Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multi-host pathogen transmission and the disease-diversity relationship

How diseases are transmitted within a multi-host community is a complex biological process with important ecological and societal consequences. The intricacies of interspecific disease transmission determine when a disease can spread to a novel host, including humans (zoonosis), and the severity of emerging epidemics. Interspecific disease transmission also mediates long-term disease prevalence within a multi-host community which is at the core of the disease-diversity relationship. Mathematical models play a central role in formulating predictions about spillover, prevalence, and the disease-diversity relationship. Yet, how the complexity of transmission is captured (or not) by the assumptions of these models is often unclear. Here, we decompose the transmission process into five biological stages using bovine tuberculosis (bTB) as an illustrative example of transmission in a multi-host system. We then examine the often-implicit assumptions that classic compartmental models make about this process. We use the intuition gained from this decomposition to formulate hypotheses for how transmission can mediate outbreak potential, infection prevalence, and the amplifying or diluting effects of host diversity on disease prevalence. We further illustrate the key principles and implications of transmission with a diverse array of examples of multi-host pathogens. Throughout we emphasise the role of evolution in shaping interspecific transmission, from the evolutionary relatedness of the hosts themselves to the adaptation of the pathogen to novel hosts.

Evolution of pathogen tolerance and reproductive trade-off implications

We develop an epidemic model that accounts explicitly for the pathogen pool and incorporates population variations in host defense strategy, measured in disease tolerance that is assumed to be perfectly inherited by offspring. Although the proposed model is more general, it is motivated by the devastating Batrachochytrium dendrobatidis (Bd) fungus that is responsible for severe declines in amphibians. We show that the model's basic reproduction number consists of a weighted average of individual basic reproduction numbers associated to each tolerance class. If the individual basic reproduction number associated to the highest tolerance level is less than one, then any solution converges to a (non-unique) disease-free equilibrium. We show that in the absence of a trade-off, different host defense strategies can coexist as long as the disease will go extinct eventually. In contrast, if the disease persists, the set of pandemic equilibria consists of isolated vertex equilibria, implying the survival of an individual host defense strategy. The pandemic equilibrium corresponding to the highest tolerance, i.e., lowest disease-induced death rate is the only asymptotically stable pandemic equilibrium. Additionally, to investigate the impact of a trade-off, we incorporate a tolerance cost in reproduction, whereby a higher tolerance comes at the expense of a lower reproductive rate. Now, the coexistence of host defense strategies in the absence of the disease is no longer supported. However, the set of pandemic equilibria increases in richness to contain equilibria where different tolerance classes are present.

Using mathematical modeling to study the dynamics of Legionnaires' disease and consider management options

Legionnaires' disease (LD) is a largely understudied and underreported pneumonic environmentally transmitted disease caused by the bacteria Legionella. It primarily occurs in places with poorly maintained artificial sources of water. There is currently a lack of mathematical models on the dynamics of LD. In this paper, we formulate a novel ordinary differential equation-based susceptible-exposed-infected-recovered (SEIR) model for LD. One issue with LD is the difficulty in its detection, as the majority of countries around the world lack the proper surveillance and diagnosis methods. Thus, there is not much publicly available data or literature on LD. We use parameter estimation for our model with one of the few outbreaks with time series data from Murcia, Spain in 2001. Furthermore, we apply a global sensitivity analysis to understand the contributions of parameters to our model output. To consider managing LD outbreaks, we explore implementing sanitizing individual sources of water by constructing an optimal control problem. Using our fitted model and the optimal control problem, we analyze how different parameters and controls might help manage LD outbreaks in the future.

A fractional order model for the transmission dynamics of shigellosis

Shigellosis, a highly contagious bacterial infection causing diarrhea, fever, and abdominal pain, necessitates a deep understanding of its transmission dynamics to devise effective control measures. Our study takes a novel approach, employing a fractional order framework to explore the influence of memory and control measures on Shigellosis transmission dynamics, thereby making a unique contribution to the field. The model is presented as a system of Caputo fractional differential equations capturing time constant controls. The Caputo derivatives are chosen for their inherent benefits. The qualitative features of the model, such as the solutions' existence and uniqueness, positivity, and boundedness, are thoroughly investigated. Moreover, the equilibria of the model are derived and analyzed for their stability using suitable theorems. In particular, local stability was proved through Routh's criteria, while global stability results were established in the Ulam-Hyers sense. The model is then solved numerically with the help of the predict-evaluate-correct-evaluate method of Adams-Bashforth-Moulton. The numerical results underscore the significant impact of memory on disease evolution, highlighting the novelty of integrating memory-related aspects into the meticulous planning of effective disease control strategies. Using fractional-order derivatives is more beneficial for understanding the dynamics of Shigellosis transmission than integral-order models. The advantage of fractional derivatives is their ability to provide numerous degrees of freedom, allowing for a broader range of analysis of the system's dynamic behaviour, including nonlocal solutions. Also, an investigation on the impacts of control measures via parameter variation is done; the findings show that applying treatment and sanitation minimizes disease eruption.

Global analysis of a diffusive Cholera model with multiple transmission pathways, general incidence and incomplete immunity in a heterogeneous environment

With the consideration of the complexity of the transmission of Cholera, a partially degenerated reaction-diffusion model with multiple transmission pathways, incorporating the spatial heterogeneity, general incidence, incomplete immunity, and Holling type Ⅱ treatment was proposed. First, the existence, boundedness, uniqueness, and global attractiveness of solutions for this model were investigated. Second, one obtained the threshold condition $ \mathcal{R}_{0} $ and gave its expression, which described global asymptotic stability of disease-free steady state when $ \mathcal{R}_{0} < 1 $, as well as the maximum treatment rate as zero. Further, we obtained the disease was uniformly persistent when $ \mathcal{R}_{0} > 1 $. Moreover, one used the mortality due to disease as a branching parameter for the steady state, and the results showed that the model undergoes a forward bifurcation at $ \mathcal{R}_{0} $ and completely excludes the presence of endemic steady state when $ \mathcal{R}_{0} < 1 $. Finally, the theoretical results were explained through examples of numerical simulations.

