Multiple endemic equilibria in an environmentally-transmitted disease with three disease stages.

We construct, analyze and interpret a mathematical model for an environmental transmitted disease characterized for the existence of three disease stages: acute, severe and asymptomatic. Besides, we consider that severe and asymptomatic cases may present relapse between them. Transmission dynamics driven by the contact rates only occurs when a parameter R∗>1, as normally occur in directly-transmitted or vector-transmitted diseases, but it will not adequately correspond to a basic reproductive number as it depends on environmental parameters. In this case, the forward transcritical bifurcation that exists for R∗<1, becomes a backward bifurcation, producing multiple steady-states, a hysteresis effect and dependence on initial conditions. A threshold parameter for an epidemic outbreak, independent of R∗ is only the ratio of the external contamination inflow shedding rate to the environmental clearance rate. R∗ describes the strength of the transmission to infectious classes other than the I-(acute) type infections. The epidemic outbreak conditions and the structure of R∗ appearing in this model are both responsible for the existence of endemic states.

Modeling a traffic light warning system for acute respiratory infections.

The high morbidity of acute respiratory infections constitutes a crucial global health burden. In particular, for SARS-CoV-2, non-pharmaceutical intervention geared to enforce social distancing policies, vaccination, and treatments will remain an essential part of public health policies to mitigate and control disease outbreaks. However, the implementation of mitigation measures directed to increase social distancing when the risk of contagion is a complex enterprise because of the impact of NPI on beliefs, political views, economic issues, and, in general, public perception. The way of implementing these mitigation policies studied in this work is the so-called traffic-light monitoring system that attempts to regulate the application of measures that include restrictions on mobility and the size of meetings, among other non-pharmaceutical strategies. Balanced enforcement and relaxation of measures guided through a traffic-light system that considers public risk perception and economic costs may improve the public health benefit of the policies while reducing their cost. We derive a model for the epidemiological traffic-light policies based on the best response for trigger measures driven by the risk perception of people, instantaneous reproduction number, and the prevalence of a hypothetical acute respiratory infection. With numerical experiments, we evaluate and identify the role of appreciation from a hypothetical controller that could opt for protocols aligned with the cost due to the burden of the underlying disease and the economic cost of implementing measures. As the world faces new acute respiratory outbreaks, our results provide a methodology to evaluate and develop traffic light policies resulting from a delicate balance between health benefits and economic implications.

Structure-based identification of sensor species for anticipating critical transitions.

Ecological systems can undergo sudden, catastrophic changes known as critical transitions. Anticipating these critical transitions remains challenging in systems with many species because the associated early warning signals can be weakly present or even absent in some species, depending on the system dynamics. Therefore, our limited knowledge of ecological dynamics may suggest that it is hard to identify those species in the system that display early warning signals. Here, we show that, in mutualistic ecological systems, it is possible to identify species that early anticipate critical transitions by knowing only the system structure-that is, the network topology of plant-animal interactions. Specifically, we leverage the mathematical theory of structural observability of dynamical systems to identify a minimum set of "sensor species," whose measurement guarantees that we can infer changes in the abundance of all other species. Importantly, such a minimum set of sensor species can be identified by using the system structure only. We analyzed the performance of such minimum sets of sensor species for detecting early warnings using a large dataset of empirical plant-pollinator and seed-dispersal networks. We found that species that are more likely to be sensors tend to anticipate earlier critical transitions than other species. Our results underscore how knowing the structure of multispecies systems can improve our ability to anticipate critical transitions.

The trade-off between mobility and vaccination for COVID-19 control: a metapopulation modelling approach.

November 2020 received a string of encouraging results from leading vaccine developers raising hopes for the imminent availability of an effective and safe vaccine against the SARS-CoV-2. In the present work, we discuss the theoretical impact of introducing a vaccine across a range of scenarios. In particular, we investigate how vaccination coverage, efficacy and delivery time affect the control of the transmission dynamics in comparison to mobility restrictions. The analysis is based on a metapopulation epidemic model structured by risk. We perform a global sensitivity analysis using the Sobol method. Our analysis suggest that the reduction of mobility among patches plays a significant role in the mitigation of the disease close to the effect of immunization coverage of 30% achieved in four months. Moreover, for an immunization coverage between 20% and 50% achieved in the first half of 2021 with a vaccine efficacy between 70% and 95%, the percentage reduction in the total number of SARS-CoV-2 infections is between 30% and 50% by the end of 2021 in comparison with the no vaccination scenario.

