Reproduction Number Research Articles

Generalized contact matrices allow integrating socioeconomic variables into epidemic models.

Variables related to socioeconomic status (SES), including income, ethnicity, and education, shape contact structures and affect the spread of infectious diseases. However, these factors are often overlooked in epidemic models, which typically stratify social contacts by age and interaction contexts. Here, we introduce and study generalized contact matrices that stratify contacts across multiple dimensions. We demonstrate a lower-bound theorem proving that disregarding additional dimensions, besides age and context, might lead to an underestimation of the basic reproductive number. By using SES variables in both synthetic and empirical data, we illustrate how generalized contact matrices enhance epidemic models, capturing variations in behaviors such as heterogeneous levels of adherence to nonpharmaceutical interventions among demographic groups. Moreover, we highlight the importance of integrating SES traits into epidemic models, as neglecting them might lead to substantial misrepresentation of epidemic outcomes and dynamics. Our research contributes to the efforts aiming at incorporating socioeconomic and other dimensions into epidemic modeling.

TRANSMISSION AND CONTROL DYNAMICS OF ROTAVIRUS DIARRHEA MODEL WITH DOUBLE DOSE VACCINATION

This study introduces a six-compartmental mathematical model (S, , , E, I, R) to examine the impact of administering a double dose vaccine on the dynamic spread of diarrhea within a community. The mathematical analysis shows the existence of equilibrium points for both disease-free and endemic states in the model. The basic reproduction number was determined using the Next Generation Matrix. Analysis has shown that the basic reproduction number which indicates the disease-free equilibrium point is locally asymptotically stable. Also, using a suitable Lyapunov functional for the model system expressed in state variables and parameters defining the dynamic characteristics of spread and control strategies of the rotavirus diarrhea to obtain the global stability of disease-free equilibrium point over time. A numerical simulation was carried out by Wolfram Mathematica to show the effect of a second-dose vaccine. The inclusion of a double-dose vaccine has been found to have a significant effect on completely eliminating the outbreak of diarrhea. This is evidenced by the local and global stability results, which indicate that effective measures have been taken to prevent the reintroduction or transmission of the disease, and if there may be a risk of outbreaks or reemergence of the disease, very little continuous monitoring and intervention strategies are required to maintain control as this should be taken seriously by medical practitioners or policy health makers.

Stability and Bifurcation Analysis of a Symmetric Fractional-Order Epidemic Mathematical Model with Time Delay and Non-Monotonic Incidence Rates for Two Viral Strains

This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system’s solutions, the research ensures the model’s well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R01 and R02 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model’s applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (τ), individual heterogeneity (m, k), and memory effects (θ), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models.

The role of asymptomatic cattle for leptospirosis dynamics in a herd with imperfect vaccination

Leptospirosis is an emerging zoonotic disease with high health and economic damage. In this study, we developed a deterministic mathematical model that describes the dynamics of leptospirosis transmission within a cattle herd, incorporating asymptomatic infected and vaccinated compartments. The study examined the transmission role of asymptomatic cattle that contaminate herds without farmers’ knowledge. We proved the well-posedness of the proposed model and found the basic reproduction number using the next-generation matrix. Analytically, we demonstrated that the disease-free equilibrium point is locally and globally asymptotically stable when R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {R}}_0$$\\end{document} is less than unity and is otherwise unstable. Graphically, we further established the local asymptotic stability of disease-free and endemic equilibria. Sensitivity analysis showed that the contact rate with asymptomatic infected cattle, βA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _A$$\\end{document}, is the most sensitive parameter in the stated model, followed by the recovery rate of asymptomatic infected cattle, σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}, and the vaccination rate of susceptible cattle, τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Numerical simulations revealed that a reduction in contact rate with asymptomatic infected cattle significantly reduced pathogen Leptospira transmission in the herd. In addition, fostering the recovery rate of asymptomatic infected cattle can significantly reduce new infections in the herd. Furthermore, augmenting the vaccination rate among susceptible cattle resulted in a notable decrease in disease prevalence within the herd. Findings of this study underscore the remarkable importance of targeted interventions, such as reducing contact rates with asymptomatic infected cattle, increasing recovery rates using proper treatments, and enhancing vaccination efforts to manage leptospirosis transmission in cattle herds.

