Обучение по природни науки и върхови технологии

https://doi.org/10.53656/nat2025-1.04

2025/1, стр. 71 - 93

MATHEMATICAL MODELLING OF THE TRANSMISSION DYNAMICS OF PNEUMONIA AND MENINGITIS COINFECTION WITH VACCINATION

Резюме: Pneumonia and meningitis pose substantial threats to global public health due to their high morbidity and mortality rates. This study investigates the dynamics of these diseases with a focus on coinfection and evaluates the effectiveness of vaccination as a control measure. Using a mathematical model, the transmission dynamics are explored and the basic reproduction number is derived to identify conditions for disease-free and endemic states. Numerical simulations analyze the impact of varying vaccination compliance levels, demonstrating that higher compliance significantly reduces the number of susceptible and infected individuals while increasing the vaccinated population. The study emphasizes the need for integrated public health strategies combining vaccination campaigns, ecient vaccine distribution, and supportive medical care to mitigate the burden of pneumonia and meningitis coinfections. These findings provide a framework for designing effective interventions aimed at reducing disease prevalence and improving public health outcomes.

Ключови думи: Pneumonia; Meningitis; Coinfection; Mathematical Modelling; Transmission Dynamics, Vaccination; Sensitivity Analysis

https://doi.org/10.53656/nat2025-1.04

2025/1, стр. 71 - 93

MATHEMATICAL MODELLING OF THE TRANSMISSION DYNAMICS OF PNEUMONIA AND MENINGITIS COINFECTION WITH VACCINATION

Deborah O. Daniel, Sefiu A. Onitilo, Omolade B. Benjamin, Ayoola A. Olasunkanmi Olabisi Onabanjo University Ago Iwoye – Ogun State, Nigeria

Abstract. Pneumonia and meningitis pose substantial threats to global public health due to their high morbidity and mortality rates. This study investigates the dynamics of these diseases with a focus on coinfection and evaluates the effectiveness of vaccination as a control measure. Using a mathematical model, the transmission dynamics are explored and the basic reproduction number is derived to identify conditions for disease-free and endemic states. Numerical simulations analyze the impact of varying vaccination compliance levels, demonstrating that higher compliance significantly reduces the number of susceptible and infected individuals while increasing the vaccinated population. The study emphasizes the need for integrated public health strategies combining vaccination campaigns, ecient vaccine distribution, and supportive medical care to mitigate the burden of pneumonia and meningitis coinfections. These findings provide a framework for designing effective interventions aimed at reducing disease prevalence and improving public health outcomes.

Keywords: Pneumonia; Meningitis; Coinfection; Mathematical Modelling; Transmission Dynamics, Vaccination; Sensitivity Analysis

1. Introduction

Infectious diseases remain a persistent challenge for public health systems worldwide, causing substantial illness and death. Among these, pneumonia and meningitis are particularly concerning due to their prevalence and severe consequences. Pneumonia, primarily caused by pathogens such as Streptococcus pneumoniae, is a major cause of death, particularly in young children and older adults (McLuckie 2009). Meningitis, commonly caused by bacteria like Neisseria meningitidis and Haemophilus inuenzae, is a life-threatening condition that affects the brain and spinal cord, often resulting in significant fatalities or long-term disabilities (Fresnadillo Martínez et al. 2013; Afolabi et al. 2021). Addressing these diseases is a key priority for reducing their burden on global health.

Pneumonia affects the respiratory system and is transmitted through airborne particles. It is responsible for millions of deaths annually, with Streptococcus pneumoniae being a leading cause (World Health Organization, 1991). This bacterium is also linked to other serious infections, including meningitis and sinusitis (Opatowski et al. 2013; Kotola et al. 2022). Meningitis primarily targets the central nervous system and is associated with high mortality and long-term neurological complications (Zunt et al. 2018; Tilahun 2019). When pneumonia and meningitis occur together, they pose significant challenges to clinical management and public health efforts. This combination increases the severity of illness and places additional strain on healthcare systems (Tabatabaei et al. 2022; Chukwu et al. 2020). Despite these challenges, the mechanisms of coinfection remain insuciently studied, leaving gaps in our understanding of effective treatment and prevention strategies.

Coinfection, where two or more pathogens infect a person simultaneously, has significant implications for disease progression, transmission, and treatment outcomes (Kehr & Engelmann, 2015). The combined occurrence of pneumonia and meningitis is particularly concerning, as it may worsen patient outcomes and complicate medical interventions (Obi et al. 2010; Kotola & Mekonnen 2022). Understanding the interplay between these infections is critical for developing effective public health strategies and improving clinical outcomes.

Mathematical modeling plays a pivotal role in understanding and addressing infectious diseases by offering frameworks to simulate disease dynamics and assess intervention strategies (Tilahun et al. 2018; Asamoah et al. 2018). According to Bailey (1975), the primary goal of mathematical modeling in epidemiology is to support informed decision-making. Models help evaluate the most costeffective approaches to minimize the adverse impacts of diseases (Di Liddo 2016; Kizito & Tumwiine 2018). Researchers have developed models for pneumonia and meningitis individually, enhancing our understanding of their spread and informing control measures (Blyuss 2016; Joseph 2012; Musa et al. 2020). Pneumonia, for instance, has been the subject of extensive modeling efforts due to its significant global burden (McLuckie 2009; Joseph 2012). Tilahun (2019) utilized the SIR model to explore the co-dynamics of pneumonia and meningitis, contributing valuable insights into their interactions.

