SIR-type Model Research Articles

A stochastic epidemic model of COVID-19 disease

To model the evolution of diseases with extended latency periods and the presence of asymptomatic patients like COVID-19, we define a simple discrete time stochastic SIR-type epidemic model. We include both latent periods as well as the presence of quarantine areas, to capture the evolutionary dynamics of such diseases.

Assessing the role of human mobility on malaria transmission

South Sudan accounts for a large proportion of all annual malaria cases in Africa. In recent years, the country has witnessed an unprecedented number of people on the move, refugees, internally displaced people, people who have returned to their counties or areas of origin, stateless people and other populations of concern, posing challenges to malaria control. Thus, one can claim that human mobility is one of the contributing factors to the resurgence of malaria. The aim of this paper is to assess the impact of human mobility on the burden of malaria disease in South Sudan. For this, we formulate an SIR-type model that describes the transmission dynamics of malaria disease between multiple patches. The proposed model is a system of stochastic differential equations consisting of ordinary differential equations perturbed by a stochastic Wiener process. For the deterministic part of the model, we calculate the basic reproduction number. Concerning the whole stochastic model, we use the maximum likelihood approach to fit the model to weekly malaria data of 2011 from Central Equatoria State, Western Bahr El Ghazal State and Warrap State. Using the parameters estimated on the fitted model, we simulate the future observation of the disease pattern. The disease was found to persist in the low transmission patches when there is human inflow in these patches and although the intervention coverage reaches 75%.

Space dependent models for studying the spread of some diseases

In this paper we present some extensions of the classical SIR model with non-symmetric spatial dependence. SIR-type models usually describe the epidemic in a population, which is split into three categories, namely the susceptibles (S), the infected (I) and the recovered (R). The proposed model yields a system of partial integro-differential equations. Two methods handling the integrals of the equations are presented. We give numerical examples which show that the discrete models preserve the basic qualitative properties of the original biological process, and these results are supported by the oretical theorems.

Stochastic models of influenza outbreaks on a college campus

Disease outbreaks on residential college campuses provide an ideal opportunity for mathematical modelling. Unfortunately, publicly available data are rare and many of these outbreaks are relatively small, confounding traditional data-fitting techniques such as least-squares. Using data from three outbreaks during the 2015 and 2017 flu seasons at Trinity College, we fit several SIR-type stochastic models by approximating the likelihood of each model. We find that stochasticity is a key driver in determining the size of the outbreak, and that it strongly depends on the amount of time between the start of the outbreak and the next school holiday. Our results indicate that in order to prevent or limit the size of an outbreak, school closure is likely to be more effective than increasing the vaccination rate. As influenza is a leading cause of negative academic outcomes, these results offer important guidance for school administrators.

SIR-Type Epidemic Models as Block-Structured Markov Processes

This paper proposes a block-structured Markov process to describe the spread of epidemics of Susceptible-Infected-Removed (SIR) type. Our purpose is to determine the distribution of the final state of the process and of some other interesting measures of the dimension of the epidemic. The followed modeling approach proves to be simple and systematic. Its flexibility is underlined by the presentation of several specific models that extend the classical general epidemic. Finally, two numerical examples illustrate some of the results obtained.

Optimal control for Malaria disease through vaccination

Malaria is a disease caused by an amoeba (single-celled animal) type of plasmodium where anopheles mosquito serves as the carrier. This study examines the optimal control problem of malaria disease spread based on Aron and May (1982) SIR type models and seeks the optimal solution by minimizing the prevention of the spreading of malaria by vaccine. The aim is to investigate optimal control strategies on preventing the spread of malaria by vaccination. The problem in this research is solved using analytical approach. The analytical method uses the Pontryagin Minimum Principle with the symbolic help of MATLAB software to obtain optimal control result and to analyse the spread of malaria with vaccination control.

Mathematical Analysis of an SIQR Influenza Model with Imperfect Quarantine.

