epidemiology and informs public health policies aimed at enhancing vaccination coverage and DAT availability to achieve sustainable disease control and prevention.
... Mathematical models of infectious diseases provide significant insights into understanding the epidemiological processes, the transmission dynamics of infectious diseases and the implementation of control interventions [13]. It can also serve as a means of making policymakers comprehend and forecast the dynamics of an infectious disease under many different scenarios [14]. ...
... The state and the control variables of Eqs. (11) are non-negative, as established in Subsection (3.1), and the condition in Eq. (13); this implies that the set 1 is closed, convex and exists. The optimal control exists by applying Corollary 4.1 of Pages 68-69 in [40] as implemented in [41]. ...
... Discrete time models were also widely used for modeling systems' behaviors [34][35][36][37][38]. MTL has been widely discussed in the literature. ...
Metric temporal logic (MTL) is a popular real-time extension of linear temporal logic (LTL). This paper presents a new simple SAT-based bounded model-checking (SAT-BMC) method for MTL interpreted over discrete infinite timed models generated by discrete timed automata with digital clocks. We show a new translation of the existential part of MTL to the existential part of linear temporal logic with a new set of atomic propositions and present the details of the new translation. We compare the new method’s advantages to the old method based on a translation of the hard reset LTL (HLTL). Our method does not need new clocks or new transitions. It uses only one path and requires a smaller number of propositional variables and clauses than the HLTL-based method. We also implemented the new method, and as a case study, we applied the technique to analyze several systems. We support the theoretical description with the experimental results demonstrating the method’s efficiency.
... In the past two decades, the theory of matrix population model has been constantly developed, and it has also become a powerful tool to simulate the dynamics of infectious diseases in discrete disease states [Allen & Van den Driessche, 2008]. Based on this theory, a great number of researches have obtained rich dynamic behaviors of discrete models [Hu et al., 2014;Liz, 2007;Muroya et al., 2012;Oli et al., 2006;Wang & Fečkan, 2020;Yousef et al., 2018;Zhou et al., 2014;Zhu et al., 2020]. However, there are still few studies that directly use discrete models to describe tick-borne diseases. ...
Severe fever with thrombocytopenia syndrome (SFTS) is an acute tick-borne disease caused by SFTS virus (SFTSV). In this paper, we use difference equations to establish a discrete tick-borne disease model with systemic and transovarial transmission. Using the method of the next generation of matrix, we get the basic reproduction number R0 to determine whether SFTS will die out. Furthermore, we analyze the existence and stability of equilibrium points by R0. In addition, the transcritical bifurcation property at the disease-free equilibrium point is discussed by deriving a equation describing the flow on the center manifold. Finally, we perform numerical simulations to verify the theoretical results.
... We use a discrete-time epidemiological model (e.g. Bjørnstad et al., 2002;Oli et al., 2006) Amoebidium parasiticum reproduces by the production of spores (Kuno, 1973) that can remain viable for a long time (Decaestecker et al., 2004;Ebert, 1995). As we kept one third of the old medium when cleaning the aquaria twice a week, there might have been a build-up of spores in the aquaria through time. ...
Global warming challenges the persistence of local populations, not only through heat‐induced stress, but also through indirect biotic changes. We study the interactive effects of temperature, competition and parasitism in the water flea Daphnia magna.
We carried out a common garden experiment monitoring the dynamics of Daphnia populations along a temperature gradient. Halfway through the experiment, all populations became infected with the ectoparasite Amoebidium parasiticum, enabling us to study the interactive effects of temperature and parasite dynamics. We combined Integral Projection Models with epidemiological models, parameterized using the experimental data on the performance of individuals within dynamic populations. This enabled us to quantify the contribution of different vital rates and epidemiological parameters to population fitness across temperatures and Daphnia clones originating from two latitudes.
Interactions between temperature and parasitism shaped competition, where Belgian clones performed better under infection than Norwegian clones. Infected Daphnia populations performed better at higher than at lower temperatures, mainly due to an increased host capability of reducing parasite loads. Temperature strongly affected individual vital rates, but effects largely cancelled out on a population‐level. In contrast, parasitism strongly reduced fitness through consistent negative effects on all vital rates. As a result, temperature‐mediated parasitism was more important than the direct effects of temperature in shaping population dynamics. Both the outcome of the competition treatments and the observed extinction patterns support our modelling results.
