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SIR model. Schematic representation, differential equations, and plot for the basic SIR (susceptible, infectious, and recovered) model. Model parameters are b , the transmission rate ( b = 0.0005), and c , the recovery rate ( c = 0.05). There is initially one infection in a population of 1,000 individuals. doi:10.1371/journal.pntd.0000761.g001
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Neglected tropical diseases affect more than one billion people worldwide. The populations most impacted by such diseases are typically the most resource-limited. Mathematical modeling of disease transmission and cost-effectiveness analyses can play a central role in maximizing the utility of limited resources for neglected tropical diseases. We re...
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... modeling of vector-borne infectious diseases originated with Sir Ronald Ross’s study of malaria transmission in 1916 [1]. Ross recognized that vector-borne infections are governed by nonlinear dynamics, which makes intuitive assessment of the natural trajectory of an epidemic and intervention effectiveness difficult, if not impossible, without mathematical modeling. Mathematical models can play important roles in the study of infectious diseases. Models help explain the dynamics of an infectious disease within a host or a population, and they facilitate comparisons among competing control strategies that can inform policy decisions. The use of mathematical models has been gaining momentum in recent decades. Models are being used to address an ever- expanding number of diseases and public health questions, as well as to explore the importance of biological and ecological details on disease transmission [2]. For example, to realistically incorporate the population dynamics of mosquitoes, there is a need to take into account age structure, seasonality, and density-dependent mortality [3–5]. The realistic incorporation of vectors then improves the evaluation of the long-term impact of control strategies [6,7]. In this review, we address model parameterization and sensitivity analysis, two important steps in the building and analysis of mathematical models. We discuss specific features of neglected vector-borne diseases that should be incorporated into quantitative methods that analyze control strategies for such diseases. We also review the areas in which models have been and can be most useful, including drug, vaccine, vector, and alternative control strategies, as well as cost-effectiveness analysis. We searched PubMed, Web of Science, and SciELO using the terms ‘‘mathematical model’’, ‘‘modeling’’, ‘‘cost-effectiveness analysis’’, and ‘‘economic analysis’’. For the theoretical literature, we included studies that, in our opinion, addressed important technicalities of mathematical models applied to infectious diseases, including, for example, dynamic modeling, sensitivity analysis, and cost-effectiveness analysis. Modeling analyses conducted for diseases not considered to be neglected vector-borne diseases were excluded. We report on studies that evaluate the impact of interventions on vector-borne neglected tropical diseases. Model development involves several steps and considerations. Once the modeler identifies the essential components of the biological processes necessary to address the questions of interest, the information needs to be translated into equations that describe the transmission dynamics. The most popular mathematical model is the SIR model, which divides hosts into compartments on the basis of whether they are susceptible, infectious, or recovered/ immune (Figure 1). A susceptible individual ( S ) who contracts disease becomes infectious ( I ) and then recovers ( R ) to become immune. The parameters of the SIR model are the rate at which susceptible hosts become infected ( b ) and the rate at which infectious individuals recover ( c ). For vector-borne diseases, the rate at which hosts become infected ( b ) depends on vector competence and abundance. Model parameterization can be achieved using published studies and from fitting a model to observed data [2]. As parameter values are only estimates of ‘‘true’’ values [8], modelers need to perform sensitivity analysis in order to explore which parameters have the greatest impact on model predictions. Different approaches to sensitivity analysis vary in their simplicity and applicability to specific models. Univariate sensitivity analysis measures the impact of the variation of one parameter on the outcome of the model while all other parameters are held constant. One method for conducting univariate sensitivity analysis is to change the value of each model parameter by a certain percent, and then measure the percent change in the value of the outcome. Such sensitivity analysis can be represented on a tornado plot, which is a graphical way of showing which parameters most strongly influence the outcome of a model [9]. For vector-borne diseases, it is common that demographic parameters, such as the vector’s life span and reproductive capacity, have the greatest impact on the population dynamics of the vector. For example, the mortality rate of a vector usually has a particularly pronounced influence on disease transmission [10]. Multivariate