The identification of mechanisms responsible for recurrent epidemic outbreaks, such as age structure, cross-immunity and variable delays in the infective classes, has challenged and fascinated epidemiologists and mathematicians alike. This paper addresses, motivated by mathematical work on influenza models, the impact of imperfect quarantine on the dynamics of SIR-type models. A susceptible-infectious-quarantine-recovered (SIQR) model is formulated with quarantined individuals altering the transmission dynamics process through their possibly reduced ability to generate secondary cases of infection. Mathematical and numerical analyses of the model of the equilibria and their stability have been carried out. Uniform persistence of the model has been established. Numerical simulations show that the model supports Hopf bifurcation as a function of the values of the quarantine effectiveness and other parameters. The upshot of this work is somewhat surprising since it is shown that SIQR model oscillatory behavior, as shown by multiple researchers, is in fact not robust to perturbations in the quarantine regime.

Optimal combinations of control strategies and cost-effective analysis for visceral leishmaniasis disease transmission.

Visceral leishmaniasis (VL) is a deadly neglected tropical disease that poses a serious problem in various countries all over the world. Implementation of various intervention strategies fail in controlling the spread of this disease due to issues of parasite drug resistance and resistance of sandfly vectors to insecticide sprays. Due to this, policy makers need to develop novel strategies or resort to a combination of multiple intervention strategies to control the spread of the disease. To address this issue, we propose an extensive SIR-type model for anthroponotic visceral leishmaniasis transmission with seasonal fluctuations modeled in the form of periodic sandfly biting rate. Fitting the model for real data reported in South Sudan, we estimate the model parameters and compare the model predictions with known VL cases. Using optimal control theory, we study the effects of popular control strategies namely, drug-based treatment of symptomatic and PKDL-infected individuals, insecticide treated bednets and spray of insecticides on the dynamics of infected human and vector populations. We propose that the strategies remain ineffective in curbing the disease individually, as opposed to the use of optimal combinations of the mentioned strategies. Testing the model for different optimal combinations while considering periodic seasonal fluctuations, we find that the optimal combination of treatment of individuals and insecticide sprays perform well in controlling the disease for the time period of intervention introduced. Performing a cost-effective analysis we identify that the same strategy also proves to be efficacious and cost-effective. Finally, we suggest that our model would be helpful for policy makers to predict the best intervention strategies for specific time periods and their appropriate implementation for elimination of visceral leishmaniasis.

An elaborated SIR model for haemonchosis in sheep in South Africa under a targeted selective anthelmintic treatment regime

Infection with the abomasal nematode Haemonchus contortus is responsible for considerable production loss in small ruminants globally, and especially in warm, summer-rainfall regions. Previous attempts to predict infection levels have followed the traditional framework for macroparasite models, i.e. tracking parasite population sizes as a function of host and climatic factors. Targeted treatment strategies, in which patho-physiological indices are used to identify the individuals most affected by parasites, could provide a foundation for alternative, incidence-based epidemiological models. In this paper, an elaboration of the classic susceptible-infected-recovered (SIR) model framework for microparasites was adapted to haemonchosis and used to predict disease in Merino sheep on a commercial farm in South Africa. Incidence was monitored over a single grazing season using the FAMACHA scoring system for conjunctival mucosal coloration, which indicates high burdens of H. contortus, and used to fit the model by estimating transmission parameters. The model predicted force of infection (FOI) between sequential FAMACHA monitoring events in groups of dry, pregnant and lactating ewes, and related FOI to factors including climate (temperature, rainfall and rainfall entropy), using a random effects model with reproductive status group as the cluster variable. Temperature and rainfall in the seven days prior to monitoring significantly predicted the interval FOI (p≤0.002), while rainfall entropy did not (p=0.289). Differences across the three groups accounted for approximately 90% of the variability in the interval FOI over the period of investigation. Maintained FOI during targeted treatment of cases of haemonchosis suggests strong underlying transmission from sub-clinically infected animals, and/or limited impact on pre-existing pasture contamination by removal of clinical worm burdens later in the grazing season. The model has the potential to contribute to the sustainable management of H. contortus by predicting periods of heightened risk, and hence to focus and optimise limited resources for monitoring and treatment. SIR-type model frameworks are an alternative to classic abundance-based compartmental models of macroparasite epidemiology, and could be useful where incidence data are available. Significant challenges remain, however, in the ability to calibrate such models to field data at spatial and temporal scales that are useful for decision support at farm level.