Mathematical modeling of cholera dynamics with intrinsic growth considering constant interventions

A mathematical model that describes the dynamics of bacterium vibrio cholera within a fixed population considering intrinsic bacteria growth, therapeutic treatment, sanitation and vaccination rates is developed. The developed mathematical model is validated against real cholera data. A sensitivity analysis of some of the model parameters is also conducted. The intervention rates are found to be very important parameters in reducing the values of the basic reproduction number. The existence and stability of equilibrium solutions to the mathematical model are also carried out using analytical methods. The effect of some model parameters on the stability of equilibrium solutions, number of infected individuals, number of susceptible individuals and bacteria density is rigorously analyzed. One very important finding of this research work is that keeping the vaccination rate fixed and varying the treatment and sanitation rates provide a rapid decline of infection. The fourth order Runge-Kutta numerical scheme is implemented in MATLAB to generate the numerical solutions.

Climate-dependent effectiveness of nonpharmaceutical interventions on COVID-19 mitigation

Environmental factors have a significant impact on the transmission of infectious diseases. Existing results show that the novel coronavirus can persist outside the host. We propose a susceptible-exposed-presymptomatic-infectious-asymptomatic-recovered-susceptible (SEPIARS) model with a vaccination compartment and indirect incidence to explore the effect of environmental conditions, temperature and humidity, on the transmission of the SARS-CoV-2 virus. Using climate data and daily confirmed cases data in two Canadian cities with different atmospheric conditions, we evaluate the mortality rates of the SARS-CoV-2 virus and further estimate the transmission rates by the inverse method, respectively. The numerical results show that high temperature or humidity can be helpful in mitigating the spread of COVID-19 during the warm summer months. Our findings verify that nonpharmaceutical interventions are less effective if the virus can persist for a long time on surfaces. Based on climate data, we can forecast the transmission rate and the infection cases up to four weeks in the future by a generalized boosting machine learning model.

Edge-based compartmental modeling for the spread of cholera on random networks: A case study in Somalia

Cholera remains a major public health problem that threatens human health worldwide and its severity is continuing. In this paper, an edge-based model for cholera transmission on random networks is proposed and investigated. The model assumes that two communities share a common water source and includes three transmission routes, namely intra- and inter-community human-to-human transmission as well as water-to-human transmission. Intra-community human-to-human contacts are modeled through a random contact network, while both inter-community and water-to-human transmission are modeled through external nodes that reach each individual in the network to the same extent. The basic reproduction number and the equations of the final epidemic size are obtained. In addition, our study considers the cholera situation in Banadir, which is one of the most severely infected regions in Somalia, during the period (2019-2021). According to the geographical location, two adjacent districts are selected and our model fits well with the real data on the monthly cumulative cholera cases of these two districts during the above-mentioned period. From the perspective of network topology, cutting off high-risk contacts by supervising, isolating, quarantining and closing places with high-degree cholera-infected individuals to reduce degree heterogeneity is an effective measure to control cholera transmission. Our findings might offer some useful insights on cholera control.

The effect of governance structures on optimal control of two-patch epidemic models

Infectious diseases continue to pose a significant threat to the health of humans globally. While the spread of pathogens transcends geographical boundaries, the management of infectious diseases typically occurs within distinct spatial units, determined by geopolitical boundaries. The allocation of management resources within and across regions (the "governance structure") can affect epidemiological outcomes considerably, and policy-makers are often confronted with a choice between applying control measures uniformly or differentially across regions. Here, we investigate the extent to which uniform and non-uniform governance structures affect the costs of an infectious disease outbreak in two-patch systems using an optimal control framework. A uniform policy implements control measures with the same time varying rate functions across both patches, while these measures are allowed to differ between the patches in a non-uniform policy. We compare results from two systems of differential equations representing transmission of cholera and Ebola, respectively, to understand the interplay between transmission mode, governance structure and the optimal control of outbreaks. In our case studies, the governance structure has a meaningful impact on the allocation of resources and burden of cases, although the difference in total costs is minimal. Understanding how governance structure affects both the optimal control functions and epidemiological outcomes is crucial for the effective management of infectious diseases going forward.

Rich dynamics of a bidirectionally linked immuno-epidemiological model for cholera

Cholera is an environmentally driven disease where the human hosts both ingest the pathogen from polluted environment and shed the pathogen to the environment, generating a two-way feedback cycle. In this paper, we propose a bidirectionally linked immuno-epidemiological model to study the interaction of within- and between-host cholera dynamics. We conduct a rigorous analysis for this multiscale model, with a focus on the stability and bifurcation properties of each feasible equilibrium. We find that the parameter that represents the bidirectional connection is a key factor in shaping the rich dynamics of the system, including the occurrence of the backward bifurcation and Hopf bifurcation. Numerical results illustrate a practical application of our model and add new insight into the prevention and intervention of cholera epidemics.

An immuno-epidemiological model linking between-host and within-host dynamics of cholera

Cholera, a severe gastrointestinal infection caused by the bacterium Vibrio cholerae, remains a major threat to public health, with a yearly estimated global burden of 2.9 million cases. Although most existing models for the disease focus on its population dynamics, the disease evolves from within-host processes to the population, making it imperative to link the multiple scales of the disease to gain better perspectives on its spread and control. In this study, we propose an immuno-epidemiological model that links the between-host and within-host dynamics of cholera. The immunological (within-host) model depicts the interaction of the cholera pathogen with the adaptive immune response. We distinguish pathogen dynamics from immune response dynamics by assigning different time scales. Through a time-scale analysis, we characterise a single infected person by their immune response. Contrary to other within-host models, this modelling approach allows for recovery through pathogen clearance after a finite time. Then, we scale up the dynamics of the infected person to construct an epidemic model, where the infected population is structured by individual immunological dynamics. We derive the basic reproduction number ($ \mathcal{R}_0 $) and analyse the stability of the equilibrium points. At the disease-free equilibrium, the disease will either be eradicated if $ \mathcal{R}_0 < 1 $ or otherwise persists. A unique endemic equilibrium exists when $ \mathcal{R}_0 > 1 $ and is locally asymptotically stable without a loss of immunity.

Heterogeneity in network structure switches the dominant transmission mode of infectious diseases

Several recent emerging diseases have exhibited both sexual and nonsexual transmission modes (Ebola, Zika, and mpox). In the recent mpox outbreaks, transmission through sexual contacts appears to be the dominant mode of transmission. Motivated by this, we use an SIR-like model to argue that an initially dominant sexual transmission mode can be overtaken by casual transmission at later stages, even if the basic casual reproduction number is less than one. Our results highlight the risk of intervention designs which are informed only by the early dynamics of the disease.

