We set up and analyse a mathematical model, the Serious Crime Model, which describes the interaction of mild and serious offenders and potential criminals. However we get more complete results for a simpler version of this model, the Mild Crime Model, with no serious offenders. For the full Serious Crime Model there are two key parameters R 10 and R 20 corresponding to the basic reproduction number in the mathematics of infectious diseases, which determine the behaviour of the system. For the Simpler Mild Crime Model there is just one such parameter R 10 . Both forward and backward bifurcation can occur for this second model with two subcritical non-trivial equilibria possible for R 10 < 1 in the backwards case. For backwards bifurcation there is another threshold value R*0 such that the upper non-trivial equilibrium is unstable for R 10 < R*0 and stable for R 10 > R*0. For for-wards bifurcation there is a second additional threshold value R 0** such that the stability of the unique non-trivial equilibrium switches from unstable to stable as R 10 passes through R ** 0 . At the end we return to the full Serious Crime Model and discuss the behaviour of this model. The results are meaningful and interesting because in all of the other epidemiolog-ical and sociological models of which we are aware, analogous thresholds to R*0 and R ** 0 do not exist. For forwards bifurcation the unique non-trivial equilibrium, and for backwards bifurcation with two subcritical endemic equilibria the higher non-trivial equilibrium, are also usually always locally asymptotically stable. So our models exhibit unusual and inter-esting behaviour.(c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license( http://creativecommons.org/licenses/by/4.0/ )

Mathematical models have a long history in epidemiological research, and as the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. Mathematical models describing dengue fever epidemiological dynamics are found back from 1970. Dengue fever is a viral mosquito-borne infection caused by four antigenically related but distinct serotypes (DENV-1 to DENV-4). With 2.5 billion people at risk of acquiring the infection, it is a major international public health concern. Although most of the cases are asymptomatic or mild, the disease immunological response is complex, with severe disease linked to the antibody-dependent enhancement (ADE) - a disease augmentation phenomenon where pre-existing antibodies to previous dengue infection do not neutralize but rather enhance the new infection. Here, we present a 10-year systematic review on mathematical models for dengue fever epidemiology. Specifically, we review multi-strain frameworks describing host-to-host and vector-host transmission models and within-host models describing viral replication and the respective immune response. Following a detailed literature search in standard scientific databases, different mathematical models in terms of their scope, analytical approach and structural form, including model validation and parameter estimation using empirical data, are described and analyzed. Aiming to identify a consensus on infectious diseases modeling aspects that can contribute to public health authorities for disease control, we revise the current understanding of epidemiological and immunological factors influencing the transmission dynamics of dengue. This review provide insights on general features to be considered to model aspects of real-world public health problems, such as the current epidemiological scenario we are living in.

Objectives: We developed a simple mathematical model to describe criminality and the justice system composed of the police investigation and court trial. The model assessed two features of organized crime – the cost-benefit analysis done by the crime-susceptible to commit a crime and the whistleblowing of the law offenders. ∗Corresponding author: tel. + 55 19 3521-6031

When a Susceptible-Infective-Recovered (SIR) model with a constant contact rate is used to describe the dynamics of directly transmitted infections, oscillations, which decay exponentially with time, are obtained. Due to damped oscillations, intermittent vaccination schemes can be designed in order to reduce or even eliminate the infection. A simple intermittent vaccination can be described by a series of pulses, i.e., a proportion of susceptible individuals is vaccinated intermittently at every fixed period of time. Analysis of the model is done by numerical simulations in order to determine the trajectories in the phase space. It is observed that as the proportion of vaccinated individuals increases, closed orbits with multiple cycles appear, even irregular trajectories arise occasionally. These results can be understood by comparing with bifurcations occurring in a discrete logistic model describing a single population. Further, bifurcations occurring in epidemiological models that use periodic functions to mimic seasonal variations in the disease transmission are discussed.

We propose a mathematical model to investigate the antibody-dependent enhancement (ADE) phenomenon during secondary dengue infection. The model consists of an ODE system that describes the interaction of the dengue virus with macrophages and memory B and T cell role during the infection. The qualitative model analysis is done in terms of memory B and T cell cloning parameters and the basic reproduction number R0. In the absence of memory B and T cell cloning, if R0<1 the dengue virus population extinguishes, while for R0>1, it tends asymptotically to a positive equilibrium. However, when we consider the memory B cell cloning, it is possible to occur dengue infection even when R0<1. Memory T cells have an essential role in eliminating the possibility of ADE occurrence when R0<1.