A simple mathematical model for Ebola in Africa

ABSTRACTWe deal with the following question: Can the consumption of contaminated bush meat, the funeral practices and the environmental contamination explain the recurrence and persistence of Ebola virus disease outbreaks in Africa? We develop an SIR-type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses. We prove that the full model has one (endemic) equilibrium which is locally asymptotically stable whereas, it is globally asymptotically stable in the absence of the Ebola virus shedding in the environment. For the sub-model without the provision of Ebola viruses, the disease dies out or stabilizes globally at an endemic equilibrium. At the endemic level, the number of infectious is larger for the full model than for the sub-model without provision of Ebola viruses. We design a nonstandard finite difference scheme, which preserves the dynamics of the model. Numerical simulations are provided.

On a stochastic epidemic SEIHR model and its diffusion approximation

To model the evolution of diseases with extended latency periods like varicella, we generalise a simple discrete time stochastic SIR-type epidemic model of Tuckwell and Williams. We include both latent periods as well as the presence of quarantine areas, to capture the evolutionary dynamics of such diseases. Analytical results of the proposed Markovian model are provided. The basic reproduction number \(R_0\) of the former and new models as well as a diffusion approximation are derived, leading to a stochastic differential equation with multiple delays in a natural way.

Dynamics of high-risk nonvaccine human papillomavirus types after actual vaccination scheme.

Human papillomavirus (HPV) has been identified as the main etiological factor in the developing of cervical cancer (CC). This finding has propitiated the development of vaccines that help to prevent the HPVs 16 and 18 infection. Both genotypes are associated with 70% of CC worldwide. In the present study, we aimed to determine the emergence of high-risk nonvaccine HPV after actual vaccination scheme to estimate the impact of the current HPV vaccines. A SIR-type model was used to study the HPV dynamics after vaccination. According to the results, our model indicates that the application of the vaccine reduces infection by target or vaccine genotypes as expected. However, numerical simulations of the model suggest the presence of the phenomenon called vaccine—induced pathogen strain replacement. Here, we report the following replacement mechanism: if the effectiveness of cross-protective immunity is not larger than the effectiveness of the vaccine, then the high-risk nonvaccine genotypes emerge. In this scenario, further studies of infection dispersion by HPV are necessary to ascertain the real impact of the current vaccines, primarily because of the different high-risk HPV types that are found in CC.

Smoking epidemic eradication in a eco-epidemiological dynamical model

Smoking is perceived as a major epidemic with regard to mortality. Modelling is a major tool used to obtain insight in the dynamics and possible solutions to decrease or even eradicate this epidemic. Most models on smoking consider the epidemiological context explicitly, in which smoking is regarded as an ‘infectious disease’, in which individuals ‘infect’ each other. However, the population dynamics are often ignored, while these occur at roughly the same timescale as smoking, and hence should explicitly be considered in the modelling of smoking. We present a simple but dynamical eco-epidemiological model. The model formulation consists of a resource-population dynamic part coupled to an epidemiological part resembling a SIR type model for the three compartments: non-smokers, smokers and ex-smokers. The coupling is via birth of non-smokers and death of the three classes with different death rates. The final four-dimensional system of ordinary differential equations are studied using brute force simulations for the short term dynamics and bifurcation analysis for the long-term dynamics. Due to a feed-back mechanism of the two coupling terms there is a codim-two tangent-transcritical bifurcation. This leads to bi-stability of one smoker endemic interior equilibrium and a smoker free boundary equilibrium. Changing parameters beyond the emerging tangent bifurcation leads on the short term to eradicating smoking. We consider the Netherlands in this paper for parametrization, but the modelling approach may be generally applicable.