Power and sample size calculations for testing the ratio of reproductive values in phylogenetic samples.

The quality of the inferences we make from pathogen sequence data is determined by the number and composition of pathogen sequences that make up the sample used to drive that inference. However, there remains limited guidance on how to best structure and power studies when the end goal is phylogenetic inference. One question that we can attempt to answer with molecular data is whether some people are more likely to transmit a pathogen than others. Here we present an estimator to quantify differential transmission, as measured by the ratio of reproductive numbers between people with different characteristics, using transmission pairs linked by molecular data, along with a sample size calculation for this estimator. We also provide extensions to our method to correct for imperfect identification of transmission linked pairs, overdispersion in the transmission process, and group imbalance. We validate this method via simulation and provide tools to implement it in an R package, phylosamp.

Host plant suitability for the biological performance of Tetranychus urticae Koch and its predator, Phytoseiulus persimilis Athias-Henriot

The life table parameters provide the most extensive information on the biological features of pests and natural enemies, facilitating the potential employment of biological agents in pest management programs. This study investigated how cucumber, strawberry, eggplant, and tomato plants influence the reproduction and life-table parameters of Tetranychus urticae Koch and its predator, Phytoseiulus persimilis Athias-Henriot under laboratory conditions. It also explored the relationship between these parameters and leaf trichome density and the levels of polyphenol compounds. The mites’ biological properties varied based on the morphological and chemical differences in host plant leaves. Both T. urticae and P. persimilis exhibited higher development and reproduction rates on strawberry and cucumber leaves compared to eggplant and tomato plants. Eggplant leaves had the highest polyphenol content (3826.99 mg/kg), followed by tomato leaves (2406.86 mg/kg). The lowest total amount of polyphenol compounds was (704.34 mg/kg) and (273.743 mg/kg) in strawberry and cucumber leaves, respectively. The life table parameters of P. persimilis significantly differed when it fed on T. urticae that had been reared on strawberry, cucumber, eggplant, and tomato leaves. Overall, the results showed that the net (R0) and gross (GRR) reproductive rates were reduced when the predator preyed T. urticae that had been reared on eggplant and tomato leaves. Consequently, the intrinsic (rm) and finite (λ) rates of increase were affected. As a result, the performance of P. persimilis was negatively affected on these plants. The availability of information on the biological performance of P. persimilis is necessary for promoting biological control implementation.

Prenatal phthalate exposure and pubertal development in 16-year-old daughters: reproductive hormones and number of ovarian follicles.

Is there a possible association between prenatal phthalate exposure and late effects in teenage daughters with respect to reproductive hormone levels, uterine volume, and number of ovarian follicles? Our study showed subtle associations between phthalate metabolite concentrations in maternal serum from pregnancy or cord blood and LH and insulin-like growth factor 1 (IGF-1) levels as well as uterine volume in their daughters 16 years later. Endocrine-disrupting environmental chemicals may adversely affect human reproductive health, and many societies have experienced a trend toward earlier puberty and an increasing prevalence of infertility in young couples. The scientific evidence of adverse effects of foetal exposure to a large range of chemicals, including phthalates, on male reproductive health is growing, but very few studies have explored effects on female reproduction. This follow-up study included 317 teenage daughters who were part of the Copenhagen Mother-Child Cohort, a population-based longitudinal birth cohort of 1210 females born between 1997 and 2002. A total of 317 female participants (median age 16 years) were examined for weight, height, and menstrual pattern. A serum sample was analysed for concentrations of reproductive hormones, and trans-abdominal 3D ultrasonography was performed to obtain the number of ovarian follicles, ovarian and uterine size. Prenatal maternal serum samples were available for 115 females, and cord blood samples were available for 118 females. These were analysed for concentrations of 32 phthalate metabolites. Weighted quantile sum regression was used for modelling associations of combined prenatal phthalate exposure with the reproductive outcomes in post-menarcheal females. In bivariate correlation analyses, negative significant associations were found between several prenatal phthalate metabolite concentrations and serum hormone concentrations (testosterone, 17-OH-progesterone, and IGF-1) as well as number of ovarian follicles in puberty. Positive significant correlations were found between prenatal phthalate exposure and FSH and sex hormone-binding globulin concentrations. Combined analyses of phthalate exposure (weighted quantile sums) showed significant negative associations with IGF-1 concentration and uterine volume as well as a significant positive association with LH concentration. Phthalate metabolites were measured in serum from single prenatal maternal blood samples and cord blood samples. Potential concomitant exposure to other endocrine-disrupting environmental chemicals before or after birth was not controlled for. The study population size was limited. Our results support the need for further research into possible adverse effects of environmental chemicals during foetal development of the female reproductive system. The work was supported by The Center on Endocrine Disruptors (CeHoS) under The Danish Environmental Protection Agency and The Ministry of Environment and Food (grant number: MST-621-00 065). No conflicts of interest are declared. N/A.