We propose a mathematical model to study the antibody-dependent enhancement (ADE) phenomenon. Here, we explore the interaction between macrophages, dengue virus and plasma cells, especially the effect of a limitation on plasma cell proliferation, which occurs due to immunological memory. The model has up to three equilibrium points: one virus-free equilibrium and two virus-presence equilibrium, depending on the value of two thresholds. We determine the existence regions for the model equilibrium points and their stability, a sensitivity analysis was performed in the model thresholds. Numerical simulations illustrate that ADE can occur even when the basic reproduction number is less than one.

Schistosomiasis is a parasite infection that affects millions of people around the world. It is endemic in 13 different states in Brazil and responsible for increasing morbidity in the population. One of its main characteristics is a heterogeneous distribution of worm burden in the human population, which makes the diagnosis difficult. We aimed to investigate how the sensitivity of the diagnostic method may contribute to successful control interventions against infections in a population. In order to do that, we present an ordinary differential equations model that considers three levels of worm burden in the human population, a snail population, and a miracidium reservoir. Through a steady-state analysis and its local stability, we show how this worm-burden heterogeneity can be responsible for the persistence of infection, especially due to reinfection in the highest level of worm burden. The analysis highlights sensitive diagnosis, besides treatment and sanitary improvements, as a key factor for schistosomiasis transmission control.

Coronavirus disease 2019 (CoViD-19), with the fatality rate in elder (60 years old or more) being much higher than young (60 years old or less) patients, was declared a pandemic by the World Health Organization on March 11, 2020. A mathematical model considering young and elder subpopulations under different fatality rates was formulated based on the natural history of CoViD-19 to study the transmission of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The model considered susceptible, exposed, asymptomatic, pre-symptomatic, mild CoViD-19, severe CoViD-19, and recovered compartments, besides compartments of isolated individuals and those who were caught by test. This model was applied to study the epidemiological scenario resulting from the adoption of quarantine (isolation or lockdown) in many countries to control the rapid propagation of CoViD-19. We chose as examples the isolation adopted in São Paulo State (Brazil) in the early phase but not at the beginning of the epidemic, and the lockdown implemented in Spain when the number of severe CoViD-19 cases was increasing rapidly. Based on the data collected from São Paulo State and Spain, the model parameters were evaluated, and we obtained a higher estimation for the basic reproduction number R0 (9.24 for São Paulo State, and 8 for Spain) compared to the currently accepted estimation of R0 around 2 using the SEIR (susceptible, exposed, infectious, and recovered compartments) model. In comparison with the lockdown in Spain, the relatively early adoption of the isolation in São Paulo State resulted in enlarging the period of the first wave of the epidemic and delaying its peak. The model allowed to explain the flattening of the epidemic curves by quarantine when associated with the protective measures (face mask, washing hands with alcohol and gel, and social distancing) adopted by the population. The description of the epidemic under quarantine and protections can be a background to foreseen the epidemiological scenarios from the release strategies, which can help guide public health policies by decision-makers.

Background: The continuous SARS-CoV-2 transmission in several countries could contribute to the mutations' appearance. The circulation of more virulent variants may increase the number of severe CoViD-19 needing hospital care and fatalities hugely. Methods: The partial quarantine in Sao Paulo State and further relaxation associated with the mutations are explained by a mathematical model based on the CoViD-19 natural history encompassing the age-dependent fatality. The model parameters were fitted considering the observed data from Sao Paulo State. Results: The partial quarantine was explained by the less virulent SARS-CoV-2 transmission, but the relaxation alone could not explain the epidemic observed in Sao Paulo State. However, more virulent variants plus the transmission among isolated individuals explained the increased CoViD-19 fatalities. Conclusions: The model described the CoViD-19 epidemic in Sao Paulo State by considering the partial quarantine, relaxation and mutations. The model provided a potential epidemiological scenario in the absence of mass vaccination.

We developed a mathematical model to describe the new coronavirus transmission in São Paulo State, Brazil. The model divided a community into subpopulations composed of young and elder persons considering a higher risk of fatality among elder persons with severe CoViD-19. From the data collected in São Paulo State, we estimated the transmission and additional mortality rates. Based on the estimated model parameters, we calculated the basic reproduction number $R_{0}$, and we retrieved the number of deaths due to CoViD-19, which was three times lower than those found in the literature. Considering isolation as a control mechanism, we varied the isolation rates in the young and elder subpopulations to assess the epidemiological impacts. The epidemiological scenarios focused mainly on evaluating the reduction in the number of severe CoViD-19 cases and deaths due to this disease when isolation is introduced in a population.