Modeling and Statistical Analysis of the Spatio-Temporal Patterns of Seasonal Influenza in Israel

BackgroundSeasonal influenza outbreaks are a serious burden for public health worldwide and cause morbidity to millions of people each year. In the temperate zone influenza is predominantly seasonal, with epidemics occurring every winter, but the severity of the outbreaks vary substantially between years. In this study we used a highly detailed database, which gave us both temporal and spatial information of influenza dynamics in Israel in the years 1998–2009. We use a discrete-time stochastic epidemic SIR model to find estimates and credible confidence intervals of key epidemiological parameters.FindingsDespite the biological complexity of the disease we found that a simple SIR-type model can be fitted successfully to the seasonal influenza data. This was true at both the national levels and at the scale of single cities.The effective reproductive number Re varies between the different years both nationally and among Israeli cities. However, we did not find differences in Re between different Israeli cities within a year. R e was positively correlated to the strength of the spatial synchronization in Israel. For those years in which the disease was more “infectious”, then outbreaks in different cities tended to occur with smaller time lags. Our spatial analysis demonstrates that both the timing and the strength of the outbreak within a year are highly synchronized between the Israeli cities. We extend the spatial analysis to demonstrate the existence of high synchrony between Israeli and French influenza outbreaks.ConclusionsThe data analysis combined with mathematical modeling provided a better understanding of the spatio-temporal and synchronization dynamics of influenza in Israel and between Israel and France. Altogether, we show that despite major differences in demography and weather conditions intra-annual influenza epidemics are tightly synchronized in both their timing and magnitude, while they may vary greatly between years. The predominance of a similar main strain of influenza, combined with population mixing serve to enhance local and global influenza synchronization within an influenza season.

Studying the recovery procedure for the time-dependent transmission rate(s) in epidemic models

Determining the time-dependent transmission function that exactly reproduces disease incidence data can yield useful information about disease outbreaks, including a range potential values for the recovery rate of the disease and could offer a method to test the “school year” hypothesis (seasonality) for disease transmission. Recently two procedures have been developed to recover the time-dependent transmission function, β(t), for classical disease models given the disease incidence data. We first review the β(t) recovery procedures and give the resulting formulas, using both methods, for the susceptible-infected-recovered (SIR) and susceptible-exposed-infected-recovered (SEIR) models. We present a modification of one procedure, which is then shown to be identical to the other. Second, we explore several technical issues that appear when implementing the procedure for the SIR model; these are important when generating the time-dependent transmission function for real-world disease data. Third, we extend the recovery method to heterogeneous populations modeled with a certain SIR-type model with multiple time-dependent transmission functions. Finally, we apply the β(t) recovery procedure to data from the 2002–2003 influenza season and for the six seasons from 2002–2003 through 2007–2008, for both one population class and for two age classes. We discuss the consequences of the technical conditions of the procedure applied to the influenza data. We show that the method is robust in the heterogeneous cases, producing comparable results under two different hypotheses. We perform a frequency analysis, which shows a dominant 1-year period for the multi-year influenza transmission function(s).

Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem

The transmission rate of many acute infectious diseases varies significantly in time, but the underlying mechanisms are usually uncertain. They may include seasonal changes in the environment, contact rate, immune system response, etc. The transmission rate has been thought difficult to measure directly. We present a new algorithm to compute the time-dependent transmission rate directly from prevalence data, which makes no assumptions about the number of susceptible or vital rates. The algorithm follows our complete and explicit solution of a mathematical inverse problem for SIR-type transmission models. We prove that almost any infection profile can be perfectly fitted by an SIR model with variable transmission rate. This clearly shows a serious danger of overfitting such transmission models. We illustrate the algorithm with historic UK measles data and our observations support the common belief that measles transmission was predominantly driven by school contacts.

On the infinitesimal dispersion of multivariate Markov counting systems

We provide a multivariate extension of a recent result for univariate Markov counting processes: necessity and sufficiency of compoundness for infinitesimal over-dispersion. As an illustration, we show that infinitesimally over-dispersed epidemiological SIR-type compartment models must rely on compound counting processes.

Age-related Toxoplasma gondii seroprevalence in Dutch wild boar inconsistent with lifelong persistence of antibodies.