sensitivity analysis consists of simultaneously mea- suring the impact of multiple parameters. One approach for multivariate sensitivity analysis is through Monte Carlo simulations. For this procedure, probability distributions are assigned to parameters and the values of those parameters are sampled repeatedly from these distributions. Model simulations with each set of these parameters are then computed to generate a distribution of model outcomes from which summary statistics can be calculated. Statistical regression models can then be used to determine which parameters most strongly influence model outcome [11]. Mathematical models and cost-effectiveness analysis have been used to assess the impact of various control strategies for a wide range of neglected tropical diseases, which we review here. Many neglected vector-borne diseases can be treated and controlled with drugs [12]. However, this control strategy imposes selection for drug resistance [13,14]. To maximize long-term drug utility, such evolutionary consequences should be taken into account. Currently, several vector control programs advocate widespread administration of drugs to avoid mass screening for detection of infected individuals, because diagnostic tests can be costly and imperfect [15]. However, mass administration of drugs results in the unnecessary treatment of uninfected individuals, a practice leading to higher rates of adverse effects and faster selection for drug resistance [16]. For example, the dilemma of mass versus targeted drug administration for onchocerciasis, a disease usually treated with the drug ivermectin, has been explored using a model that incorporates heterogeneity in human exposure [16]. It was found that targeted ivermectin interventions can reduce the onchocerciasis health burden using only 20%–25% of the doses required for mass drug administration, thus resulting in decreased costs, a smaller proportion of adverse effects, and a lower probability of spread of ivermectin resistance [16]. This example illustrates the positive impact on treatment approaches that modeling public health interventions can have to reduce both the spread of disease and the development of resistance. Vector control [12], which relies on the use of insecticides, is the primary control method of neglected vector-borne diseases. The basic reproduction number ( R 0 ) is the number of secondary infections generated from a single infected individual introduced into a susceptible population. In order to curtail transmission, vector control efforts need to decrease the value of R 0 below the critical value of 1. For example, R 0 was used to determine the extent of vector control necessary to eliminate the transmission of Chagas disease in Brazil [17]. Given that the R 0 for Chagas disease in Brazil is 1.25, it was shown that a 25% increase in vector control mortality induced by insecticides was sufficient to reduce R 0 below 1. Nonetheless, a differential equation model showed that a vector control strategy that reduced R 0 just below 1 would require more than half a century to achieve disease eradication due to disease persistence in chronically infected individuals. Two models incorporating vector control have also evaluated insecticide-based vector control strategies for dengue prevention [18,19]. The ...
Citations
... Regarding mathematical modeling for lymphatic filariasis, a number of deterministic and statistical models have been formulated and analyzed. These include [8,[16][17][18][19][20][21][22][23][24][25][26][27][28] and [29]. Although asymptomatic individuals play a significant role in disease transmission, none of the aforementioned studies have considered the simultaneous inclusion of asymptomatic, acute, and chronically infected populations in their modeling approaches. ...
Lymphatic filariasis is a neglected tropical disease which poses public health concern and socio-economic challenges in developing and low-income countries. In this paper, we formulate a deterministic mathematical model for transmission dynamics of lymphatic filariasis to generate data by white noise and use least square method to estimate parameter values. The validity of estimated parameter values is tested by Gaussian distribution method. The residuals of model outputs are normally distributed and hence can be used to study the dynamics of Lymphatic filariasis. After deriving the basic reproduction number, R0 by the next generation matrix approach, the Partial Rank Correlation Coefficient is employed to explore which parameters significantly affect and most influential to the model outputs. The analysis for equilibrium states shows that the Lymphatic free equilibrium is globally asymptotically stable when the basic reproduction number is less a unity and endemic equilibrium is globally asymptotically stable when R0≥1. The findings reveal that rate of human infection, recruitment rate of mosquitoes increase the average new infections for Lymphatic filariasis. Moreover, asymptomatic individuals contribute significantly in the transmission of Lymphatic filariasis.
... Mathematical modelling with control and optimization also exposes the authors [42] to sugar beet pests. The articles of the authors [43], [44], [45], [46], [47], and many others are inscribed on the same coordinates. ...