Computation of the basic reproduction numbers for reaction-diffusion epidemic models

We consider a class of $ k $-dimensional reaction-diffusion epidemic models ($ k = 1, 2, \cdots $) that are developed from autonomous ODE systems. We present a computational approach for the calculation and analysis of their basic reproduction numbers. Particularly, we apply matrix theory to study the relationship between the basic reproduction numbers of the PDE models and those of their underlying ODE models. We show that the basic reproduction numbers are the same for these PDE models and their associated ODE models in several important scenarios. We additionally provide two numerical examples to verify our analytical results.

Recent Advances in Triboelectric Nanogenerators: From Technological Progress to Commercial Applications

Serious climate changes and energy-related environmental problems are currently critical issues in the world. In order to reduce carbon emissions and save our environment, renewable energy harvesting technologies will serve as a key solution in the near future. Among them, triboelectric nanogenerators (TENGs), which is one of the most promising mechanical energy harvesters by means of contact electrification phenomenon, are explosively developing due to abundant wasting mechanical energy sources and a number of superior advantages in a wide availability and selection of materials, relatively simple device configurations, and low-cost processing. Significant experimental and theoretical efforts have been achieved toward understanding fundamental behaviors and a wide range of demonstrations since its report in 2012. As a result, considerable technological advancement has been exhibited and it advances the timeline of achievement in the proposed roadmap. Now, the technology has reached the stage of prototype development with verification of performance beyond the lab scale environment toward its commercialization. In this review, distinguished authors in the world worked together to summarize the state of the art in theory, materials, devices, systems, circuits, and applications in TENG fields. The great research achievements of researchers in this field around the world over the past decade are expected to play a major role in coming to fruition of unexpectedly accelerated technological advances over the next decade.

Wastewater surveillance beyond COVID-19: a ranking system for communicable disease testing in the tri-county Detroit area, Michigan, USA

Introduction

Throughout the coronavirus disease 2019 (COVID-19) pandemic, wastewater surveillance has been utilized to monitor the disease in the United States through routine national, statewide, and regional monitoring projects. A significant canon of evidence was produced showing that wastewater surveillance is a credible and effective tool for disease monitoring. Hence, the application of wastewater surveillance can extend beyond monitoring SARS-CoV-2 to encompass a diverse range of emerging diseases. This article proposed a ranking system for prioritizing reportable communicable diseases (CDs) in the Tri-County Detroit Area (TCDA), Michigan, for future wastewater surveillance applications at the Great Lakes Water Authority's Water Reclamation Plant (GLWA's WRP).

Methods

The comprehensive CD wastewater surveillance ranking system (CDWSRank) was developed based on 6 binary and 6 quantitative parameters. The final ranking scores of CDs were computed by summing the multiplication products of weighting factors for each parameter, and then were sorted based on decreasing priority. Disease incidence data from 2014 to 2021 were collected for the TCDA. Disease incidence trends in the TCDA were endowed with higher weights, prioritizing the TCDA over the state of Michigan.

Results

Disparities in incidences of CDs were identified between the TCDA and state of Michigan, indicating epidemiological differences. Among 96 ranked CDs, some top ranked CDs did not present relatively high incidences but were prioritized, suggesting that such CDs require significant attention by wastewater surveillance practitioners, despite their relatively low incidences in the geographic area of interest. Appropriate wastewater sample concentration methods are summarized for the application of wastewater surveillance as per viral, bacterial, parasitic, and fungal pathogens.

Discussion

The CDWSRank system is one of the first of its kind to provide an empirical approach to prioritize CDs for wastewater surveillance, specifically in geographies served by centralized wastewater collection in the area of interest. The CDWSRank system provides a methodological tool and critical information that can help public health officials and policymakers allocate resources. It can be used to prioritize disease surveillance efforts and ensure that public health interventions are targeted at the most potentially urgent threats. The CDWSRank system can be easily adopted to geographical locations beyond the TCDA.

Analysis of an environmental epidemic model based on voluntary vaccination policy

With the worsening of the environment and the increasing international trade, indirect transmission from exposure to contaminants in the surrounding environment has become an overlooked mode of transmission. This paper proposes a new game-theoretic model considering voluntary vaccination against imperfection and the unique integration of human-to-human and virus-to-human transmission routes. Based on the individual-based risk assessment update rule (IB-RA), the strategy-based risk assessment update rule (SB-RA), and the direct commitment update rule (DC), the different effects of individuals' behaviors on disease prevalence are analyzed. To find the effect of indirect transmission on epidemic transmission, we compare our model with the traditional SVIR model. Finally, it can be seen that indirect transmission mechanisms will aggravate the spread of epidemics.

Reframing Optimal Control Problems for Infectious Disease Management in Low-Income Countries

Optimal control theory can be a useful tool to identify the best strategies for the management of infectious diseases. In most of the applications to disease control with ordinary differential equations, the objective functional to be optimized is formulated in monetary terms as the sum of intervention costs and the cost associated with the burden of disease. We present alternate formulations that express epidemiological outcomes via health metrics and reframe the problem to include features such as budget constraints and epidemiological targets. These alternate formulations are illustrated with a compartmental cholera model. The alternate formulations permit us to better explore the sensitivity of the optimal control solutions to changes in available budget or the desired epidemiological target. We also discuss some limitations of comprehensive cost assessment in epidemiology.

Dynamics of a diffusion epidemic SIRI system in heterogeneous environment

This paper studies the dynamical behaviors of a diffusion epidemic SIRI system with distinct dispersal rates. The overall solution of the system is derived by using L p theory and the Young's inequality. The uniformly boundedness of the solution is obtained for the system. The asymptotic smoothness of the semi-flow and the existence of the global attractor are discussed. Moreover, the basic reproduction number is defined in a spatially uniform environment and the threshold dynamical behaviors are obtained for extinction or continuous persistence of disease. When the spread rate of the susceptible individuals or the infected individuals is close to zero, the asymptotic profiles of the system are studied. This can help us to better understand the dynamic characteristics of the model in a bounded space domain with zero flux boundary conditions.