The application of mathematical models extends to evaluating vaccination and treatment strategies. For example, Tilahun et al. (2017) developed an SVCIR model to investigate cost-effective control strategies for pneumonia, concluding that a combination of prevention and treatment yields the most significant impact. Similarly, Onyinge et al. (2016) formulated models to analyze pneumonia coinfection with HIV/AIDS, highlighting the importance of combined intervention strategies. These efforts underscore the utility of mathematical frameworks in optimizing resource allocation for disease management.

In the case of meningitis, mathematical modeling has been instrumental in understanding its epidemiology and guiding control efforts. Meningitis remains a severe global health threat, particularly in resource-limited settings (Zunt et al. 2018; Ghia & Rambhad 2021; Abdullahi Baba et al. 2020). Outbreaks of bacterial meningitis, often caused by pathogens like Streptococcus pneumoniae and Neisseria meningitidis, demand urgent interventions (Van De Beek et al. 2010; Scarborough & Thwaites, 2008; Türkün et al. 2023). Models have been used to predict outbreak dynamics and assess the impact of vaccination programs (Jayaraman et al. 2018; Oordt-Speets et al. 2018; Kotola et al. 2022). For instance, the introduction of vaccines targeting Haemophilus inuenzae type b and Streptococcus pneumoniae has significantly reduced the incidence of bacterial meningitis in highincome countries, though the burden persists in low- and middle-income regions (McIntyre et al. 2012; Peter et al. 2021).

Despite these advances, the combined dynamics of pneumonia and meningitis remain underexplored, particularly in the context of vaccination. Few studies have addressed their coinfection using mathematical models. Tilahun (2019) developed a model incorporating both diseases and emphasized the need for integrated approaches to improve intervention outcomes. Such efforts are essential for addressing the complexities of coinfection, where interactions between diseases may amplify severity and complicate treatment (Kotola & Mekonnen 2022; Chukwu et al. 2020). Still, studies on their combined dynamics remain limited, and existing models often fail to capture the combined effects of these diseases or explore the role of vaccination in reducing their impact. This highlights the need for a comprehensive approach to modeling pneumonia-meningitis coinfection, particularly in resource-constrained settings. This study seeks to address these gaps by constructing a mathematical model to examine the dynamics of pneumoniameningitis coinfection considering the impact of vaccination.

2. Mathematical Formulation

In this study, we present a deterministic mathematical model designed to capture the transmission dynamics of pneumonia and meningitis coinfection within a population. The model divides the population into seven distinct compartments. These compartments include susceptible individuals (denoted as \(S\) ), who are healthy but can be infected by pneumonia, meningitis, or both; vaccinated individuals (\(V\) ), who have received protection through vaccination; pneumonia-infected individuals (Ip), who are infected solely with pneumonia and can transmit the disease to others; meningitis-infected individuals (Im), who are infected solely with meningitis; coinfected individuals (Ipm), who are infected by both pneumonia and meningitis; and recovered individuals (\(R\) ), representing those who have recovered from pneumonia, meningitis, or both diseases, or those who have removed themselves from the transmission cycle through vaccination.

The total population at any time \(t\) is the sum of all these compartments, given by \[ N(t)=S(t)+I_{p}(t)+I_{m}(t)+I_{p m}(t)+R(t)+V(t) . \]

The dynamics of each compartment are governed by a system of differential equations that describe how individuals move between compartments due to factors such as infection, vaccination, recovery, and death. The rate of change for each compartment is determined by various transmission, recovery, and death rates.

The susceptible population, denoted by \(S(t)\), is inuenced rate \(\Lambda\) , and it is impacted by the force of infection due to contact with infected individuals, which is represented by the transmission rates \(\sigma_{1}, \sigma_{2}\) and \(\sigma_{3}\) for pneumonia, meningitis, and coinfection, respectively. The susceptible compartment also decreases due to natural deaths at a rate \(\mu\), as well as due to recovery from either pneumonia or meningitis. Furthermore, there is a transfer from the vaccinated group to the susceptible group, modeled by the rate \(\chi\), as vaccination immunity wanes. This is balanced by a natural recruitment rate of new susceptible individuals, with vaccination and recovery contributing to the pool of new individuals who might be susceptible again at rate \(\theta\).

The vaccinated individuals, represented by \(V(t)\), are recruited from the susceptible population through vaccination, but they are also susceptible to infection at a reduced rate due to the effectiveness of the vaccine. The vaccination rate is affected by the contact with infected individuals as well as the natural death rate. Additionally, some vaccinated individuals may become susceptible again if immunity wanes,which is modeled by the transfer rate \(\chi\).