Our study highlights that shifts in biotic interactions can be equally or more important for responses to warming than direct physiological effects of warming, emphasizing that we need to include such interactions in our studies to predict the competitive ability of natural populations experiencing global warming.
A free Plain Language Summary can be found within the Supporting Information of this article.
... In (5)-(9), the contact terms are normalized by the living population, whereas such a normalization does not occur in (19)- (23). In the frequency-dependent formulation (5)-(9), this normalization implies that the contagion is independent of population density, while this is not the case in the density-dependent formulation, as the name may suggest [32,38]. Both models may be valid and deliver accurate results, depending on the physical situation, and we will show computations performed with both formulations in the present work. ...
The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of an epidemic in space. Such research on PDE models within a Susceptible, Infected, Exposed, Recovered, and Deceased framework has led to promising results in reproducing COVID-19 contagion dynamics. In this paper, we assess the robustness of this modeling framework by considering different geometries over more extended periods than in other similar studies. We first validate our code by reproducing previously shown results for Lombardy, Italy. We then focus on the U.S. state of Georgia and on the Brazilian state of Rio de Janeiro, one of the most impacted areas in the world. Our results show good agreement with real-world epidemiological data in both time and space for all regions across major areas and across three different continents, suggesting that the modeling approach is both valid and robust.
... In (5)-(9), the contact terms are normalized by the living population, whereas such a normalization does not occur in (19)- (23). In the frequency-dependent formulation (5)- (9), this normalization implies that the contagion is independent of population density, while this is not the case in the density-dependent formulation, as the name may suggest [28,29]. Both models may be valid and deliver accurate results, depending on the physical situation, and we will show computations performed with both formulations in the present work. ...
The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation (ODE) models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of an epidemic in space. Such research on PDE models within a Susceptible, Infected, Exposed, Recovered, and Deceased (SEIRD) framework has led to promising results in reproducing COVID-19 contagion dynamics. In this paper, we assess the robustness of this modeling framework by considering different geometries over more extended periods than in other similar studies. We first validate our code by reproducing previously shown results for Lombardy, Italy. We then focus on the U.S. state of Georgia and on the Brazilian state of Rio de Janeiro, one of the most impacted areas in the world. Our results show good agreement with real-world epidemiological data in both time and space for all regions across major areas and across three different continents, suggesting that the modeling approach is both valid and robust.
... Given the great extent of IOD caused by P. cinnamomi, the relative scarcity of quantitative studies is somewhat surprising. The mathematical modelling of infectious diseases can provide important insights into such processes and has therefore become an indispensable tool for predicting the transmission dynamics in a host population (Madden, 2006;Oli, 2006). Such insights are a key element to guide strategic decisions aimed at controlling the spread of infections (Plantegenest et al., 2007). ...
The pathogen Phytophthora cinnamomi is considered a main driver of Iberian oak decline (IOD), a forest disease which decimates holm oaks (Quercus ilex) and cork oaks (Quercus suber) in a multipurpose, silvo‐pastoral and seminatural ecosystem of 3.1 million hectares in the south‐west of Europe. Little is known about the spatial dynamics of Phytophthora cinnamomi and how forest stand characteristics affect the IOD epidemic. Here, we analyse IOD spread over several decades in one such ecosystem by means of a multilevel approach based on (a) identification of diseased sites via repeated aerial imagery at landscape scale, (b) confirmation by subsampling of soil and roots, and iii) an epidemic model accounting for host population heterogeneities. We use a 'self‐exciting' spatiotemporal point process with two additive risk components: an epidemic component represents the inoculum pressure from nearby disease foci, and a background component describes sporadic disease transmission over larger distances or from unobserved sources. Both risk components are found to increase over time, and a lagged power‐law spatial kernel provides the best fit for the observed disease pattern. We estimate that 49% of the secondary infections triggered by a primary source occur within a distance of 250 m. Our results also highlight the role of density and diversity of the host population; we find that the rate of sporadic infections in silvo‐pastoral systems (dehesas) is lower than in forests, and higher in mixed stands and shrub encroached oak lands than in pure stands. These results have direct implications for IOD management, for example the estimated spatial kernel may guide the definition of suitable target areas for localized control measures and help to quantify their success. Our results also suggest that silviculture treatments aimed at controlling the density and species composition of oak stands, as well as the abundance of shrubs, are crucial to the containment of IOD.