A fractional SVIR-B epidemic model for Cholera with imperfect vaccination and saturated treatment

Given the limited medical resources in most cholera endemic countries, in this paper, a nonlinear fractional SVIR-B (Susceptible-Vaccinated-Infected-Recovered, Bacterial) cholera epidemic model with imperfect vaccination and saturated treatment is proposed and investigated. The model performs well-posed in both epidemiologically and mathematically, especially, we demonstrate the positivity and boundedness of all solutions using the generalized mean value theorem. The control reproduction number R vt is derived using the next generation matrix method and both local and global stability analyses for the disease-free equilibrium is performed by analyzing the characteristic equation and using Lyapunov functional. Then, the existence and stability of endemic equilibria are further addressed. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also derived. Subsequently, an optimal control problem with vaccination, media coverage, treatment, and sanitation as control strategies is proposed and analyzed, providing a rationale for cholera control and prevention. In addition, several numerical examples are employed to illustrate our theoretical results.

Leveraging infectious disease models to interpret randomized controlled trials: Controlling enteric pathogen transmission through water, sanitation, and hygiene interventions

Randomized controlled trials (RCTs) evaluate hypotheses in specific contexts and are often considered the gold standard of evidence for infectious disease interventions, but their results cannot immediately generalize to other contexts (e.g., different populations, interventions, or disease burdens). Mechanistic models are one approach to generalizing findings between contexts, but infectious disease transmission models (IDTMs) are not immediately suited for analyzing RCTs, since they often rely on time-series surveillance data. We developed an IDTM framework to explain relative risk outcomes of an infectious disease RCT and applied it to a water, sanitation, and hygiene (WASH) RCT. This model can generalize the RCT results to other contexts and conditions. We developed this compartmental IDTM framework to account for key WASH RCT factors: i) transmission across multiple environmental pathways, ii) multiple interventions applied individually and in combination, iii) adherence to interventions or preexisting conditions, and iv) the impact of individuals not enrolled in the study. We employed a hybrid sampling and estimation framework to obtain posterior estimates of mechanistic parameter sets consistent with empirical outcomes. We illustrated our model using WASH Benefits Bangladesh RCT data (n = 17,187). Our model reproduced reported diarrheal prevalence in this RCT. The baseline estimate of the basic reproduction number R 0 for the control arm (1.10, 95% CrI: 1.07, 1.16) corresponded to an endemic prevalence of 9.5% (95% CrI: 7.4, 13.7%) in the absence of interventions or preexisting WASH conditions. No single pathway was likely able to sustain transmission: pathway-specific R 0 s for water, fomites, and all other pathways were 0.42 (95% CrI: 0.03, 0.97), 0.20 (95% CrI: 0.02, 0.59), and 0.48 (95% CrI: 0.02, 0.94), respectively. An IDTM approach to evaluating RCTs can complement RCT analysis by providing a rigorous framework for generating data-driven hypotheses that explain trial findings, particularly unexpected null results, opening up existing data to deeper epidemiological understanding.

Host movement, transmission hot spots, and vector-borne disease dynamics on spatial networks

We examine how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics. Specifically, we consider a Ross-Macdonald-type disease model on n spatial locations that are coupled by host movement on a strongly connected, weighted, directed graph. We derive a closed form approximation to the domain reproduction number using a Laurent series expansion, and use this approximation to compute sensitivities of the basic reproduction number to model parameters. To illustrate how these results can be used to help inform mitigation strategies, as a case study we apply these results to malaria dynamics in Namibia, using published cell phone data and estimates for local disease transmission. Our analytical results are particularly useful for understanding drivers of transmission when mobility sinks and transmission hot spots do not coincide.

Mathematical Models for Cholera Dynamics-A Review

Cholera remains a significant public health burden in many countries and regions of the world, highlighting the need for a deeper understanding of the mechanisms associated with its transmission, spread, and control. Mathematical modeling offers a valuable research tool to investigate cholera dynamics and explore effective intervention strategies. In this article, we provide a review of the current state in the modeling studies of cholera. Starting from an introduction of basic cholera transmission models and their applications, we survey model extensions in several directions that include spatial and temporal heterogeneities, effects of disease control, impacts of human behavior, and multi-scale infection dynamics. We discuss some challenges and opportunities for future modeling efforts on cholera dynamics, and emphasize the importance of collaborations between different modeling groups and different disciplines in advancing this research area.

COVID-19 outbreak: a predictive mathematical study incorporating shedding effect

In this paper, a modified SEIR epidemic model incorporating shedding effect is proposed to analyze transmission dynamics of the COVID-19 virus among different individuals' classes. The direct impact of pathogen concentration over susceptible populations through the shedding of COVID-19 virus into the environment is investigated. Moreover, the threshold value of shedding parameters is computed which gives information about their significance in decreasing the impact of the disease. The basic reproduction number ( R 0 ) is calculated using the next-generation matrix method, taking shedding as a new infection. In the absence of disease, the condition for the equilibrium point to be locally and globally asymptotically stable withR 0 < 1 are established. It has been shown that the unique endemic equilibrium point is globally asymptotically stable under the conditionR 0 > 1 . Bifurcation theory and center manifold theorem imply that the system exhibit backward bifurcation atR 0 = 1 . The sensitivity indices of R 0 are computed to investigate the robustness of model parameters. The numerical simulation is demonstrated to illustrate the results.

The backward bifurcation of an age-structured cholera transmission model with saturation incidence

In this paper, we consider an age-structured cholera model with saturation incidence, vaccination age of vaccinated individuals, infection age of infected individuals, and biological age of pathogens. First, the basic reproduction number is calculated. When the basic reproduction number is less than one, the disease-free equilibrium is locally stable. Further, the existence of backward bifurcation of the model is obtained. Numerically, we also compared the effects of various control measures, including basic control measures and vaccination, on the number of infected individuals.

Simulation applications to support teaching and research in epidemiological dynamics

BACKGROUND

An understanding of epidemiological dynamics, once confined to mathematical epidemiologists and applied mathematicians, can be disseminated to a non-mathematical community of health care professionals and applied biologists through simple-to-use simulation applications. We used Numerus Model Builder RAMP ^Ⓡ (Runtime Alterable Model Platform) technology, to construct deterministic and stochastic versions of compartmental SIR (Susceptible, Infectious, Recovered with immunity) models as simple-to-use, freely available, epidemic simulation application programs.