The compartment for pneumonia-infected individuals, \(I_{p}(t)\), represents those infected only with pneumonia. These individuals become infected through contact with susceptible individuals or those vaccinated but not fully protected. The rate of change of this group is influenced by the transmission rate for pneumonia,\(\sigma_{p}\), as well as the rate of recovery, denoted by \(\gamma_{1}\) , and the death rate due to pneumonia complications, represented by \(\mu\). These individuals either recover or die from pneumonia, and they may transition to the recovered state at a rate \(\delta_{l}\) , which reects the rate of recovery from pneumonia.

Similarly, the meningitis-infected individuals, \(I_{m}(t)\) , represent those infected solely with meningitis. The rate of change of this compartment depends on the transmission rate \(\sigma_{2}\) and the interaction between susceptible and vaccinated individuals. These individuals also recover from meningitis at a rate \(\gamma_{2}\) and may transition to the recovered state at a rate \(\delta_{2}\) , while some individuals may die from meningitis at a rate \(\mu\).

The coinfected individuals, \(I_{p m}(t)\) , represent those who are infected with both pneumonia and meningitis. The dynamics of this group are inuenced by the transmission rates of both pneumonia and meningitis. The rate of change in this compartment depends on the interaction between the pneumonia and meningitis infected individuals, and the recovery rate from coinfection is denoted by \(\gamma_{3}\). Coinfected individuals also transition to the recovered state at a rate defined by \(\delta_{1}\) and \(\delta_{2}\) , and some may die due to the complications of having both infections.

The recovered individuals, \(R(t)\), represent those who have recovered from pneumonia, meningitis, or coinfection. Their rate of recovery is governed by the recovery rates \(\gamma_{1}, \gamma_{2}\) and \(\gamma_{3}\) , corresponding to pneumonia, meningitis, and coinfection recovery, respectively. However, these individuals can be removed from the recovered group due to natural deaths or loss of immunity, as modeled by the natural death rate \(\mu\) and the immunity loss rate \(\theta\). The system of differential equations describing these dynamics is given by:

where the parameters are defined as follows:

\(\left(\sigma_{3}=\sigma_{1}+\sigma_{2}, \sigma_{4}=\sigma_{2}\left(I_{m}+I_{p m}\right), \sigma_{5}=\sigma_{l}\left(I_{p}+I_{p m}\right)\right.\)

The mathematical model proposed in this study aims to capture the transmission dynamics of pneumonia and meningitis coinfection, with a particular focus on the role of vaccination as a public health intervention. To keep the model tractable while still maintaining its ability to reect real-world scenarios, several assumptions have been made which include the following:

The population is homogeneous, with equal contact rates between individuals.

Vaccination is modeled as a constant rate, affecting the overall population without individual-level variation.

No cross-protection is assumed between pneumonia and meningitis.

Vaccine effectiveness is constant across the entire population.

These assumptions balance the need for simplicity and computational feasibility with the necessity of accurately representing the underlying epidemiological processes.

2.1. Qualitative Analysis

2.1.1. Invariant Region

The invariant region defines the domain where the solutions of the coinfection model are both biologically and mathematically valid. It is essential to demonstrate that the region \(\Omega\), where the model is feasible, remains positively invariant for all \(t>0\). This ensures that the system described by equation (1) is well-posed.

Theorem 1. The region \(\Omega=\left\{\left(S, V, I_{p}, I_{m}, I_{p m}, R\right) \in \mathbb{R}_{+}^{6}: \mathcal{N}(t) \leq \frac{\Lambda}{\mu}\right\}\) is positively

invariant. Proof. The total population at any time \({ }^{t}\) is defined as: \(\mathcal{N}(t)=S(t)+V(t)+I_{p}(t)+I_{m}(t)+I_{p m}(t)+R(t)\). The rate of change of the total population is given by:

\(\frac{d \mathcal{N}}{d t}=\frac{d S}{d t}+\frac{d V}{d t}+\frac{d I_{p}}{d t}+\frac{d I_{m}}{d t}+\frac{d I_{p m}}{d t}+\frac{d R}{d t}\).

Substituting from equation (1), we get:

\(\frac{d \mathcal{N}}{d t}=\Lambda-\mu \mathcal{N}\).

Integrating this differential equation yields: \(\mathcal{N}(t) \leq \frac{\Lambda}{\mu}+C e^{-\mu t}\),

where \(C\) is a constant of integration.

As \(t \rightarrow \infty\), the total population \(\mathcal{N}(t)\) approaches:

\(\mathcal{N}(t) \leq \frac{\Lambda}{\mu}\).

Hence, the feasible region for the solution of the system is within \(\Omega\), making the region positively invariant and ensuring that the model is biologically meaningful.

2.1.2. Positivity of the Solutions

The positivity of the solutions ensures that all model variables remain nonnegative over time. Since the coinfection model represents human populations, it is assumed that all variables and parameters are positive for \(t \geq 0\) .

Theorem 2. Given initial conditions \(S(0), V(0), I_{p}(0), I_{m}(0), I_{p m}(0), R(0)>0\), the solutions of the system (1) remain positive for all \(t \geq 0\) .