A simple criterion to design optimal non-pharmaceutical interventions for mitigating epidemic outbreaks.

For mitigating the COVID-19 pandemic, much emphasis is made on implementing non-pharmaceutical interventions to keep the reproduction number below one. However, using that objective ignores that some of these interventions, like bans of public events or lockdowns, must be transitory and as short as possible because of their significant economic and societal costs. Here, we derive a simple and mathematically rigorous criterion for designing optimal transitory non-pharmaceutical interventions for mitigating epidemic outbreaks. We find that reducing the reproduction number below one is sufficient but not necessary. Instead, our criterion prescribes the required reduction in the reproduction number according to the desired maximum of disease prevalence and the maximum decrease of disease transmission that the interventions can achieve. We study the implications of our theoretical results for designing non-pharmaceutical interventions in 16 cities and regions during the COVID-19 pandemic. In particular, we estimate the minimal reduction of each region's contact rate necessary to control the epidemic optimally. Our results contribute to establishing a rigorous methodology to design optimal non-pharmaceutical intervention policies for mitigating epidemic outbreaks.

Electro-magnetically modulated self-propulsion of swimming sperms via cervical canal.

The purpose of this study is to theoretically investigate the electro-magneto-biomechanics of the swimming of sperms through cervical canal in the female reproductive system. During sexual intercourse, millions of sperms migrate into the cervix in large groups, hence we can approximately model their movement activity by a swimming sheet through the electrically-conducting biofluid. The Eyring-Powell fluid model is considered as the base fluid to simulate male's semen with self-propulsive sperms. An external magnetic field is applied on the flow in transverse direction. The governing partial differential system of equations is analytically solved. Creeping flow regimen is employed throughout the channel due to self-propulsion of swimmers along with long wavelength approximation. Solutions for the stream function, velocity profile, and pressure gradient (above and below the swimming sheet) are obtained and plotted with the pertinent parameters. The prominent features of pumping characteristics are also investigated. Results indicate that the propulsive velocity is reduced with an increase in the electric field which is an important feature that can be used in controlling the transport of spermatozoa inside the cervical canal. Not only is the present analysis valid for living micro-organisms, but also valid for artificially designed electro-magnetic micro-swimmers which is further utilized in electro-magnetic therapy taking place in female's lubricous cervical canal filled with mucus.

In-host Mathematical Modelling of COVID-19 in Humans.

COVID-19 pandemic has underlined the impact of emergent pathogens as a major threat to human health. The development of quantitative approaches to advance comprehension of the current outbreak is urgently needed to tackle this severe disease. Considering different starting times of infection, mathematical models are proposed to represent SARS-CoV-2 dynamics in infected patients. Based on the target cell limited model, the within-host reproductive number for SARS-CoV-2 is consistent with the broad values of human influenza infection. The best model to fit the data was including immune cell response, which suggests a slow immune response peaking between 5 to 10 days post-onset of symptoms. The model with the eclipse phase, time in a latent phase before becoming productively infected cells, was not supported. Interestingly, model simulations predict that SARS-CoV-2 may replicate very slowly in the first days after infection, and viral load could be below detection levels during the first 4 days post infection. A quantitative comprehension of SARS-CoV-2 dynamics and the estimation of standard parameters of viral infections is the key contribution of this pioneering work. These models can serve for future evaluation of control theoretical approaches to tailor new drugs against COVID-19.

Lifting mobility restrictions and the effect of superspreading events on the short-term dynamics of COVID-19.