Overcoming bias in estimating epidemiological parameters with realistic history-dependent disease spread dynamics

Epidemiological parameters such as the reproduction number, latent period, and infectious period provide crucial information about the spread of infectious diseases and directly inform intervention strategies. These parameters have generally been estimated by mathematical models that involve an unrealistic assumption of history-independent dynamics for simplicity. This assumes that the chance of becoming infectious during the latent period or recovering during the infectious period remains constant, whereas in reality, these chances vary over time. Here, we find that conventional approaches with this assumption cause serious bias in epidemiological parameter estimation. To address this bias, we developed a Bayesian inference method by adopting more realistic history-dependent disease dynamics. Our method more accurately and precisely estimates the reproduction number than the conventional approaches solely from confirmed cases data, which are easy to obtain through testing. It also revealed how the infectious period distribution changed throughout the COVID-19 pandemic during 2020 in South Korea. We also provide a user-friendly package, IONISE, that automates this method.

Biophysical risk in 189 Italian prisons: an inference analysis from the second to the fourth wave of the COVID-19 pandemic

Previous studies have highlighted how SARS-CoV-2 has entered Italian prisons, causing a slightly higher number of infections compared to the Italian population. This result has not been confirmed, on the contrary, with regard to the values of "Reproductive number over time" (R(t)) and "Prevalence of hospitalisation" (H). In particular, R(t), i.e. the average number of secondary transmissions caused by an infected person, is a useful parameter for measuring the infectivity of bioagents and the effectiveness of public health measures to control the spread of the infectious disease. Previous results have been obtained from data collected from all Italian prisons and therefore represent averages of many and varied realities. On the contrary, the aim of this study is to analyse and compare the Infected (I) and the R(t), in each prison, and to search for the side variables that may favour the spread of the virus in these places. The R(t) evaluation is obtained using a differential equation based on the SIR model, but unlike its usual application, no simplification has been made (i.e. population (N) ≠ susceptible (S)). Preliminary results focus on the period between September 2020 and March 2022. They show that during the second and third waves, the infection rate followed a gradient from north to south and to the islands, while the maximum R(t) values were randomly distributed across the peninsula. Moreover, we describe as the maximum R(t) values are poorly correlated with the crowding rate of each prison and the same was obtained when examining other variables such as the type of prison and the presence/absence of health isolation sections. On the contrary, there is a significant correlation when the independent variable is the presence of staff. This analysis will provide the basis for the development of other linear and nonlinear physical models that could be helpful in predicting the spread of infectious diseases in closed and controlled places.

An immuno-epidemiological model with non-exponentially distributed disease stage on complex networks

Most of epidemic models assume that duration of the disease phase is distributed exponentially for the simplification of model formulation and analysis. Actually, the exponentially distributed assumption on the description of disease stages is hard to accurately approximate the interplay of drug concentration and viral load within host. In this article, we formulate an immuno-epidemiological epidemic model on complex networks, which is composed of ordinary differential equations and integral equations. The linkage of within- and between-host is connected by setting that the death caused by the disease is an increasing function in viral load within host. Mathematical analysis of the model includes the existence of the solution to the epidemiological model on complex networks, the existence and stability of equilibrium, which are completely determined by the basic reproduction number of the between-host system. Numerical analysis are shown that the non-exponentially distributions and the topology of networks have significant roles in the prediction of epidemic patterns.