Proof. Starting with the first equation in the system (1):

\[ \frac{d S}{d t}=(1-\kappa) \Lambda-\left(\sigma_{1} I_{p}+\sigma_{2} I_{m}+\sigma_{3} I_{p m}\right) S+\theta R-(\zeta+\mu) S \]

Rewriting, we have:

\[ \frac{d S}{d t} \geq-\left(\sigma_{1} I_{p}+\sigma_{2} I_{m}+\sigma_{3} I_{p m}\right) S-(\zeta+\mu) S \]

Using separation of variables and integrating: \[ S(t)=S(0) e^{-\left(\sigma_{1} l_{p}+\sigma_{2} l_{m}+\sigma_{3} l_{p m}+\zeta+\mu\right) t} \]

Since \(S(0)>0\), it follows that \(S(t)>0\) for all \(t \geq 0\).

A similar approach can be used for \(V(t), I_{p}(t), I_{m}(t), I_{p m}(t)\), and \(R(t)\), showing that

all state variables remain positive over time. Thus, the solutions of the system are positive for all t ≥ 0.

2.2. Disease-Free Equilibrium

The disease-free equilibrium (DFE) of a coinfection model represents the steadystate solutions when no infections are present in the population. Let \(E_{0}\) denote the disease-free equilibrium. Setting \(I_{p}(t), I_{m}(t)\), and \(I_{p m}(t)\), to zero in equation (1) and solving for \(S(t)\), we obtain: \[ E_{0}=\left(S_{0}, V_{0}, 0,0,0,0\right) \] where

\[ S_{0}=\frac{(1-\kappa) \Lambda}{\zeta+\mu}, V_{0}=\frac{\left(\kappa \Lambda+\zeta S_{0}\right)}{\chi+\mu} \]

2.2.1. The Effective Reproduction Number

The effective reproduction number, \(R_{E}\) , is a critical threshold that determines whether an infection can invade and persist in a population in the presence of intervention. For this coinfection model, \(R_{E}\) is calculated as the dominant eigenvalue of the next-generation matrix, defined as: \[ R_{E}=\rho\left(F V^{-1}\right) \] where \(\rho\) is the spectral radius of the matrix product \(F V^{-1}\) . The matrices \(F\) and \(V\) are defined as follows:

\[ \begin{aligned} & F=\left(\begin{array}{ccc} \sigma_{1}\left(S_{0}+\phi V_{0}\right) & 0 & \sigma_{1}\left(S_{0}+\phi V_{0}\right) \\ 0 & \sigma_{2}\left(S_{0}+V_{0}\right) & \sigma_{2}\left(S_{0}+V_{0}\right) \\ 0 & 0 & 0 \end{array}\right) \\ & V=\left(\begin{array}{ccc} \left(\gamma_{1}+\delta_{1}+\mu\right) & 0 & 0 \\ 0 & \left(\gamma_{2}+\delta_{2}+\mu\right) & 0 \\ 0 & 0 & \left(\delta_{1}+\delta_{2}+\gamma_{3}+\mu\right) \end{array}\right) \end{aligned} \]

The next-generation matrix is given by:

\(F V^{-1}=\left(\begin{array}{ccc}\frac{\sigma_{1}\left(S_{0}+\phi V_{0}\right)}{\gamma_{1}+\delta_{1}+\mu} & 0 & \frac{\sigma_{1}\left(S_{0}+\phi V_{0}\right)}{\delta_{1}+\delta_{2}+\gamma_{3}+\mu} \\ 0 & \frac{\sigma_{2}\left(S_{0}+V_{0}\right)}{\gamma_{2}+\delta_{2}+\mu} & \frac{\sigma_{2}\left(S_{0}+V_{0}\right)}{\delta_{1}+\delta_{2}+\gamma_{3}+\mu} \\ 0 & 0 & 0\end{array}\right)\).
\(R_{p}=\frac{\sigma_{1}((\kappa \mu+\zeta) \Lambda \phi+(1-\kappa)(\chi+\mu) \Lambda)}{(\zeta+\mu)(\chi+\mu)\left(\gamma_{1}+\delta_{1}+\mu\right)}\),
\(R_{m}=\frac{\sigma_{2}((\kappa \mu+\zeta) \Lambda \phi+(1-\kappa)(\chi+\mu) \Lambda)}{(\zeta+\mu)(\chi+\mu)\left(\gamma_{2}+\delta_{2}+\mu\right)}\).

Thus, the overall reproduction number is: \(R_{E}=\max \left\{R_{p}, R_{m}\right\}\). 2.2.2. Local Stability of the Disease-Free Equilibrium Theorem 3. The disease-free equilibrium (DFE) of the system (1) is locally asymptotically stable if \(R_{\mathrm{E}}<1\) and unstable if \(R_{E}>1\) . Proof. The Jacobian matrix at the DFE is given by:

The eigenvalues of this matrix include:

\(-\left(\gamma_{1}+\delta_{1}+\mu\right),-\left(\gamma_{2}+\delta_{2}+\mu\right),-\left(\delta_{1}+\delta_{2}+\gamma_{3}+\mu\right),-(\theta+\mu)\). The remaining eigenvalues are determined from the submatrix:

\(\left|\begin{array}{cc}-(\zeta+\mu)-\lambda & \chi \\ \zeta & -(\chi+\mu)-\lambda\end{array}\right|=0\).