SARS-CoV-2 has now infected 15 million people and produced more than six hundred thousand deaths around the world. Due to high transmission levels, many governments implemented social distancing and confinement measures with different levels of required compliance to mitigate the COVID-19 epidemic. In several countries, these measures were effective, and it was possible to flatten the epidemic curve and control it. In others, this objective was not or has not been achieved. In far too many cities around the world, rebounds of the epidemic are occurring or, in others, plateaulike states have appeared, where high incidence rates remain constant for relatively long periods of time. Nonetheless, faced with the challenge of urgent social need to reactivate their economies, many countries have decided to lift mitigation measures at times of high incidence. In this paper, we use a mathematical model to characterize the impact of short duration transmission events within the confinement period previous but close to the epidemic peak. The model also describes the possible consequences on the disease dynamics after mitigation measures are lifted. We use Mexico City as a case study. The results show that events of high mobility may produce either a later higher peak, a long plateau with relatively constant but high incidence or the same peak as in the original baseline epidemic curve, but with a post-peak interval of slower decay. Finally, we also show the importance of carefully timing the lifting of mitigation measures. If this occurs during a period of high incidence, then the disease transmission will rapidly increase, unless the effective contact rate keeps decreasing, which will be very difficult to achieve once the population is released.

Using posterior predictive distributions to analyse epidemic models: COVID-19 in Mexico City.

Epidemiological models usually contain a set of parameters that must be adjusted based on available observations. Once a model has been calibrated, it can be used as a forecasting tool to make predictions and to evaluate contingency plans. It is customary to employ only point estimators of model parameters for such predictions. However, some models may fit the same data reasonably well for a broad range of parameter values, and this flexibility means that predictions stemming from them will vary widely, depending on the particular values employed within the range that gives a good fit. When data are poor or incomplete, model uncertainty widens further. A way to circumvent this problem is to use Bayesian statistics to incorporate observations and use the full range of parameter estimates contained in the posterior distribution to adjust for uncertainties in model predictions. Specifically, given an epidemiological model and a probability distribution for observations, we use the posterior distribution of model parameters to generate all possible epidemic curves, whose information is encapsulated in posterior predictive distributions. From these, one can extract the worst-case scenario and study the impact of implementing contingency plans according to this assessment. We apply this approach to the evolution of COVID-19 in Mexico City and assess whether contingency plans are being successful and whether the epidemiological curve has flattened.

Modeling behavioral change and COVID-19 containment in Mexico: A trade-off between lockdown and compliance.

Sanitary Emergency Measures (SEM) were implemented in Mexico on March 30th, 2020 requiring the suspension of non-essential activities. This action followed a Healthy Distance Sanitary action on March 23rd, 2020. The aim of both measures was to reduce community transmission of COVID-19 in Mexico by lowering the effective contact rate. Using a modification of the Kermack-McKendrick SEIR model we explore the effect of behavioral changes required to lower community transmission by introducing a time-varying contact rate, and the consequences of disease spread in a population subject to suspension of non-essential activities. Our study shows that there exists a trade-off between the proportion of the population under SEM and the average time an individual is committed to all the behavioral changes needed to achieve an effective social distancing. This trade-off generates an optimum value for the proportion of the population under strict mitigation measures, significantly below 1 in some cases, that minimizes maximum COVID-19 incidence. We study the population-level impact of three key factors: the implementation of behavior change control measures, the time horizon necessary to reduce the effective contact rate and the proportion of people under SEM in combating COVID-19. Our model is fitted to the available data. The initial phase of the epidemic, from February 17th to March 23rd, 2020, is used to estimate the contact rates, infectious periods and mortality rate using both confirmed cases (by date of symptoms initiation), and daily mortality. Data on deaths after March 23rd, 2020 is used to estimate the mortality rate after the mitigation measures are implemented. Our simulations indicate that the most likely dates for maximum incidence are between late May and early June, 2020 under a scenario of high SEM compliance and low SEM abandonment rate.

Transmission dynamics of acute respiratory diseases in a population structured by age.