Toxoplasma gondii is an important zoonotic pathogen that is best known as a cause of abortion or abnormalities in the newborn after primary infection during pregnancy. Our aim was to determine the prevalence of T. gondii in wild boar to investigate the possible role of their meat in human infection and to get an indication of the environmental contamination with T. gondii. The presence of anti-T. gondii antibodies was determined by in-house ELISA in 509 wild boar shot in 2002/2003 and 464 wild boar shot in 2007. Most of the boar originated from the “Roerstreek” (n = 673) or the “Veluwe” (n = 241). A binormal mixture model was fitted to the log-transformed optical density values for wild boar up to 20 months old to estimate the optimal cut-off value (−0.685) and accompanying sensitivity (90.6%) and specificity (93.6%). The overall seroprevalence was estimated at 24.4% (95% CI: 21.1–27.7%). The prevalence did not show variation between sampling years or regions, indicating a stable and homogeneous infection pressure from the environment. The relation between age and seroprevalence was studied in two stages. Firstly, seroprevalence by age group was determined by fitting the binary mixture model to 200 animals per age category. The prevalence showed a steep increase until approximately 10 months of age but stabilized at approximately 35% thereafter. Secondly, we fitted the age-dependent seroprevalence data to several SIR-type models, with seropositives as infected (I) and seronegatives as either susceptible (S) or resistant (R). A model with a recovery rate (SIS) was superior to a model without a recovery rate (SI). This finding is not consistent with the traditional view of lifelong persistence of T. gondii infections. The high seroprevalence suggests that eating undercooked wild boar meat may pose a risk of infection with T. gondii.

The effect of host movement on viral transmission dynamics in a vector-borne disease system

Many vector-borne pathogens whose primary vectors are generalists, such as Ixodid ticks, can infect a wide range of host species and are often zoonotic. Understanding their transmission dynamics is important for the development of disease management programmes. Models exist to describe the transmission dynamics of such diseases, but are necessarily simplistic and generally limited by knowledge of vector population dynamics. They are typically deterministic SIR-type models, which predict disease dynamics in a single, non-spatial, closed patch. Here we explore the limitations of such a model of louping-ill virus dynamics by challenging it with novel field data. The model was only partially successful in predicting Ixodes ricinus density and louping-ill virus prevalence at 6 Scottish sites. We extend the existing multi-host model by forming a two-patch model, incorporating the impact of roaming hosts. This demonstrates that host movement may account for some of the discrepancies between the original model and empirical data. We conclude that insights into the dynamics of multi-host vector-borne pathogens can be gained by using a simple two-patch model. Potential improvements to the model, incorporating aspects of spatial and temporal heterogeneity, are outlined.

Antigenic distance and cross-immunity, invasibility and coexistence of pathogen strains in an epidemiological model with discrete antigenic space

In models of pathogen interaction and evolution discrete genotypes in the form of bit strings may be mapped to points in a discrete phenotype space based on similarity in antigenic structure. Cross-immunity between strains, that is the reduction in susceptibility to strain A conferred to a host by infection with strain B, can then be defined for pairs of points in the antigenic space by a specified function. Analysis of an SIR type model shows that, if two strains are at equilibrium, the shape of the cross-immunity function has a strong influence on the invasion and coexistence of a third strain and, consequently, the expected evolutionary pathway. A function that is constant except for discontinuities at the end points is expected to result in the accumulation of diversity until a pair of discordant strains occurs that can, depending on parameter values, exclude all other strains. For a function of the form f ( h ) = h q , where h is the antigenic distance between two strains, invasion and coexistence is always possible if q ≤ 1 and little antigenic structure is expected in the pathogen population. However, if q > 1 invasion and coexistence may be impossible, depending on parameter values, and the pathogen population is expected to show significant antigenic structuring. In addition to illuminating the role of cross-immunity in pathogen evolution, this analysis indicates that the choice of cross-immunity function, the representation of immunity acquired from multiple previous infections and the number of elements used to characterize the antigenic space must be carefully considered in the development and interpretation of more sophisticated models of pathogen dynamics and evolution.