A two-strain COVID-19 co-infection model with strain 1 vaccination

COVID-19 has caused substantial morbidity and mortality worldwide. Previous models of strain 1 vaccination with re-infection when vaccinated, as well as infection with strain 2 did not consider co-infected classes. To fill this gap, a two co-circulating COVID-19 strains model with strain 1 vaccination, and co-infected is formulated and theoretically analyzed. Sufficient conditions for the stability of the disease-free equilibrium and single-strain 1 and -strain 2 endemic equilibria are obtained. Results show as expected that (1) co-infected classes play a role in the transmission dynamics of the disease (2) a high efficacy vaccine could effectively help mitigate the spread of co-infection with both strain 1 and 2 compared to the low-efficacy vaccine. Sensitivity analysis reveals that the main drivers of the effective reproduction number Re are primarily the effective contact rate for strain 2 (β2), the strain 2 recovery rate (τ2), and the vaccine efficacy or infection reduction due to the vaccine (η). Thus, implementing intervention measures to mitigate the spread of COVID-19 should not ignore the co-infected individuals who can potentially spread both strains of the disease.

The impact of lethal and sub-lethal exposure of emamectin benzoate on populations of Spodoptera litura (Lepidoptera: Noctuidae) under laboratory conditions

Emamectin benzoate is an avermectin bio-insecticide commonly used for managing several insect pests including Spodoptera litura (Fabricius) (Lepidoptera: Noctuidae), a major polyphagous pest of many cultivated crops. The current study was conducted to evaluate the effects of emamectin benzoate on the fitness of S. litura populations exhibiting differential susceptibility to insecticide. The selection process and all the bioassays were carried out using 6-day-old 2nd instar larvae of S. litura. A field-collected population of S. litura was divided into two groups: one selected with emamectin benzoate for eight generations (EB-Sel) and the other kept unexposed (Unsel-Lab) to insecticide in the laboratory. An increase in resistance ratio from 1.71-fold in the F1 generation to 22.54-fold in the F8 generation of the EB-Sel population was observed compared to the Unsel-Lab (F8) population. The EB-Sel and Unsel-Lab populations were treated with their respective lethal and sub-lethal concentrations which resulted in an extended development period, decreased larval survival, and adult emergence along with increased morphological abnormalities in adults. Significant reductions were observed in both male and female longevity, fecundity, egg hatching, net reproductive rate (R0), intrinsic rate of increase (rm), and finite rate of increase (λ) in EB-Sel and Unsel-Lab populations. Higher concentrations of the insecticide also reduced the relative fitness (Rf) of S. litura larvae, with maximum effect at LC50 of the EB-Sel population where the Rf value was 0.32 compared to the Unsel-Lab population. Both populations have been affected by emamectin benzoate exposure, however, the impact was more pronounced in the EB-Sel population indicating fitness costs. Our results suggested the fitness cost linked to emamectin benzoate resistance in S. litura which might favor managing insecticide resistance by reducing the frequency of resistant alleles by removing selection pressure. Consequently, our research provides significant insights to devise better pest management strategies for S. litura.

Modeling the impact of screening awareness and vaccination on human papilloma virus transmission

In this work, a two‐sex deterministic model for human papillomavirus (HPV) is developed to assess the impact of screening awareness and vaccination on the transmission dynamics. We provide a detailed analysis of the HPV model and established results for boundedness of the solutions and existence of a positively invariant and attracting set. The basic reproduction number is characterized, and it is shown that the disease‐free equilibrium is locally asymptotically stable if and the system is uniformly persistent if . Sensitivity analysis of reveals that HPV vaccination can mitigate HPV transmission, and HPV vaccination along with screening can further reduce the spread of HPV infection. Numerical simulations showed that the public awareness of screening after being vaccinated is contributing to reduce the spread of epidemic, and the role of screening awareness in controlling the disease was significantly greater among men than women. Besides, the numerical simulations reveal that increasing vaccination rates for male is effective in controlling HPV transmission. Hence, it is necessary to enhance the screening for men. Moreover, increasing vaccination rates is more effective than increasing screening awareness in eliminating the epidemic.