The article presents a mathematical model for experiments evaluating the effectiveness of diatomaceous earth treatments against the bean weevil, Acanthoscelides obtectus. The proposed mathematical model is of the differential type, inspired by the category of prey-predator models. The system of equations is nonlinear and is solved numerically. A systemic characterization of the bean weevil treatment process is used to describe the model, which uses three functions of time: the number of beans, the pest population, and the amount of diatomaceous earth. The three functions offer users four applications: forecasting, control, formulation of treatment efficacy estimators, and simulation of different types of pest control. The model is built for closed (isolated) experiments typical of laboratories, but this feature makes it extensible to other treatments to combat bean weevils in closed spaces characteristic of the storage of beans in silos.
... In compartmental modeling, such as in Susceptible-Infectious-Removed (SIR) framework [34], heterogeneity is represented by incorporating the k parameter in the differential equation model. For example, we can represent the change in the S population due to the disease by [2]: ...
In many host–parasite systems, overdispersion in the distribution of macroparasites leads to parasite aggregation in the host population. This overdispersed distribution is often characterized by the negative binomial or by the power law. The aggregation is shown by a clustering of parasites in certain hosts, while other hosts have few or none. Here, I present a theory behind the overdispersion in complex spatiotemporal systems as well as a computational analysis for tracking the behavior of transmissible diseases with this kind of dynamics. I present a framework where heterogeneity and complexity in host–parasite systems are related to aggregation. I discuss the problem of focusing only on the average parasite burden without observing the risk posed by the associated variance; the consequences of under- or overestimation of disease transmission in a heterogenous system and environment; the advantage of including the network of social interaction in epidemiological modeling; and the implication of overdispersion in the management of health systems during outbreaks.
... In Figure 8, we compared the effect between chemical and vaccination control strategies modeled as pulse inputs, where a vaccination campaign is performed as a long-term strategy in a specific time interval instead of considering it as a constant value over time. As shown in literature, the vaccination is a better option than chemical control in long-term implementation [54] because the disease incidence could disappear from the population if we inoculate enough individuals. ...
Some deterministic models deal with environmental conditions and use parameter estimations to obtain experimental parameters, but they do not consider anthropogenic or environmental disturbances, e.g., chemical control or climatic conditions. Even more, they usually use theoretical or measured in-lab parameters without worrying about uncertainties in initial conditions, parameters, or changes in control inputs. Thus, in this study, we estimate parameters (including chemical control parameters) and confidence contours under uncertainty conditions using data from the municipality of Bello (Colombia) during 2010–2014, which includes two epidemic outbreaks. Our study shows that introducing non-periodic pulse inputs into the mathematical model allows us to: (i) perform parameter estimation by fitting real data of consecutive dengue outbreaks, (ii) highlight the importance of chemical control as a method of vector control, and (iii) reproduce the endemic behavior of dengue. We described a methodology for parameter and sub-contour box estimation under uncertainties and performed reliable simulations showing the behavior of dengue spread in different scenarios.
... Mathematical modelling can contribute by recommending improved or optimised intervention strategies for various (vector-borne) diseases, but there are many challenges that modellers have to overcome to provide policy-relevant insights [54,55]. One must find a balance between creating easy-to-use models and ensuring that biological simplifications do not alter the resultant policy recommendations by over or underestimating the impact of different intervention measures. ...
Mathematical models of vector-borne infections, including malaria, often assume age-independent mortality rates of vectors, despite evidence that many insects senesce. In this study we present survival data on insecticide-resistant Anopheles gambiae s.l . from experiments in Côte d’Ivoire. We fit a constant mortality function and two age-dependent functions (logistic and Gompertz) to the data from mosquitoes exposed (treated) and not exposed (control) to insecticide-treated nets (ITNs), to establish biologically realistic survival functions. This enables us to explore the effects of insecticide exposure on mosquito mortality rates, and the extent to which insecticide resistance might impact the effectiveness of ITNs. We investigate this by calculating the expected number of infectious bites a mosquito will take in its lifetime, and by extension the vectorial capacity. Our results show that the predicted vectorial capacity is substantially lower in mosquitoes exposed to ITNs, despite the mosquitoes in the experiment being highly insecticide-resistant. The more realistic age-dependent functions provide a better fit to the experimental data compared to a constant mortality function and, hence, influence the predicted impact of ITNs on malaria transmission potential. In models with age-independent mortality, there is a great reduction for the vectorial capacity under exposure compared to no exposure. However, the two age-dependent functions predicted an even larger reduction due to exposure, highlighting the impact of incorporating age in the mortality rates. These results further show that multiple exposures to ITNs had a considerable effect on the vectorial capacity. Overall, the study highlights the importance of including age dependency in mathematical models of vector-borne disease transmission and in fully understanding the impact of interventions.