Solving, we find: \(\lambda^{2}+\lambda(\zeta+\chi+2 \mu)+(\zeta+\mu)(\chi+\mu)-\zeta \chi=0\).

Since all coecients are positive, the roots of the characteristic equation have negative real parts, ensuring local stability when \(R_{E}<1\) .

2.2.3. Global Stability of the Disease-Free Equilibrium

Theorem 4. The disease-free equilibrium point is globally asymptotically stable

if \(R_{\mathrm{E}}<1\)Proof. Let . Otherwise unstable. \(F\) and \(V\) denote the Jacobian matrices for the new infection rates \(F_{i}\) and net transition rates \(V_{i}=V_{i}^{-}-V_{i}^{+}\), evaluated at the disease-free equilibrium \(x_{0}\) :

\[ F=\left(\frac{\partial F_{i}\left(x_{0}\right)}{\partial x_{i}}\right), V=\left(\frac{\partial V_{i}\left(x_{0}\right)}{\partial x_{i}}\right), 1 \leq i, j \leq n . \]

The dynamics of the infected compartments (\(I_{p}, I_{m}, I_{p m}\) ) are bounded by:

\[ \frac{d}{d t}\left(\begin{array}{c} I_{p} \\ I_{m} \\ I_{p m} \end{array}\right) \leq(F-V)\left(\begin{array}{c} I_{p} \\ I_{m} \\ I_{p m} \end{array}\right) \]

where \(F\) captures new infections and \(V\) transitions between compartments, with:

\[ \begin{aligned} & =\left(\begin{array}{ccc} \frac{\beta_{1}(1-\rho) \Gamma[\psi+d+\alpha(\rho+\epsilon)]}{(\epsilon+d)(\psi+d)} & 0 & \frac{\beta_{1}(1-\rho) \Gamma[\psi+d+\alpha(\rho d+\epsilon)]}{(\epsilon+d)(\psi+d)} \\ 0 & \beta_{2}\left(S_{0}+\alpha V_{0}\right) & \frac{\beta_{2}(1-\rho) \Gamma[\psi+d+\alpha(\rho d+\epsilon)]}{(\epsilon+d)(\psi+d)} \\ 0 & 0 & 0 \end{array}\right), V \\ & =\operatorname{diag}\left(\eta_{1}+k_{1}+d, \eta_{2}+k_{2}+d, \eta_{3}+k_{3}+d\right) . \end{aligned} \]

\[ \left(I_{p}, I_{m}, I_{p m}\right) \rightarrow(0,0,0) \text { as } t \rightarrow \infty . \]

Consequently, the susceptible population \(S(t)\) approaches the steady state:

\[ S(t) \rightarrow \frac{(1-\rho) \Gamma}{\epsilon+d} \]

Thus, the disease-free equilibrium \(E_{0}\) is globally asymptotically stable when \(R_{p}<1\) and \(R_{m}<1\)

2.3. The Endemic Equilibria

The endemic equilibria represent the steady states where one or both infections persist in the population. We examine three cases: the pneumonia endemic equilibrium, the meningitis endemic equilibrium, and the coexistence equilibrium.

2.3.1. Pneumonia Endemic Equilibrium

The pneumonia endemic equilibrium occurs when pneumonia persists in the population (\(I_{p} \neq 0\) ), but meningitis infection is absent (\(I_{m}=0\) ). Substituting into system (1) and solving, the system reduces to:

\[ \begin{aligned} (1-\kappa) \Lambda-\sigma_{1} I_{p} S+\theta R-(\zeta+\mu) S+\chi V & =0 \\ \kappa \Lambda+\zeta S-\phi \sigma_{1} I_{p} V-(\chi+\mu) V & =0 \\ \sigma_{1} I_{p}(S+\phi V)-\left(\gamma_{1}+\delta_{1}+\mu\right) I_{p} & =0 \\ \gamma_{1} I_{p}-(\theta+\mu) R & =0 \end{aligned} \]

The pneumonia endemic equilibrium is given by \(E_{p}=\left(S^{p}, V^{p}, I_{p}^{p}, 0,0, R^{p}\right)\), where: \[ \begin{aligned} S^{p} & =\frac{(1-\kappa) \Lambda+\theta R^{p}}{\sigma_{1} I_{p}^{p}+\zeta+\mu} \\ V^{p} & =\frac{\kappa \Lambda+\zeta S^{p}}{\phi \sigma_{1} I_{p}^{p}+\chi+\mu} \\ R^{p} & =\frac{\gamma_{1} I_{p}^{p}}{\theta+\mu} \end{aligned} \]

Here, \(I_{p}^{p}\) is obtained numerically or analytically by solving: \[ \sigma_{1} I_{p}^{p}\left(S^{p}+\phi V^{p}\right)=\left(\gamma_{1}+\delta_{1}+\mu\right) I_{p}^{p} \]