Genetic analysis for reproductive and longevity traits in Landlly sows

The rate of reproduction and lifetime productivity of the animals determine the success of a herd. As these features significantly impact sow productivity, welfare, and profitability, there is considerable interest in global swine breeding programmes to genetically select better sows for these attributes. In this scenario, reliable heritability estimates and genetic correlations between traits are essential for the success of genetic selection for such traits. The goals of the current study were to examine the reproductive and longevity traits and to estimate the genetic and phenotypic parameters for these traits in Landlly sows kept at a swine production farm, Izatnagar, Uttar Pradesh, India. The effect of non-genetic factors on different reproductive traits was estimated using SPSS version 16.0 and Bayesian approach which were utilized to estimate the genetic parameters for reproductive and longevity traits. The current study’s finding that the year and season of birth significantly affect reproductive traits generally suggests that there is potential for boosting productivity by optimising managemental practices. The heritability estimates were low for the reproductive and longevity traits, indicating a very little additive genetic variance in these traits, and individual selection will not be helpful for improving them. As some traits, such as LSB_FF & LSW_FF and LSB _SF & LSW_SF, have a moderate genetic correlation, indirect selection can be used to improve these two pairs of features.

Reproductive Coercion: Prevalence and Risk Factors Related to Relationship Health Knowledge and Skills.

Reproductive coercion is any behavior that limits a person's reproductive decision-making and can lead to negative health and safety outcomes. Previous research has explored reproductive coercion prevalence rates in clinical samples, as well as demographic risk factors for experiencing reproductive coercion. The purpose of this study is to assess the prevalence rates of two specific forms of reproductive coercion, pregnancy coercion and condom manipulation, in an ethnically and racially diverse sample of young females. We also explore the association between relationship health knowledge and skills with reproductive coercion. We used a sample of 143 females with previous sexual activity. Participants were diverse in terms of race, ethnicity, sexual orientation, and family immigration status. Descriptive statistics and logistic regression analyses were used to determine prevalence rates of pregnancy coercion and condom manipulation and the likelihood of experiencing reproductive coercion based on participants' knowledge and skills related to relationship health. Results revealed that 16.1% of the sample had experienced reproductive coercion, with all participants in this group reporting lifetime experiences of pregnancy coercion. Lifetime experiences of condom manipulation were reported by 6.3% of the sample. The most common form of reproductive coercion experienced by participants was being told by a partner not to use any birth control. Furthermore, results indicate that higher relationship health knowledge may be a protective factor for pregnancy coercion and condom manipulation. Likewise, higher decision-making skills in relationships and higher confidence in relationships may also protect against condom manipulation. Results from this study suggest implications for sexual and relationship health programming that expands education around consent, choice, decision-making, and communication around the use of contraception.

A Between-Host Cholera Mathematical Model Incorporating Temperature Dependence

This paper establishes a between-host cholera model with temperature dependent parameter. This is done using system of ODEs to analyse the effect of temperature change on cholera disease. The model analysis reveals that when R0 < 1, the disease free equilibrium point is locally and globally asymptotically. It is also noticed that if R0 > 1, the endemic equilibrium point is locally and globally asymptotically stable. The sensitivity analysis of model parameters shows that R0 ​depends intensively on infection rate of pathogen α1​ normalized with temperature. An increase in infection rate of pathogen α1​ that is dependent on temperature by 10% would increase R0​ by 10% and decreasing it by 10% reduces R0 by 10%; hence, increasing the temperature of the environment where the pathogen lives would help reduce the rate of infection of the pathogen, thus reducing the reproduction number R0​. We conducted numerical simulation of the model in response to temperature changes, and the results indicate that Vibrio cholerae pathogens multiply faster at 23°C but between 23°C < T ≤ 43°C the pathogen multiplication is hindered, therefore, at 23°C, more pathogens are active to cause infection compared to high temperatures.

Ecological and epidemiological ramification of fear: Exploring deterministic and stochastic dynamics in a predator–prey system with predator switching and harvesting

In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels.

The optimal spatially-dependent control measures to effectively and economically eliminate emerging infectious diseases.