... For instance, the works of Kermack and McKendrick began with the simple SIR model, which creates three compartments: susceptible, infected, and recovered. [18] However, the model has since been adapted and expanded upon, leading to the creation of new compartmental models, such as the SEIR model-which considers incubation periods-and the MSIR model which accounts for maternally-derived immunity [19,20]. Some compartmental models also account for variable contact rates, which are more realistic as they reflect changes in the infection rate as an outbreak progresses [21]. ...
As of December 2020, the COVID-19 pandemic has infected over 75 million people, making it the deadliest pandemic in modern history. This study develops a novel compartmental epidemiological model specific to the SARS-CoV-2 virus and analyzes the effect of common preventative measures such as testing, quarantine, social distancing, and vaccination. By accounting for the most prevalent interventions that have been enacted to minimize the spread of the virus, the model establishes a paramount foundation for future mathematical modeling of COVID-19 and other modern pandemics. Specifically, the model expands on the classic SIR model and introduces separate compartments for individuals who are in the incubation period, asymptomatic, tested-positive, quarantined, vaccinated, or deceased. It also accounts for variable infection, testing, and death rates. I first analyze the outbreak in Santa Clara County, California, and later generalize the findings. The results show that, although all preventative measures reduce the spread of COVID-19, quarantine and social distancing mandates reduce the infection rate and subsequently are the most effective policies, followed by vaccine distribution and, finally, public testing. Thus, governments should concentrate resources on enforcing quarantine and social distancing policies. In addition, I find mathematical proof that the relatively high asymptomatic rate and long incubation period are driving factors of COVID-19's rapid spread.
... Mathematical modelling with control and optimization completion also exposes the authors [42], for sugar beet pests. The articles of the authors [43], [44], [45], [46], [47] and many others are inscribed on the same coordinates. ...
... It is defined 40 in [11] as "the expected number of infective mosquito bites that would eventually 41 arise from all the mosquitoes that would bite a single fully infectious person on 42 a single day", i.e. the average number of humans that get infected due to one 43 infectious human per day. These metrics have been used to study the dynamics 44 of vector-borne diseases and also quantitatively assess the possible impact of 45 interventions to control them. 46 Contact pesticides were being used at the time Macdonald was researching 47 vector control, and that is when he realised that transmission potential was 48 affected by two important factors relating to mosquito longevity [8]: 49 (a) a mosquito which is infected will only become infectious if it survives the 50 time needed for the pathogen to develop, commonly known as the extrinsic 51 incubation period (EIP), and 52 (b) once the mosquito is infectious it must take a blood-meal in order to 53 transmit the infection on to a host. ...
Mathematical models of vector-borne infections, including malaria, often assume age-independent mortality rates of vectors, despite evidence that many insects senesce. In this study we present survival data on insecticide-resistant Anopheles gambiae s.l. from field experiments in Côte d’Ivoire. We fit a constant mortality function and two age-dependent functions (logistic and Gompertz) to the data from mosquitoes exposed (treated) and not exposed (control) to insecticidetreated nets (ITNs), to establish biologically realistic survival functions. This enables us to explore the effects of insecticide exposure on mosquito mortality rates, and the extent to which insecticide resistance might impact the effectiveness of ITNs. We investigate this by calculating the expected number of infectious bites a mosquito will take in its lifetime, and by extension the vectorial capacity. Our results show that the predicted vectorial capacity is substantially lower in mosquitoes exposed to ITNs, despite the mosquitoes in the experiment being highly insecticide-resistant. The more realistic age-dependent functions provide a better fit to the experimental data compared to a constant mortality function and, hence, influence the predicted impact of ITNs on malaria transmission potential. In models with age-independent mortality, there is a reduction of 56.52% (±14.66) for the vectorial capacity under exposure compared to no exposure. However, the two age-dependent functions predicted a larger reduction due to exposure: for the logistic function the reduction is 74.38% (±9.93) and for the Gompertz 74.35% (±7.11), highlighting the impact of incorporating age in the mortality rates. These results further show that multiple exposures to ITNs had a considerable effect on the vectorial capacity. Overall, the study highlights the importance of including age dependency in mathematical models of vector-borne disease transmission and in fully understanding the impact of interventions.