... For two continuous time fitted models of host-virus associations from our Table 2 subset which did not report R 0 , we calculated R 0 from the system of equations and best-fit parameters reported in the corresponding article using a Next Generation Matrix (NGM) approach [22][23][24] (Supplementary Appendix 1 and 2). For two discrete time fitted models from our Table 2 subset that did not report R 0 [both from 25], we calculated R 0 following a discrete time approximation of the NGM approach, again using best-fit parameters reported in the corresponding article [24,26] (Supplementary Appendix 3). All maps and summary figures were generated using R v. 4.0.0 for Macintosh. ...
... • Support recovered for models allowing bats to become transiently or persistently infectious, while maintaining an option to progress to recovery and return to susceptibility * R 0 values for [25,40,47] were not reported in the original articles but were calculated for this paper. For [40,47], we used a Next Generation Matrix approach [23] (Supplementary Appendix 1 and 2), and for [25], we used the discrete time approximation of this methodology [22,24,26] Colored pie-charts indicate each model's approach to data (by percentage) for each locality, whether purely theoretical, simulated using data-derived parameters, or explicitly fitted to data. For theoretical models that did not specify a distinct locality, the continent (i.e. ...
The emergence of SARS-CoV-2, a coronavirus with suspected bat origins, highlights a critical need for heightened understanding of the mechanisms by which bats maintain potentially zoonotic viruses at the population level and transmit these pathogens across species. We review mechanistic models, which test hypotheses of the transmission dynamics that underpin viral maintenance in bat systems. A search of the literature identified only twenty-five mechanistic models of bat-virus systems published to date, derived from twenty-three original studies. Most models focused on rabies and related lyssaviruses (eleven), followed by Ebola-like filoviruses (seven), Hendra and Nipah-like henipaviruses (five), and coronaviruses (two). The vast majority of studies has modelled bat virus transmission dynamics at the population level, though a few nested within-host models of viral pathogenesis in population-level frameworks, and one study focused on purely within-host dynamics. Population-level studies described bat virus systems from every continent but Antarctica, though most were concentrated in North America and Africa; indeed, only one simulation model with no associated data was derived from an Asian bat-virus system. In fact, of the twenty-five models identified, only ten population-level models were fitted to data – emphasizing an overall dearth of empirically derived epidemiological inference in bat virus systems. Within the data fitted subset, the vast majority of models were fitted to serological data only, highlighting extensive uncertainty in our understanding of the transmission status of a wild bat. Here, we discuss similarities and differences in the approach and findings of previously published bat virus models and make recommendations for improvement in future work.
... Disease dynamics can be described theoretically using a variety of mathematical approaches. Strategies range from simple equations (Anderson and May, 1979), susceptible, infected, removed models, complex network analysis (Adams et al., 2012), population matrix models (Oli et al., 2006), or individual-based models with disease components (Willem et al., 2017). However, to develop a theoretical framework integrating the most important processes, disease dynamics are often explained by the pathogen reproduction number (Anderson and May, 1979). ...
Salmon farming has multiplied from a side business of coastal farmers to one of the world's major aquaculture species. This has dramatically altered the disease dynamics between farmed and wild salmonids. As salmon fish farming has increased, new restrictions have been enforced to combat emerging density-dependent impacts of pathogen spillover. In most northern and arctic regions, the effects of pathogens from fish farms on wild salmonids have been minimal for two key reasons: (i) relative low density of fish farms in the north and (ii) cold water temperatures. However, both factors are set to change dramatically. On one side, there is an increasing interest in utilizing northern areas for fish farming due to limited capacity for expansion in mid-latitude regions. On the other side, climate change is rapidly changing these northern ecosystems. High-latitude regions inhabit some of the largest remaining wild Atlantic salmon populations in the world along with sea trout and Arctic charr. Wild salmonids in the north have most likely seldom been exposed to high infection pressure, and we question how these populations will cope with changes that are coming. We identify 12 research questions emerging from these imminent changes and discuss methodologies for addressing them. We conclude that policies related to fish farming must consider uncertainties with respect to pathogen dynamics in the north until these research questions are fully addressed.