Platelets have been seen traditionally as fragments of blood mediating coagulation. However, evidence during malaria infection suggests that platelets also act against merozoites, an infectious form of malaria in the bloodstream, and megakaryocytes can release giant platelets with a larger volume than normal platelets. We propose a mathematical model to study the interaction between red blood cells, merozoites, and platelets during malaria infection. We analyzed two cases of the interaction of platelets with malaria infection. In the first one, we considered the isolated action of normal platelets and, in the second one, the joint antiparasitic action of both normal and giant platelets. Numerical simulations were performed to evaluate the stability of the equilibrium points of the system of equations. The model showed that the isolated antiparasitic action of normal platelets corroborates malaria infection control. However, the system can converge to a presence-merozoite equilibrium point, or an oscillatory behavior may appear. The joint antiparasitic action of both normal and giant platelets eliminated the oscillatory behavior and drove the dynamics to converge to lower parasitic concentration than the case of isolated action of normal platelets. Moreover, the joint antiparasitic action of platelets proved more easily capable of eliminating the infection.

Background: At the beginning of 2020, SARS-CoV-2 spread to all continents, and since then, mutations have appeared in different regions of the world. The appearance of more virulent mutations leads to asseverate that they are also more transmissible. We analyzed the lower and higher virulent SARS-CoV-2 epidemics to establish a relationship between transmissibility and virulence based on a mathematical model. Methods: A compartmental mathematical model based on the CoViD-19 natural history encompassing the age-dependent fatality was applied to evaluate the SARS-CoV-2 transmissibility and virulence. The transmissibility was measured by the basic reproduction number R0 and the virulence by the proportion of asymptomatic individuals. The model parameters were fitted considering the observed data from Sao Paulo State. Results: The numbers of severe CoViD-19 and deaths are three times higher, but R0 is 25% lower in more virulent SARS-CoV-2 transmission than in a less virulent one. However, the number of more virulent SARS-CoV-2 transmitting individuals is 25% lower, mainly due to symptomatic individuals' isolation, explaining the increased transmission in lower virulence. Conclusions: The quarantine study in Sao Paulo State showed that the more virulent SARS-CoV-2 resulted in a higher number of fatalities but less transmissible than the less virulent one. One possible explanation for the number of deaths surpassing that predicted by the low virulent SARS-CoV-2 infection could be the transmission of more virulent variant(s).

At the beginning of 2020, SARS-CoV-2 spread to all continents, and many countries adopted quarantine to flatten the COVID-19 outbreak, resulting in the containment of transmission. However, the persistent transmission of SARS-CoV-2, an RNA virus, resulted in the appearance of mutations in the original virus. The appearance of more virulent mutations in different regions of the world resulted in an increased number of severe COVID-19 cases. A mathematical model based on the COVID-19 natural history encompassing the mutations was applied to evaluate the SARS-CoV-2 epidemic. The model parameters were fitted against the observed data from São Paulo State (Brazil) considering quarantine, relaxation, and mutation. The estimated data from São Paulo State showed that the original SARS-CoV-2 transmission better explained the COVID-19 epidemic during the quarantine. However, the increased number of fatalities beyond that predicted by the relaxation indicated the occurrence of mutations. Hence, the relaxation in association with the transmission of a more virulent SARS-CoV-2 better estimated the COVID-19 epidemic from February 26, 2020, to June 15, 2021, in São Paulo State.

In epidemiological modeling, the basic reproduction number is usually defined as being the spectral radius of the next-generation matrix evaluated at the trivial equilibrium. The global stability of the trivial equilibrium point was determined by the left eigenvector associated with that next-generation matrix. More recently, the fraction of susceptible individuals was also obtained from the next generation matrix. The gross reproduction number and the fraction of susceptible individuals were calculated by revisiting the drug-sensitive and resistant tuberculosis model. Hence, the next-generation matrices shed light on the evolution of the dynamics: the beginning of the epidemic via the basic reproduction number and approaching the epidemic's steady-state via the susceptible individuals' asymptotic fraction.

O câncer é um problema de saúde mundial e uma das mais relevantes causas de mortalidade e morbidade em adultos e crianças. Neste trabalho desenvolvemos um modelo matemático para representar a resposta imunológica adaptativa contra tumores baseando-se em um sistema de equações diferenciais ordinárias não-linear, com o intuito de avaliar a dinâmica hospedeiro-tumorsistema imune.

Dengue disease is caused by an infected mosquito bite and manifests in different clinical symptoms. The complexity of the pathogenesis of dengue virus and the limitations of biological knowledge have been barriers to completely understanding the progress of this disease. To address this concern, we developed a mathematical model of the immune response to eliminate dengue virus. The model considered both cellular and humoral immune responses, and we evaluated their contributions to the clearance of dengue virus.We also performed global sensitivity analysis and parameters estimation using clinical data. We found the global stability for virus-free equilibrium and for the virus-presence equilibrium, we concluded that to avoid oscillations in the model and to control the viral load, a strong proliferation of cytotoxic cells must prevail. However, if there exists a weak proliferation of cytotoxic cells, the way to avoid instabilities is to either inhibit the differentiation of T-CD4+ helper cells in Th1 cells or increase the proliferation of B cells.