RESULTS

We take the reader through simulations used to demonstrate the following concepts: 1) disease prevalence curves of unmitigated outbreaks have a single peak and result in epidemics that 'burn' through the population to become extinguished when the proportion of the susceptible population drops below a critical level; 2) if immunity in recovered individuals wanes sufficiently fast then the disease persists indefinitely as an endemic state, with possible dampening oscillations following the initial outbreak phase; 3) the steepness and initial peak of the prevalence curve are influenced by the basic reproductive value R₀, which must exceed 1 for an epidemic to occur; 4) the probability that a single infectious individual in a closed population (i.e. no migration) gives rise to an epidemic increases with the value of R₀>1; 5) behavior that adaptively decreases the contact rate among individuals with increasing prevalence has major effects on the prevalence curve including dramatic flattening of the prevalence curve along with the generation of multiple prevalence peaks; 6) the impacts of treatment are complicated to model because they effect multiple processes including transmission, recovery and mortality; 7) the impacts of vaccination policies, constrained by a fixed number of vaccination regimens and by the rate and timing of delivery, are crucially important to maximizing the ability of vaccination programs to reduce mortality.

CONCLUSION

Our presentation makes transparent the key assumptions underlying SIR epidemic models. Our RAMP simulators are meant to augment rather than replace classroom material when teaching epidemiological dynamics. They are sufficiently versatile to be used by students to address a range of research questions for term papers and even dissertations.

Timescale reverses the relationship between host density and infection risk

Host density shapes infection risk through two opposing phenomena. First, when infective stages are subdivided among multiple hosts, greater host densities decrease infection risk through 'safety in numbers'. Hosts, however, represent resources for parasites, and greater host availability also fuels parasite reproduction. Hence, host density increases infection risk through 'density-dependent transmission'. Theory proposes that these phenomena are not disparate outcomes but occur over different timescales. That is, higher host densities may reduce short-term infection risk, but because they support parasite reproduction, may increase long-term risk. We tested this theory in a zooplankton-disease system with laboratory experiments and field observations. Supporting theory, we found that negative density-risk relationships (safety in numbers) sometimes emerged over short timescales, but these relationships reversed to 'density-dependent transmission' within two generations. By allowing parasite numerical responses to play out, time can shift the consequences of host density, from reduced immediate risk to amplified future risk.

Why the Spectral Radius? An intuition-building introduction to the basic reproduction number

The basic reproduction number [Formula: see text] is a fundamental concept in mathematical epidemiology and infectious disease modeling. Loosely speaking, it describes the number of people that an infectious person is expected to infect. The basic reproduction number has profound implications for epidemic trajectories and disease control strategies. It is well known that the basic reproduction number can be calculated as the spectral radius of the next generation matrix, but why this is the case may not be intuitively obvious. Here, we walk through how the discrete, next generation process connects to the ordinary differential equation disease system of interest, linearized at the disease-free equilibrium. Then, we use linear algebra to develop a geometric explanation of why the spectral radius of the next generation matrix is an epidemic threshold. Finally, we work through a series of examples that help to build familiarity with the kinds of patterns that arise in parameter combinations produced by the next generation method. This article is intended to help new infectious disease modelers develop intuition for the form and interpretation of the basic reproduction number in their disease systems of interest.

Shigellosis seasonality and transmission characteristics in different areas of China: A modelling study

Objective

In China, the burden of shigellosis is unevenly distributed, notably across various ages and geographical areas. Shigellosis temporal trends appear to be seasonal. We should clarify seasonal warnings and regional transmission patterns.

Method

This study adopted a Logistic model to assess the seasonality and a dynamics model to compare the transmission in different areas. The next-generation matrix was used to calculate the effective reproduction number (R eff) to quantify the transmissibility.

Results

In China, the rate of shigellosis fell from 35.12 cases per 100,000 people in 2005 to 7.85 cases per 100,000 people in 2017, peaking in June and August. After simulation by the Logistic model, the 'peak time' is mainly concentrated from mid-June to mid-July. China's 'early warning time' is primarily focused on from April to May. We predict the 'peak time' of shigellosis is the 6.30th month and the 'early warning time' is 3.87th month in 2021. According to the dynamics model results, the water/food transfer pathway has been mostly blocked off. The transmissibility of different regions varies greatly, such as the mean R eff of Longde County (3.76) is higher than Xiamen City (3.15), higher than Chuxiong City (2.52), and higher than Yichang City (1.70).

Conclusion

The 'early warning time' for shigellosis in China is from April to May every year, and it may continue to advance in the future, such as the early warning time in 2021 is in mid-March. Furthermore, we should focus on preventing and controlling the person-to-person route of shigellosis and stratified deploy prevention and control measures according to the regional transmission.

Hydrochemical characteristics of surface waters and their relationships to the Kashin-Beck Disease in Longzi County, Tibet

Although previous studies have been reported between the Kashin-Beck Disease (KBD) epidemic and the hydrochemical characteristics of surface waters, the etiology of the disease remains unclear. In the present study, we comprehensively investigated the relationship between the KBD and the hydrochemical characteristics of surface waters in Longzi County. Results show that, the pH (mean = 7.27 ± 0.30), total hardness (TH, mean = 57.08 ± 45.74 mg L-1), total dissolved solids (TDS, mean = 67.56 ± 44.00 mg L-1) and oxidation-reduction potential (ORP, mean = 84.11 ± 23.55 mV) of surface waters in KBD endemic areas are lower than those in the non-KBD endemic areas (means of pH = 7.49 ± 0.30; TH = 262.06 ± 123.29 mg L-1; TDS = 253.25 ± 100.39 mg L-1; ORP = 215.90 ± 55.99 mV). These results suggest that long-term consumption of low TDS, essential trace elements (e.g., nickel, cobalt, iron, selenium, zinc, molybdenum, and iodine) deficient, and potential toxic elements (e.g., arsenic) enriched waters by humans likely causes the KBD. Environmental factors such as the geology and geomorphology may produce biogeochemical imbalance, geomorphic, vegetation types and local climatic conditions may have significant impact on food fungi toxin poisoning and water organic compound poisoning, and these also impact the KBD occurrence.

A reaction-advection-diffusion model of cholera epidemics with seasonality and human behavior change

Cholera is a water- and food-borne infectious disease caused by V. cholerae. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-advection-diffusion model that incorporates human hosts and aquatic reservoir of V. cholerae. We derive the basic reproduction number [Formula: see text] for this system and then establish a threshold type result on its global dynamics in terms of [Formula: see text]. Further, we show that the bacterial loss at the downstream end of the river due to water flux can reduce the disease risk, and describe the asymptotic behavior of [Formula: see text] for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread.