2.3.2. Meningitis Endemic Equilibrium

The meningitis endemic equilibrium occurs when meningitis persists in the population (\(I_{m} \neq 0\) ), but pneumonia infection is absent (\(I_{p}=0\) ). Substituting \(I_{p}=0\) into system (1), the system reduces to:

\[ \begin{array}{r} (1-\kappa) \Lambda-\sigma_{2} I_{m} S+\theta R-(\zeta+\mu) S+\chi V=0 \\ \kappa \Lambda+\zeta S-\phi \sigma_{2} I_{m} V-(\chi+\mu) V=0 \\ \sigma_{2} I_{m}(S+\phi V)-\left(\gamma_{2}+\delta_{2}+\mu\right) I_{m}=0 \\ \gamma_{2} I_{m}-(\theta+\mu) R=0 \end{array} \]

The meningitis endemic equilibrium is given by \(E_{m}=\left(S^{m}, V^{m}, 0, I_{m}^{m}, 0, R^{m}\right)\), where:

\[ \begin{aligned} S^{m} & =\frac{(1-\kappa) \Lambda+\theta R^{m}}{\sigma_{2} I_{m}^{m}+\zeta+\mu} \\ V^{m} & =\frac{\kappa \Lambda+\zeta S^{m}}{\phi \sigma_{2} I_{m}^{m}+\chi+\mu} \\ R^{m} & =\frac{\gamma_{2} I_{m}^{m}}{\theta+\mu} \end{aligned} \]

Here, \(I_{m}^{m}\) is obtained numerically or analytically by solving: \[ \sigma_{2} I_{m}^{m}\left(S^{m}+\phi V^{m}\right)=\left(\gamma_{2}+\delta_{2}+\mu\right) I_{m}^{m} \]

2.3.3. Coexistence Equilibrium

The coexistence equilibrium occurs when both pneumonia and meningitis persist in the population \(\left(I_{p} \neq 0, I_{m} \neq 0\right)\). Solving system (1) with all compartments non-zero, the equilibrium is given by \(E_{m p}=\left(S^{*}, V^{*}, I_{p}^{*}, I_{m}^{*}, I_{p m}^{*}, R^{*}\right)\), where:

\[ \begin{aligned} S^{*} & =\frac{\Lambda-\left(\sigma_{1} I_{p}^{*}+\sigma_{2} I_{m}^{*}+\sigma_{3} I_{p m}^{*}\right) S^{*}+\theta R^{*}}{\zeta+\mu} \\ V^{*} & =\frac{\kappa \Lambda+\zeta S^{*}-\phi\left(\sigma_{1} I_{p}^{*}+\sigma_{2} I_{m}^{*}+\sigma_{3} I_{p m}^{*}\right) V^{*}}{\chi+\mu} \\ R^{*} & =\frac{\gamma_{1} I_{p}^{*}+\gamma_{2} I_{m}^{*}+\gamma_{3} I_{p m}^{*}}{\theta+\mu} \end{aligned} \]

Here, \(I_{p}^{*}, I_{m}^{*}\), and \(I_{p m}^{*}\) are obtained by solving the nonlinear system numerically or analytically, incorporating interaction terms between pneumonia and meningitis.

2.4. Sensitivity Analysis

In the study of epidemiology, it is critical to identify the factors that significantly contribute to disease transmission and prevalence. Sensitivity analysis serves as an essential method for assessing how variations in parameters inuence the dynamics of a disease, enabling the development of targeted strategies to reduce both morbidity and mortality.

To evaluate the impact of parameters on the outcomes of the model, sensitivity indices are employed. These indices provide a measure of how a relative change in a parameter translates into a relative change in a state variable. The normalized forward sensitivity index is particularly useful in this context, as it quantifies the proportional effect of parameter variations on a given variable.

If the state variable is a differentiable function of the parameter, the sensitivity index forward can sensitivity be calculated index of using a variable partial \(\eta\) with derivatives. respect to Specifically a parameter , \(\Omega\) the is normalized defined as: \[ \Upsilon_{\Omega}^{\eta}=\frac{\partial \eta}{\partial \Omega} \times \frac{\Omega}{\eta} \] where \(\Omega\) represents any of the fundamental parameters in the model. This formulation allows researchers to systematically determine the relative importance of different parameters, providing valuable insights into the dynamics and control of the disease.

3. Numerical Simulations

This section presents the numerical simulation of the pneumonia-meningitis co-infection model, utilizing the baseline parameter values outlined in Table 1.

Simulations were performed and visualized over time (in days) using MATLAB, with the resulting plots displayed in Figures 1 – 6.

The initial conditions for the pneumonia and meningitis coinfection model are obtained as follows: The initial number of susceptible individuals is taken to be \(S(0)=2000\). The initial number of vaccinated individuals is \(V(0)=250\). The initial number of individuals infected with pneumonia is \(I_{p}(0)=500\) . The initial number of individuals infected with meningitis is \(I_{m}(0)=700\) . The initial number of individuals coinfected with both pneumonia and meningitis is \(I_{p m}(0)=280\) . The initial number of recovered individuals is \(R(0)=200\) .