Background It is widely assumed that all mutant microorganisms present in a culture are able to grow and form colonies, provided that they express the features required for selection. Unlike wild-type Escherichia coli , PHO-constitutive mutants overexpress alkaline phosphatase and hence can hydrolyze glycerol-2-phosphate (G2P) to glycerol and form colonies on plates having G2P as the sole carbon source. These mutations mostly occur in the pst operon. However, the frequency of PHO-constitutive colonies on the G2P selective plate is exceptionally low. Results We show that the rate in which spontaneous PHO-constitutive mutations emerge is about 8.0 × 10 −6 /generation, a relatively high rate, but the growth of most existing mutants is inhibited by their neighboring wild-type cells. This inhibition is elicited only by non-mutant viable bacteria that can take up and metabolize glycerol formed by the mutants. Evidence indicates that the few mutants that do form colonies derive from microclusters of mutants on the selective plate. A mathematical model that describes the fate of the wild-type and mutant populations under these circumstances supports these results. Conclusion This scenario in which neither the wild-type nor the majority of the mutants are able to grow resembles an unavoidable “tragedy of the commons” case which results in the collapse of the majority of the population. Cooperation between rare adjacent mutants enables them to overcome the competition and eventually form mutant colonies. The inhibition of PHO-constitutive mutants provides an example of mutant frequency masked by orders of magnitude due to a competition between mutants and their ancestral wild-type cells. Similar “tragedy of the commons-like” cases may occur in other settings and should be taken into consideration while estimating true mutant frequencies and mutation rates.

AbstractSão Paulo State registered the first case of CoViD-19 on 26 February, the first death due to CoViD-19 on 16 March, and implemented the isolation of the population in non-essential activities on 24 March, which is programmed to end on 1 June. A mathematical model considering young (below 60 years old) and elder (above 60 years) subpopulations was formulated based on the natural history of CoViD-19 to study the transmission of the new coronavirus in São Paulo State, Brazil. This deterministic model used the data collected in São Paulo State to estimate the model parameters and to evaluate the effects of herd protection, that is, isolation and personal and collective protective measures. Based on the estimated parameters, we evaluated the scenarios of three releases divided in equal proportions elapsed by 14 days between releases, but beginning in three different times (the first release occurring on 1 and 23 June, and 6 July). We concluded that these three strategies of release are equivalent (little difference) in reducing the number of severe CoViD-19 if social behaviour does not change. However, if protective measures as using face mask and hygiene (washing hands, for instance) and social distancing could be massively disseminated in the population to decrease the transmission of CoViD-19 by 80%, we concluded that the health care system may not collapse with release.

AbstractTo flatten the curve of the natural epidemic of covid-19, many countries adopted lockdown or isolation resulting in the containment of the SARS-CoV-2 transmission. However, an important question arises about the strategies of release of isolated persons to avoid overloaded hospitals and increased deaths. São Paulo State (Brazil) implemented the isolation of the population in non-essential activities on March 24, and the progressive flexibilization considering the characteristics of each location (release of the isolated population) initiated on June 15. A mathematical model based on the natural history of covid-19 was applied to describe the epidemiological scenario with isolation in São Paulo State, and assess the impact of release on the covid-19 epidemic. Using data collected from São Paulo State, we estimated the model parameters to obtain the curves of the epidemic, the number of deaths, and the clinical evolution of covid-19. The epidemic under isolation was the framework to evaluate the strategies of the release, that is, how these curves are changed with the release of isolated persons. We evaluated three strategies of release. First two strategies considered four releases in the isolated population in four equal proportions, but successive releases elapsed by 14 and 21 days. In each strategy the beginning of the release was on June 29 and July 13, when the effective reproduction numberRefwas evaluated. The third strategy aimed at the protection of the elder subpopulation. We observed that the delay to begin the release and the increased elapse between successive releases resulted in a better scenario by decreasing severe covid-19 cases and, consequently, to avoid overloaded hospitals. We also observed that the release delayed to achieve lower values forRefand infectious persons retarded in several months the quick increasing phase of the forthcoming epidemic. However, this epidemic can be flattened or even suppressed by isolation of infectious persons by mass testing and/or by rigid adoption of protective measures and social distancing.

AbstractWe formulated a mathematical model considering young (below 60 years old) and elder (above 60 years) subpopulations to describe the introduction and dissemination of new coronavirus epidemics in the São Paulo State, Brazil. From the data collected in São Paulo State, we estimated the model parameters and calculated the basic reproduction number asR0= 6.828. Considering isolation as a control mechanism, we varied the releasing proportions of young and elder persons to assess their epidemiological impacts. The best scenarios were release of young persons, but maintaining elder persons isolated. To avoid the collapse of the health care system, the isolation must be at least 80%.