Dilution of Epidemic Potential of Environmentally Transmitted Infectious Diseases for Species with Partially Overlapping Habitats

AbstractSpecies diversity may play an important role in the modulation of pathogen transmission through the dilution effect. Infectious disease models can help elucidate mechanisms that may underlie this effect. While many modeling studies have assumed direct host-to-host transmission, many pathogens are transmitted through the environment. We present a mathematical modeling analysis exploring conditions under which we observe the dilution effect in systems with environmental transmission where host species interact through fully or partially overlapping habitats. We measure the strength of the dilution effect by the relative decrease in the basic reproduction number of two-species assemblages compared with that of a focal host species. We find that a dilution effect is most likely when the pathogen is environmentally persistent (frequency-dependent-like transmission). The magnitude of this effect is strongest when the species with the greater epidemic potential is relatively slow to pick up pathogens in the environment (density-dependent transmission) and the species with the lesser epidemic potential is efficient at picking up pathogens (frequency-dependent transmission). These findings suggest that measurable factors, including pathogen persistence and the host's relative efficiency of pathogen pickup, can guide predictions of when biodiversity might lead to a dilution effect and may thus give concrete direction to future ecological work.

Cryptocurrency as Epidemiologically Safe Means of Transactions: Diminishing Risk of SARS-CoV-2 Spread

In comparison with other respiratory viruses, the current COVID-19 pandemic’s rapid seizing the world can be attributed to indirect (contact) way of transmission of SARS-CoV-2 virus in addition to the regular airborne way. A significant part of indirect transmission is made through cash bank notes. SARS-CoV-2 remains on cash paper money for period around four times larger than influenza A virus and is absorbed by cash notes two and a half times more effectively than influenza A (our model). During the pandemic, cryptocurrencies have gained attractiveness as an “epidemiologically safe” means of transactions. On the basis of the authors’ gallop polls performed online with social networks users in 44 countries in 2020–2021 (the total number of clear responses after the set repair 32,115), around 14.7% of surveyed participants engaged in cryptocurrency-based transactions during the pandemic. This may be one of the reasons of significant rise of cryptocurrencies rates since mid-March 2020 till the end of 2021. The paper discusses the reasons for cryptocurrency attractiveness during the COVID-19 pandemic. Among them, there are fear of SARS-CoV-2 spread via cash contacts and the ability of the general population to mine cryptocurrencies. The article also provides a breakdown of the polled audience profile to determine the nationalities that have maximal level of trust to saving and transacting money as cryptocurrencies.

When and why direct transmission models can be used for environmentally persistent pathogens

Variants of the susceptible-infected-removed (SIR) model of Kermack & McKendrick (1927) enjoy wide application in epidemiology, offering simple yet powerful inferential and predictive tools in the study of diverse infectious diseases across human, animal and plant populations. Direct transmission models (DTM) are a subset of these that treat the processes of disease transmission as comprising a series of discrete instantaneous events. Infections transmitted indirectly by persistent environmental pathogens, however, are examples where a DTM description might fail and are perhaps better described by models that comprise explicit environmental transmission routes, so-called environmental transmission models (ETM). In this paper we discuss the stochastic susceptible-exposed-infected-removed (SEIR) DTM and susceptible-exposed-infected-removed-pathogen (SEIR-P) ETM and we show that the former is the timescale separation limit of the latter, with ETM host-disease dynamics increasingly resembling those of a DTM when the pathogen’s characteristic timescale is shortened, relative to that of the host population. Using graphical posterior predictive checks (GPPC), we investigate the validity of the SEIR model when fitted to simulated SEIR-P host infection and removal times. Such analyses demonstrate how, in many cases, the SEIR model is robust to departure from direct transmission. Finally, we present a case study of white spot disease (WSD) in penaeid shrimp with rates of environmental transmission and pathogen decay (SEIR-P model parameters) estimated using published results of experiments. Using SEIR and SEIR-P simulations of a hypothetical WSD outbreak management scenario, we demonstrate how relative shortening of the pathogen timescale comes about in practice. With atttempts to remove diseased shrimp from the population every 24h, we see SEIR and SEIR-P model outputs closely conincide. However, when removals are 6-hourly, the two models’ mean outputs diverge, with distinct predictions of outbreak size and duration.

Mathematical models of the spread and progression of communicable disease in populations are important tools in efforts to prevent and control outbreaks. A common class of disease models assume that infection is transmitted directly from infectious to susceptible individuals when they are in close proximity—so called direct transmission models. These are used widely and have proven invaluable as simplified descriptions of a wide array of infectious diseases in diverse populations. However, many pathogens spread through indirect, environmental routes of transmission, for example via contact with contaminated water sources in the case of cholera, or inhalation of infectious airborne droplets for respiratory infections, such as Covid-19.

We show that direct transmission models work well for such pathogens with short environmental lifetimes and where hosts shed pathogens into the environment at high rates. This means that we do not require information about environmental pathogen levels to understand the behaviour of outbreaks caused by these pathogens. When shedding rates are also low, e.g., with macroparasitic infections, or when variable environmental factors play a role in transmissibility, then explicit modelling of both the pathogen and environmental transmission will provide a more accurate picture than a direct transmission approximation.

A discrete-time risk-structured model of cholera infections in Cameroon

In a recent paper, Che et al. [5] used a continuous-time Ordinary Differential Equation (ODE) model with risk structure to study cholera infections in Cameroon. However, the population and the reported cholera cases in Cameroon are censored at discrete-time annual intervals. In this paper, unlike in [5], we introduce a discrete-time risk-structured cholera model with no spatial structure. We use our discrete-time demographic equation to 'fit' the annual population of Cameroon. Furthermore, we use our fitted discrete-time model to capture the annually reported cholera cases from 1987 to 2004 and to study the impact of vaccination, treatment and improved sanitation on the number of cholera infections from 2004 to 2019. Our discrete-time cholera model confirms the results of the ODE model in [5]. However, our discrete-time model predicts a decrease in the number of cholera cases in a shorter period of cholera intervention (2004-2019) as compared to the ODE model's period of intervention (2004-2022).