Author summary
Interventions against malaria are most commonly targeted on the adult mosquitoes, which transmit the infection from person to person. One of the most important interventions are bed-nets, treated with insecticides. Unfortunately, extensive exposure of mosquitoes to insecticide has led to widespread evolution of insecticide resistance, which might threaten control strategies. Piecing together the overall impact of resistance on the efficacy of insecticide-treated nets is complex, but can be informed by the use of mathematical models. However, there are some assumptions that the models frequently use which are not realistic in terms of the mosquito biology. In this paper, we formulate a model that includes age-dependent mortality rates, an important parameter in vector control since control strategies most commonly aim to reduce the lifespan of the mosquitoes. By using novel data collected using field-derived insecticide-resistant mosquitoes, we explore the effects that the presence of insecticides on nets have on the mortality rates, as well as the difference incorporating age dependency in the model has on the results. We find that including age-dependent mortality greatly alters the anticipated effects of insecticide-treated nets on mosquito transmission potential, and that ignoring this realism potentially overestimates the negative impact of insecticide resistance.
... In this context several mathematical models were used to connect the biological processes of vector dynamics and climate (Lord 2004, Lord 2007, Grassly and Fraser 2008, Luz et al. 2010, Eikenberry and Gumel 2018. To date, most epidemiological and vector population models have a deterministic nature and rely on some basic assumptions to define the various parameters of vector and disease dynamics under study (Wonham 2004, Wei at al. 2008, Lewis et al. 2010). ...
Vector born disease account for about one third of all cases of emerging diseases. Culex sp., particularly, is one of the most important mosquito vectors transmitting important diseases such as the West Nile virus, filariasis and related encephalitis. Because there are no vaccines available the most effectual means to prevent infections from the above diseases, is to target mosquitos to prevent bites and disease transmission. However, to be effective such a strategy, it is important to predict the temporal change in mosquito abundance as well as to study how it is affected by weather conditions. This dissertation is devoted on the development of new methods to predict arthropod vector dynamics and with emphasis on the development of stochastic models and computational methods for predicting Culex sp. abundance in Northern Greece. The current dissertation is divided in three parts. The first part explores the non-trivial associations between Culex sp. mosquito abundance and weather variables using traditional and straightforward novel techniques. The information from the first part was a prerequisite for developing a series of stochastic prediction models based on the most detrimental factors affecting mosquito abundance. In the second part, a series of conventional and conditional stochastic Markov chain models are applied for the first time to predict the non-linear dynamics of Culex sp. adult abundance. In the third part of the dissertation a soft computing approach is introduced to model the population dynamics of Culex sp. and a series of autoregressive artificial neural networks are implied. Finally, the information of the models is extrapolated and a machine learning algorithm is proposed to be used for predicting arthropod vector dynamics having practical implications for public health decision making. Based on the current results there was a high and positive correlation between temperature and mosquito abundance during both observation years (r = 0.6). However, a very poor correlation was observed between rain and weekly mosquito abundances (r = 0.29), as well as between wind speed (r = 0.29), respectively. Additionally, according to the multiple linear regression model the effect of temperature, was significant. The continuous power spectrum of the mosquito abundance counts and mean temperatures depict in most cases similar power for periods which are close to 1 week, indicating the point of the lowest variance of the time series, although appearing on slightly different moments of time. The cross wavelet coherent analysis showed that inter weekly cycles with a period between 2 and 3 weeks between mosquito abundance and temperature were coherent mostly during the first and the last weeks of the season. Hence, the wavelet analysis shows a progressive oscillation in mosquito occurrences with time, which is higher at the start and the end of the season. Moreover, in contrast with standard methods of analysis, wavelets can provide useful insights into the time-resolved oscillation structure of mosquito data and accompanying revealing a non-stationary association with temperature. According to the correlation results a climate-conditioned Markov Chain (CMC) model was developed and applied for the first time to predict the dynamics of vectors of important medical diseases. Temporal