Table 1. Parameters Description and Values Used in the Simulation Model

ParametersDescriptionValueSourceProportion not covered by thevaccine0.002Tilahunet al.(2017)Rate of immunity loss among re-covered individuals0.0241/dayKizito and Tumwiine(2018)Waning rate0.0025/dayTilahunet al.(2017)Contactrate:pneumonia-infectedand susceptible0.007 – 0.6Konstatin (2016)Contactrate:meningitis-infectedand susceptible0.9Fresnadilloet al.(2013)Disease-induceddeathrate:pneumonia0.006 – 0.5Tilahun (2019)Disease-induceddeathrate:meningitis0.002 – 0.2Tilahun (2019)Recovery rate: pneumonia0.9Tilahun (2019)Recovery rate: meningitis0.8Tilahun (2019)Recovery rate: coinfection0.1Tilahunet al.(2017)Proportion vaccinated againstone or both diseases0.02Swaiet al.(2021)Vaccination rate0.3Swaiet al.(2021)Natural death rate4.566e-6Tilahunet al.(2017)Birth rate150Assumed

The numerical simulation of the effect of vaccination on the population dynamics is analysed using the baseline values given in Table 1. The numerical simulation are done and plotted against time (days) using MATLAB and the results are shown in Figures 1 – 6. Figures 1 – 6 illustrates the impact of varying levels of vaccination effectiveness on the dynamics of a pneumonia-meningitis coinfection model over a period of 150 days. The figure compares scenarios with no vaccination, slightly effective vaccination (\(30 \%\) compliance), moderately effective vaccination (\(50 \%\) compliance), and highly effective vaccination (\(100 \%\) compliance). It demonstrates how these different levels of vaccination compliance affect the number of susceptible individuals (Figure 1), vaccinated individuals (Figure 2), pneumonia-infected individuals (Figure 3), meningitis-infected individuals (Figure 4), coinfected individuals (Figure 5), and recovered individuals (Figure 6). As shown in Figure 1, the number of susceptible individuals decreases over time across all levels of vaccination effectiveness. Without any vaccination (denoted by dots), the decline in susceptible individuals is the least steep. The highly effective strategy (\(100 \%\) compliance) demonstrates the steepest decline. However, slightly effective (\(30 \%\) compliance) and moderately effective(\(50 \%\) compliance) strategies also show significant reductions, indicating a gradient effect of vaccination effectiveness on reducing the number of susceptible individuals.

Figure 2 displays the increase in vaccinated individuals over time. The graph shows that with higher compliance to vaccination (\(30 \%, 50 \%\), and \(100 \%\) effectiveness), the number of vaccinated individuals increases significantly compared to no vaccination. The highly effective strategy (\(100 \%\) compliance) results in the highest number of vaccinated individuals. Interestingly, the slightly effective strategy shows a quicker initial increase compared to the moderately effective strategy. This might be due to a higher initial vaccination uptake in the slightly effective scenario, which could be attributed to increased public awareness or access to vaccines.

As displayed in Figure 3, the number of pneumonia-infected individuals decreases over time. The graph shows that with no vaccination (denoted by dots), the reduction is the slowest. Slightly effective (\(30 \%\) compliance) and moderately effective(\(50 \%\) compliance) vaccination strategies result in a more significant decrease, but the highly effective strategy (\(100 \%\) compliance) shows the most substantial reduction in pneumonia infections. The slightly effective strategy appears to reduce infections quicker than the moderately effective one. This could be due to initial variations in how the vaccine impacts different population subsets or logistical differences in vaccine distribution and uptake.

Figure 4 shows the trends for meningitis-infected individuals. Similar to pneumonia, the graph demonstrates a decrease in meningitis infections over time. The reduction is least significant without vaccination. The slightly effective strategy initially performs better than the moderately effective one, which might be due to similar reasons as discussed for pneumonia: initial vaccine impact variations and differences in vaccine distribution logistics. The highly effective strategy (\(100 \%\) compliance) leads to the most considerable decline in meningitis infections.

In Figure 5, the number of coinfected individuals (infected with both pneumonia and meningitis) increases over time across all strategies. The rate of increase is highest without any vaccination (denoted by dots) and progressively decreases with more effective vaccination strategies. The highly effective strategy (\(100 \%\) compliance) shows the slowest increase. Interestingly, the slightly effective strategy shows a lower rate of increase compared to the moderately effective strategy at certain points, potentially due to initial population dynamics and vaccine distribution eciency.

Figure 6 depicts the number of recovered individuals over time. The number of recoveries is lowest with no vaccination and increases with the effectiveness of the vaccination strategy. However, the difference in the number of recoveries among the slightly effective, moderately effective, and highly effective strategies is minimal. This small difference indicates that while vaccination does contribute to recovery, its impact on this specific outcome is less pronounced compared to its effect on reducing the number of susceptible and infected individuals.