Mathematical modeling of the influence of cultural practices on cholera infections in Cameroon

The Far North Region of Cameroon, a high risk cholera endemic region, has been experiencing serious and recurrent cholera outbreaks in recent years. Cholera outbreaks in this region are associated with cultural practices (traditional and religious beliefs). In this paper, we introduce a mathematical model of the influence of cultural practices on the dynamics of cholera in the Far North Region. Our model is an SEIR type model with a pathogen class and multiple susceptible classes based on traditional and religious beliefs. Using daily reported cholera cases from three health districts (Kaélé, Kar Hay and Moutourwa) in the Far North Region from June 25, 2019 to August 16, 2019, we estimate parameter values of our model and use Akaike information criterion (AIC) to demonstrate that our model gives a good fit for our data on cholera cases. We use sensitivity analysis to study the impact of each model parameter on the threshold parameter (control reproduction number), R_c, and the number of model predicted cholera cases. Finally, we investigate the effect of cultural practices on the number of cholera cases in the region.

Threshold dynamics of a nonlocal and delayed cholera model in a spatially heterogeneous environment

A nonlocal and delayed cholera model with two transmission mechanisms in a spatially heterogeneous environment is derived. We introduce two basic reproduction numbers, one is for the bacterium in the environment and the other is for the cholera disease in the host population. If the basic reproduction number for the cholera bacterium in the environment is strictly less than one and the basic reproduction number of infection is no more than one, we prove globally asymptotically stability of the infection-free steady state. Otherwise, the infection will persist and there exists at least one endemic steady state. For the special homogeneous case, the endemic steady state is actually unique and globally asymptotically stable. Under some conditions, the basic reproduction number of infection is strictly decreasing with respect to the diffusion coefficients of cholera bacteria and infectious hosts. When these conditions are violated, numerical simulation suggests that spatial diffusion may not only spread the infection from high-risk region to low-risk region, but also increase the infection level in high-risk region.

Modelling song popularity as a contagious process

Popular songs are often said to be 'contagious', 'infectious' or 'viral'. We find that download count time series for many popular songs resemble infectious disease epidemic curves. This paper suggests infectious disease transmission models could help clarify mechanisms that contribute to the 'spread' of song preferences and how these mechanisms underlie song popularity. We analysed data from MixRadio, comprising song downloads through Nokia cell phones in Great Britain from 2007 to 2014. We compared the ability of the standard susceptible-infectious-recovered (SIR) epidemic model and a phenomenological (spline) model to fit download time series of popular songs. We fitted these same models to simulated epidemic time series generated by the SIR model. Song downloads are captured better by the SIR model, to the same extent that actual SIR simulations are fitted better by the SIR model than by splines. This suggests that the social processes underlying song popularity are similar to those that drive infectious disease transmission. We draw conclusions about song popularity within specific genres based on estimated SIR parameters. In particular, we argue that faster spread of preferences for Electronica songs may reflect stronger connectivity of the 'susceptible community', compared with the larger and broader community that listens to more common genres.

Review of fractional epidemic models

The global impact of corona virus (COVID-19) has been profound, and the public health threat it represents is the most serious seen in a respiratory virus since the 1918 influenza A(H1N1) pandemic. In this paper, we have focused on reviewing the results of epidemiological modelling especially the fractional epidemic model and summarized different types of fractional epidemic models including fractional Susceptible-Infective-Recovered (SIR), Susceptible-Exposed-Infective-Recovered (SEIR), Susceptible-Exposed-Infective-Asymptomatic-Recovered (SEIAR) models and so on. Furthermore, we propose a general fractional SEIAR model in the case of single-term and multi-term fractional differential equations. A feasible and reliable parameter estimation method based on modified hybrid Nelder-Mead simplex search and particle swarm optimisation is also presented to fit the real data using fractional SEIAR model. The effective methods to solve the fractional epidemic models we introduced construct a simple and effective analytical technique that can be easily extended and applied to other fractional models, and can help guide the concerned bodies in preventing or controlling, even predicting the infectious disease outbreaks.

The Role of Movement Patterns in Epidemic Models on Complex Networks

In this paper, we analyze the influence of the usual movement variables on the spread of an epidemic. Specifically, given two spatial topologies, we can deduce which topology produces less infected individuals. In particular, we determine the topology that minimizes the overall number of infected individuals. It is worth noting that we do not assume any of the common simplifying assumptions in network theory such as all the links have the same diffusion rate or the movement of the individuals is symmetric. Our main conclusion is that the degree of mobility of the population plays a critical role in the spread of a disease. Finally, we derive theoretical insights to management of epidemics.

Alternative transmission pathways for guinea worm in dogs: implications for outbreak risk and control

Guinea worm (Dracunculus medinensis) has exerted a high human health burden in parts of Africa. Complete eradication of Guinea worm disease (dracunculiasis) may be delayed by the circulation of the parasite in domestic dogs. As with humans, dogs acquire the parasite by directly ingesting infected copepods, and recent evidence suggests that consuming frogs that ingested infected copepods as tadpoles may be a viable transmission route (paratenic route). To understand the relative contributions of direct and paratenic transmission routes, we developed a mathematical model that describes transmission of Guinea worm between dogs, copepods and frogs. We explored how the parasite basic reproductive number (R₀) depends on parameters amenable to actionable interventions under three scenarios: frogs/tadpoles do not consume copepods; tadpoles consume copepods but frogs do not contribute to transmission; and frogs are paratenic hosts. We found a non-monotonic relationship between the number of dogs and R₀. Generally, frogs can contribute to disease control by removing infected copepods from the waterbody even when paratenic transmission can occur. However, paratenic transmission could play an important role in maintaining the parasite when direct transmission is reduced by interventions focused on reducing copepod ingestion by dogs. Together, these suggest that the most effective intervention strategies may be those which focus on the reduction of copepods, as this reduces outbreak potential irrespective of the importance of the paratenic route.

Emerging Energy Harvesting Materials and Devices for Self-Powered Water Disinfection

Contaminated drinking water is one of the main pathogen transmission pathways making waterborne illnesses such as diarrheal diseases and gastroenteritis a huge threat to public health, especially in the areas where sanitation facilities and gird power are inadequate such as rural and disaster hit areas. Self-powered water disinfection systems are a promising solution in these cases. In this review paper, the authors provide an overview of the new and emerging methods of applying energy harvesting materials and devices as a source of power for water disinfection systems microbial disinfection in water by harnessing ambient forms of energy such as mechanical motion, light, and heat into electricity. The authors begin with a brief introduction of the different energy harvesting technologies commonly applied in water disinfection; triboelectric, piezoelectric, pyroelectric, and photovoltaic effects. Various microbial disinfection mechanisms and types of device construction are summarized. Then, a detailed discussion of the energy harvester-driven water disinfection process is provided. Finally, challenges and perspectives regarding the future development of self-powered water disinfection are described.