Determining the role of age on the transmission of an infection is a topic that has received significant attention. In this work, a dataset of acute respiratory infections structured by age from San Luis Potosí, Mexico, is analyzed to understand the age impact on this class of diseases. To do that, a compartmental SEIRS multigroup model is proposed to describe the infection dynamics among age groups. Then, a Bayesian inference approach is used to estimate relevant parameters in the model such as the probability of infection, the average time that one individual remains infectious, the average time that one individual remains immune, and the force of infection, among others. Based on those estimates, our analysis leads us to conclude that children less than 5 years old are the primary spreaders of respiratory infections in San Luis Potosí's population from 2000 to 2008 since they are more prone to get sick, remain infectious for longer periods and they are reinfected more rapidly. On the other hand, the group of young adults (20-59) is the one that differs the most from the little children's group because it does not get sick often, it remains infectious only a few days and it stays healthy for longer periods. These observations allow us to infer that the group of young adults is the one that, on average, less contributed to the spread of this class of infections during the years represented in our database.

The role of animal grazing in the spread of Chagas disease.

Chagas disease is an important neglected tropical disease which causes on average about 7000 deaths per year, and an estimated 25 million people risk of acquiring it. This illness is often found in rural areas, which are usually characterized by poverty and presence of animals which act as reservoirs of the disease. Our main objective is to study the effect of animal grazing on the disease levels of the human population. For this purpose, we consider two environments (domestic and wild) where each one has permanent residents, and there is a proportion of animals that move between both environments due to grazing. This movement is modeled through the residence time in each environment. We analyze the proposed model and finally, we discuss the influence of domestic animals residence time on the disease level of human population.

Robust qualitative estimation of time-varying contact rates in uncertain epidemics.

We will inevitably face new epidemics where the lack of long time-series data and the uncertainty about the outbreak dynamics make difficult to obtain quantitative predictions. Here we present an algorithm to qualitatively infer time-varying contact rates from short time-series data, letting us predict the start, relative magnitude and decline of epidemic outbreaks. Using real time-series data of measles, dengue, and the current zika outbreak, we demonstrate our algorithm can outperform existing algorithms based on estimating reproductive numbers.

Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: A theoretical analysis.

Three deterministic Kermack-McKendrick-type models for studying the transmission dynamics of an infection in a two-sex closed population are analyzed here. In each model it is assumed that infection can be transmitted through heterosexual contacts, and that there is a higher probability of transmission from one sex to the other than vice versa. The study is focused on understanding whether and how this bias in transmission reflects in sex differences in final attack ratios (i.e. the fraction of individuals of each sex that eventually gets infected). In the first model, where the other two transmission modes are not considered, the attack ratios (fractions of the population of each sex that will eventually be infected) can be obtained as solutions of a system of two nonlinear equations, that has a unique solution if the net reproduction number exceeds unity. It is also shown that the ratio of attack ratios depends solely on the ratio of gender-specific susceptibilities and on the basic reproductive number of the epidemic Ro, and that the gender-specific final attack-ratio is biased in the same direction as the gender-specific susceptibilities. The second model allows also for infection transmission through direct, non-sexual, contacts. In this case too, an analytical expression is derived from which the attack ratios can be obtained. The qualitative results are similar to those obtained for the previous model, but another important parameter for determining the value of the ratio between the attack ratios in the two sexes is obtained, the relative weight of direct vs. heterosexual transmission (namely, ρ). Quantitatively, the ratio of final attack ratios generally will not exceed 1.5, if non-sexual transmission accounts for most transmission events (ρ≥0.6) and the ratio of gender-specific susceptibilities is not too large (say, 5 at most). The third model considers vector-borne, instead of direct transmission. In this case, we were not able to find an analytical expression for the final attack ratios, but used instead numerical simulations. The results on final attack ratios are actually quite similar to those obtained with the second model. It is interesting to note that transient patterns can differ from final attack ratios, as new cases will tend to occur more often in the more susceptible sex, while later depletion of susceptibles may bias the ratio in the opposite direction. The analysis of these simple models, despite their lack of realism, can help in providing insight into, and assessment of, the potential role of gender-specific transmission in infections with multiple modes of transmission, such as Zika virus (ZIKV), by gauging what can be expected to be seen from epidemiological reports of new cases, disease incidence and seroprevalence surveys.

Sex, Mosquitoes and Epidemics: An Evaluation of Zika Disease Dynamics.