changes in mosquito population profiles were generated to simulate the probabilities of a high population impact. The probabilities achieved from the trained model are very near to the observed data and the CMC model satisfactorily describes the temporal evolution of the mosquito population process. In general, our numerical results indicate that it is more likely for the population system to move into a state of high population level, when the former is a state of a low population level than the opponent. Field data on frequencies of successive mosquito population levels, which were not used for the data inferred MC modeling, were assembled to obtain an empirical intensity transition matrix and the observed frequencies. The findings match to a certain degree the empirical results in which the probabilities follow analogous patterns while no significant differences were observed between the transition matrices of the CMC model and the validation data (ChiSq=14.58013, df=24, p=0.9324451). Furter, a soft system computing modeling approach was followed to simulate and predict Culex sp. abundances. Three dynamic artificial neural network (ANN) models were developed and applied to describe and predict the non-linear incidence and time evolution of a medical important mosquito species Culex sp. in Northern Greece. The first is a simple nonlinear autoregressive ANN model that used lagged population values as inputs, the second is an exogenous non-linear autoregressive recurrent neural network (NARX), which is designed to take as inputs the temperature as exogenous variable and mosquito abundance as endogenous. Finally, the third model is a focused time-delay neural network (FTD), which takes in to account only the temperature variable as input to provide forecasts of the mosquito abundance as target variable. All three models behaved well considering the non-linear nature of the adult mosquito abundance data. However, the NARX model, which takes in to account temperature, showed the best overall modelling performances. Nevertheless, although, the NARX model predicted slight better (R=0.623) compared to the FTD model (R=0.534), the advantage of the FTD over the NARX neural network model is that it can be applied in the case where past values of the population system, here mosquito abundance, are not available for their forecasting. This is very important considering that arthropod vector data are not always available as climatic data. Concluding, the proposed methods for simulating and predicting mosquito dynamics are recommended as viable for modeling vector disease population dynamics in order to make real-time recommendations utile for dynamic health policies decision making. The proposed stochastic models, as well as the current computational and machine learning techniques, of this work provide an accurate abstraction of the arthropod vector population progress observed within the dataset used for their generation. Nevertheless, the current study may consider also as a new entry point into the extensive literature of ecological modelling, medical entomology, as well as in simulating arthropod vector diseases epidemics. From a public health standpoint, the current models have the potential to be integrated into a decision support system allowing health policy makers in their planning to initiate specific management actions against the period of high activity of mosquito adults.
... For example, snakebite incidence can theoretically be modelled using snake population parameters, and the predictive ability of these models could highlight areas that require more attention from health authorities. A previous study suggested that snakebite incidence can be inferred using compartmental modelling (Bravo-Vega et al., 2019), as it is commonly done with infectious diseases (Siettos and Russo, 2013;Luz et al., 2010). In contrast to classical statistical analyses, epidemiological models rely on the processes underlying the interactions between the populations involved in disease dynamics. ...
Snakebite incidence at least partly depends on the biology of the snakes involved. However, studies of snake biology have been largely neglected in favour of anthropic factors, with the exception of taxonomy, which has been recognised for some decades to affect the design of antivenoms. Despite this, within-species venom variation and the unpredictability of the correlation with antivenom cross-reactivity has continued to be problematic. Meanwhile, other aspects of snake biology, including behaviour, spatial ecology and activity patterns, distribution, and population demography, which can contribute to snakebite mitigation and prevention, remain underfunded and understudied. Here, we review the literature relevant to these aspects of snakebite and illustrate how demographic, spatial, and behavioural studies can improve our understanding of why snakebites occur and provide evidence for prevention strategies. We identify the large gaps that remain to be filled and urge that, in the future, data and relevant metadata be shared openly via public data repositories so that studies can be properly replicated and data used in future meta-analyses.