Modeling and controlling the spread of epidemic with various social and economic scenarios

We propose a dynamical model for describing the spread of epidemics. This model is an extension of the SIQR (susceptible-infected-quarantined-recovered) and SIRP (susceptible-infected-recovered-pathogen) models used earlier to describe various scenarios of epidemic spreading. As compared to the basic SIR model, our model takes into account two possible routes of contagion transmission: direct from the infected compartment to the susceptible compartment and indirect via some intermediate medium or fomites. Transmission rates are estimated in terms of average distances between the individuals in selected social environments and characteristic time spans for which the individuals stay in each of these environments. We also introduce a collective economic resource associated with the average amount of money or income per individual to describe the socioeconomic interplay between the spreading process and the resource available to infected individuals. The epidemic-resource coupling is supposed to be of activation type, with the recovery rate governed by the Arrhenius-like law. Our model brings an advantage of building various control strategies to mitigate the effect of epidemic and can be applied, in particular, to modeling the spread of COVID-19.

Modeling the Effectiveness of TV and Social Media Advertisements on the Dynamics of Water-Borne Diseases

Cholera is a serious threat to the health of human-kind all over the world and its control is a problem of great concern. In this context, a nonlinear mathematical model to control the prevalence of cholera disease is proposed and analyzed by incorporating TV and social media advertisements as a dynamic variable. It is considered that TV and social media ads propagate the knowledge among the people regarding the severe effects of cholera disease on human health along with its precautionary measures. It is also assumed that the mode of transmission of cholera disease among susceptible individuals is due to consumption of contaminated drinking water containing Vibrio cholerae. Moreover, the propagation of knowledge through TV and social media ads makes the people aware to adopt precautionary measures and also the aware people make some effectual efforts to washout the bacteria from the aquatic environment. Model analysis reveals that increase in the washout rate of bacteria due to aware individuals causes the stability switch. It is found that TV and social media ads have the potential to reduce the number of infectives in the region and thus control the cholera epidemic. Numerical simulation is performed for a particular set of parameter values to support the analytical findings.

A dose response model for Staphylococcus aureus

Dose-response models (DRMs) are used to predict the probability of microbial infection when a person is exposed to a given number of pathogens. In this study, we propose a new DRM for Staphylococcus aureus (SA), which causes skin and soft-tissue infections. The current approach to SA dose-response is only partially mechanistic and assumes that individual bacteria do not interact with each other. Our proposed two-compartment (2C) model assumes that bacteria that have not adjusted to the host environment decay. After adjusting to the host, they exhibit logistic/cooperative growth, eventually causing disease. The transition between the adjusted and un-adjusted states is a stochastic process, which the 2C DRM explicitly models to predict response probabilities. By fitting the 2C model to SA pathogenesis data, we show that cooperation between individual SA bacteria is sufficient (and, within the scope of the 2C model, necessary) to characterize the dose-response. This is a departure from the classical single-hit theory of dose-response, where complete independence is assumed between individual pathogens. From a quantitative microbial risk assessment standpoint, the mechanistic basis of the 2C DRM enables transparent modeling of dose-response of antibiotic-resistant SA that has not been possible before. It also enables the modeling of scenarios having multiple/non-instantaneous exposures, with minimal assumptions.

A dynamic model for the COVID-19 with direct and indirect transmission pathways

Two common transmission pathways for the spread of COVID-19 virus are direct and indirect. The direct pathway refers to the person-to-person transmission between susceptibles and infectious individuals. Infected individuals shed virus on the objects, and new infections arise through touching a contaminated surface; this refers to the indirect transmission pathway. We model the direct and indirect transmission pathways with a S A D O _I R ode model. Our proposal explicitly includes compartments for the contaminated objects, susceptible individuals, asymptomatic infectious, detected infectious, and recovered individuals. We compute the basic reproduction number and epidemic growth rate of the model and determine how these fundamental quantities relate to the transmission rate of the pathways. We further study the relationship between the rate of loss of immunity and the occurrence of backward bifurcation. An efficient statistical framework is introduced to estimate the parameters of the model. We show the performance of the model in the simulation scenarios and the real data from the COVID-19 daily cases in South Korea.

Optimal Control Analysis of Cholera Dynamics in the Presence of Asymptotic Transmission

Many mathematical models have explored the dynamics of cholera but none have been used to predict the optimal strategies of the three control interventions (the use of hygiene promotion and social mobilization; the use of treatment by drug/oral re-hydration solution; and the use of safe water, hygiene, and sanitation). The goal here is to develop (deterministic and stochastic) mathematical models of cholera transmission and control dynamics, with the aim of investigating the effect of the three control interventions against cholera transmission in order to find optimal control strategies. The reproduction number Rp was obtained through the next generation matrix method and sensitivity and elasticity analysis were performed. The global stability of the equilibrium was obtained using the Lyapunov functional. Optimal control theory was applied to investigate the optimal control strategies for controlling the spread of cholera using the combination of control interventions. The Pontryagin’s maximum principle was used to characterize the optimal levels of combined control interventions. The models were validated using numerical experiments and sensitivity analysis was done. Optimal control theory showed that the combinations of the control intervention influenced disease progression. The characterisation of the optimal levels of the multiple control interventions showed the means for minimizing cholera transmission, mortality, and morbidity in finite time. The numerical experiments showed that there are fluctuations and noise due to its dependence on the corresponding population size and that the optimal control strategies to effectively control cholera transmission, mortality, and morbidity was through the combinations of all three control interventions. The developed models achieved the reduction, control, and/or elimination of cholera through incorporating multiple control interventions.

Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics

Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp. variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.

A mathematical model for Vibrio-phage interactions

A cholera model has been formulated to incorporate the interaction of bacteria and phage. It is shown that there may exist three equilibria: one disease free and two endemic equilibria. Threshold parameters have been derived to characterize stability of these equilibria. Sensitivity analysis and disease control strategies have been employed to characterize the impact of bacteria-phage interaction on cholera dynamics.

Backward bifurcation in a cholera model with a general treatment function

We consider a general cholera model with a nonlinear treatment function. The treatment function describes the saturated treatment scenario due to the limited availability of resources. The sufficient conditions for the existence of backward bifurcation have been obtained using the central manifold theory. At last, we illustrate the results by considering some special types of treatment functions.