Non-pharmaceutical interventions (NPIs) are effective in mitigating infections during the early stages of an infectious disease outbreak. However, these measures incur significant economic and livelihood costs. To address this, we developed an optimal control framework aimed at identifying strategies that minimize such costs while ensuring full control of a cross-regional outbreak of emerging infectious diseases. Our approach uses a spatial SEIR model with interventions for the epidemic process, and incorporates population flow in a gravity model dependent on gross domestic product (GDP) and geographical distance. We applied this framework to identify an optimal control strategy for the COVID-19 outbreak caused by the Delta variant in Xi'an City, Shaanxi, China, between December 2021 and January 2022. The model was parameterized by fitting it to daily case data from each district of Xi'an City. Our findings indicate that an increase in the basic reproduction number, the latent period or the infectious period leads to a prolonged outbreak and a larger final size. This indicates that diseases with greater transmissibility are more challenging and costly to control, and so it is important for governments to quickly identify cases and implement control strategies. Indeed, the optimal control strategy we identified suggests that more costly control measures should be implemented as soon as they are deemed necessary. Our results demonstrate that optimal control regimes exhibit spatial, economic, and population heterogeneity. More populated and economically developed regions require a robust regular surveillance mechanism to ensure timely detection and control of imported infections. Regions with higher GDP tend to experience larger-scale epidemics and, consequently, require higher control costs. Notably, our proposed optimal strategy significantly reduced costs compared to the actual expenditures for the Xi'an outbreak.

Stability Analysis and Optimal Control for Spreading Typhoid Fever Model with Direct and Indirect Transmissions

This article discusses a model of spreading typhoid fever with direct and indirect transmissions. There are five compartments included in the model, namely susceptible individuals, infected individuals, chronic carrier individuals, recovered individuals, and salmonella typhi bacteria in the environment. As an effort to control and minimize the numbers of infected individuals and chronic carrier individuals, we introduce simultaneously campaign for the susceptible individuals and treatment for the infected individuals and for chronic carrier individuals. The resulting model is a system of non-linear differential equations and quite challenging to analysis globally. Existence of both disease-free equilibrium point and endemic equilibrium point are analysed analytically. Stability of the equilibrium point is analysed locally by determining the eigenvalues of the associated Jacobian matrix, Routh-Hurwitz stability criteria, and basic reproduction number () via next generation matrix. The minimum Pontryagin principle and Hamiltonian equation are referred to minimize the numbers of infected and chronic carrier individuals. Simulation is carried out using suitable values of parameters. We found that the eigenvalues for endemic equilibrium point are all negative real numbers and . Optimal paths for state and constate variables are plotted using fourth order forward-backward Runge-Kutta method. In case there are no campaign and treatments, we found and endemic occurs. When campaign and treatments are included together in the model, we found optimal paths for all compartments that minimize the number of infected and chronic carrier individuals. Giving campaign and treatments increase significantly the susceptible and recovered individuals. At the same time, it reduces significantly the infected, chronic carrier individuals, and salmonella typhi bacteria in the environment. Involving campaign and treatment in the model of spreading typhoid fever can be considered as an effective strategy to minimize the numbers of infected and chronic carrier individuals.

The role of some types of erythrocytes on the growth of Entamoeba histolytica trophozoite in vitro

The parasite E.histolytica was first isolated from a stool sample, and then cultivated and maintained in vitro using Locke-egg medium (LEM) and Liver infusion agar medium (LIAM) . Then, the effect of some types of erythrocytes (human and sheep), on the growth and activity of the parasite in the two culture media was investigated. The parasite was able to ingest and lysis erythrocytes of human and sheep that were supplemented to the culture media and such manipulation was able to augment the reproduction rate of the cultivated E. histolytica, however, such consequence was media- and concentration-dependent. The reproduction rate was significantly increased (66.0, 57.5 and 58.6%, respectively) in LEM medium containing human erythrocytes types B at 0.11 x 106 cells/ml and O at 0.13 x 106 and 0.15 x 106 cells/ml. The sheep erythrocytes showed a similar enhancement (56.1%) at a concentration of 0.13 x 106 cells/ml. In contrast, adding erythrocytes to LIAM medium did not enhance the reproduction rate of the parasite significantly.