Since the first major outbreak reported on the island Yap in 2007, the Zika virus spread has alerted the scientific community worldwide. Zika is an arbovirus transmitted by Aedes mosquitoes; particularly in Central and South America, the main vector is the same mosquito that transmits dengue and chikungunya, Aedes aegypti. Seeking to understand the dynamics of spread of the Zika, in this paper, three mathematical models are presented, in which vector transmission of the virus, sexual contact transmission and migration are considered. Numerical analysis of these models allows us to have a clear view of the effects of sexual transmission and migration in the spread of the virus, showing that sexual transmission influences the magnitude of the outbreaks and migration generates outbreaks over time, each of lower intensity than the previous.

Coupled within-host and between-host dynamics and evolution of virulence.

Mathematical models coupling within- and between-host dynamics can be helpful for deriving trade-off functions between disease transmission and virulence at the population level. Such functions have been used to study the evolution of virulence and to explore the possibility of a conflict between natural selection at individual and population levels for directly transmitted diseases (Gilchrist and Coombs, 2006). In this paper, a new coupled model for environmentally-driven diseases is analyzed to study similar biological questions. It extends the model in Cen et al. (2014) and Feng et al. (2013) by including the disease-induced host mortality. It is shown that the extended model exhibits similar dynamical behaviors including the possible occurrence of a backward bifurcation. It is also shown that the within-host pathogen load and the disease prevalence at the positive stable equilibrium are increasing functions of the within- and between-host reproduction numbers (Rw0 and Rb0), respectively. Optimal parasite strategies will maximize these reproduction numbers at the two levels, and a conflict may exist between the two levels. Our results highlight the role of inter-dependence of variables and parameters in the fast and slow systems for persistence of infections and evolution of pathogens in an environmentally-driven disease. Our results also demonstrate the importance of incorporating explicit links of the within- and between-host dynamics into the computation of threshold conditions for disease control.

Towards uncertainty quantification and inference in the stochastic SIR epidemic model.

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.

Early outbreak of 2009 influenza A (H1N1) in Mexico prior to identification of pH1N1 virus.

BACKGROUND: In the aftermath of the global spread of 2009 influenza A (pH1N1) virus, still very little is known of the early stages of the outbreak in Mexico during the early months of the year, before the virus was identified. METHODOLOGY/MAIN FINDINGS: We fit a simple mathematical model, the Richards model, to the number of excess laboratory-confirmed influenza cases in Mexico and Mexico City during the first 15 weeks in 2009 over the average influenza case number of the previous five baseline years of 2004-2008 during the same period to ascertain the turning point (or the peak incidence) of a wave of early influenza infections, and to estimate the transmissibility of the virus during these early months in terms of its basic reproduction number. The results indicate that there may have been an early epidemic in Mexico City as well as in all of Mexico during February/March. Based on excess influenza cases, the estimated basic reproduction number R0 for the early outbreak was 1.59 (0.55 to 2.62) for Mexico City during weeks 5-9, and 1.25 (0.76, 1.74) for all of Mexico during weeks 5-14. CONCLUSIONS: We established the existence of an early epidemic in Mexico City and in all of Mexico during February/March utilizing the routine influenza surveillance data, although the location of seeding is unknown. Moreover, estimates of R0 as well as the time of peak incidence (the turning point) for Mexico City and all of Mexico indicate that the early epidemic in Mexico City in February/March had been more transmissible (larger R0) and peaked earlier than the rest of the country. Our conclusion lends support to the possibility that the virus could have already spread to other continents prior to the identification of the virus and the reporting of lab-confirmed pH1N1 cases in North America in April.

A model for the A(H1N1) epidemic in Mexico, including social isolation.

OBJECTIVE: We present a model for the 2009 influenza epidemic in Mexico to describe the observed pattern of the epidemic from March through the end of August (before the onset of the expected winter epidemic) in terms of the reproduction number and social isolation measures. MATERIAL AND METHODS: The model uses a system of ordinary differential equations. Computer simulations are performed to optimize trajectories as a function of parameters. RESULTS: We report on the theoretical consequences of social isolation using published estimates of the basic reproduction number. The comparison with actual data provides a reasonable good fit. CONCLUSIONS: The pattern of the epidemic outbreak in Mexico is characterized by two peaks resulting from the application of very drastic social isolation measures and other prophylactic measures that lasted for about two weeks. Our model is capable of reproducing the observed pattern.