The slightly effective strategy shows better initial performance in some graphs due to variations in how different population subsets respond to the vaccine. Some individuals might have a quicker immune response, leading to an initial rapid decline in infections or an increase in recoveries. Differences in how the vaccine is distributed and administered can impact initial results. Ecient distribution in the slightly effective scenario might lead to quicker initial results compared to a more evenly distributed but slower uptake in the moderately effective scenario. Higher public awareness or better access to vaccines in the slightly effective scenario might result in a quicker initial response, leading to better performance in the early stages. The small difference in the number of recovered individuals among the different vaccination strategies suggests that other factors, such as natural recovery rates and the effectiveness of medical treatments, play a significant role in recovery. To enhance the impact of vaccination on recovery rates, it might be necessary to combine vaccination with improved medical treatments and supportive care. Improving the logistics of vaccine distribution to ensure a more uniform and ecient rollout can help achieve better initial results across all effectiveness levels. By addressing any logistical challenges, we can ensure that vaccines reach the population more quickly and evenly. Implementing campaigns to raise awareness about the importance of vaccination can increase uptake and compliance, improving overall effectiveness. These campaigns can educate the public on the benefits of vaccination and encourage more people to get vaccinated. Regularly monitoring how different population subsets respond to the vaccine can help adjust strategies to maximize effectiveness. This monitoring can identify which groups are responding well and which may need additional support or different strategies. To enhance the impact of vaccination on recovery rates, it is important to also focus on improving medical treatments and supportive care for infected individuals. Combining vaccination with better medical care can ensure that those who contract the disease receive the best possible treatment, further reducing the overall impact of the infection. Despite some initial better performance in the slightly effective strategy, the highly effective vaccination strategy(\(100 \%\) compliance) remains the best approach in the long run. It results in the steepest decline in susceptible and infected individuals and the highest number of vaccinated individuals. While the impact on recovered individuals is less pronounced, achieving high compliance with vaccination significantly improves public health outcomes by reducing infection rates and increasing recovery in combination with effective medical care.

Figure 1. Simulation effect of vaccination on the susceptible individuals

Figure 2. Simulation effect of vaccination on the vaccinated individuals

Figure 3. Simulation effect of vaccination on the pneumonia-infected individuals

Figure 4. Simulation effect of vaccination on the meningitis-infected individuals

Figure 5. Simulation effect of vaccination on the co-infected individuals

Figure 6. Simulation effect of vaccination on the recovered individuals

3.1. Sensitivity Index

The sensitivity indices of the basic reproduction numbers are summarized in Tables 2 and 3. Parameters with positive sensitivity indices in Tables 2 and 3 indicate that increasing their values, while keeping all other parameters constant, leads to an increase in \(R_{0 p}\) and \(R_{0 m}\), respectively. These parameters play a significant role in driving the spread of pneumonia and meningitis infections.

On the other hand, parameters with negative sensitivity indices in Tables 2 and 3 suggest that increasing their values, while holding other parameters fixed, results in a decrease in \(R_{0 p}\) and \(R_{0 m}\), respectively. These parameters contribute to mitigating

the transmission of the diseases. which Thus, have the negative key parameters sensitivity for indices. effective Parameters disease control such as include \(\sigma_{1}, \sigma_{2}\) , \(\delta_{1}, \delta_{2}\) and , \(\Lambda\) exhibit and \(\zeta\) , the highest sensitivity w \(^{\text {it }}\) h positive indices,emphasizing their importance in disease propagation. Although \(\mu\) has a negative sensitivity index, biologically, increasing its value is recommended as it aids in disease control.

Table 2. Value of Sensitivity Indices of

ParametersSensitivity Index+1-0.10207-1-0.00168+0.0020103+1-0.89625-0.01996able 3.Value of Sensitivity Indices ofParametersSensitivity Index+1-0.10207-1-0.00168+0.0020103+1-0.89625-0.01996T

4. Conclusion

This study presents a compartmental deterministic mathematical model of the dynamics of pneumonia and meningitis coinfection. It highlights the important role of vaccination in controlling the spread of both diseases within a population. Our analysis demonstrates that varying levels of vaccination compliance can notably reduce the number of susceptible and infected individuals, with higher vaccination compliance leading to the most considerable reductions. Although the effect of vaccination on recovery rates is less pronounced, the combination of high vaccination coverage and improved medical care offers significant public health benefits.

The findings emphasize the need to address logistical challenges, increase public awareness, and improve vaccine accessibility to maximize the effectiveness of vaccination programs. While slightly effective strategies may show initial success, the long-term impact is most pronounced with highly effective vaccination coverage. Therefore, to optimize public health outcomes, it is crucial to expand vaccination campaigns, enhance distribution logistics, and integrate vaccination with improved medical treatments. Regular monitoring and adjustments to vaccination strategies, based on population responses, will further enhance the effectiveness of disease control measures.

While the model does not explicitly capture variations in immune responses or the logistics of vaccine distribution, it simplifies the vaccine impact by assuming that vaccination reduces susceptibility in proportion to the coverage level. The vaccine’s effectiveness is modeled as a constant factor in the transition dynamics between compartments, reecting an overall reduction in susceptibility and infection. Further refinements to the model could include the inuence of varying immune responses or more detailed vaccine administration strategies, but these aspects were not included in the current study.

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Dr. Deborah O. Daniel
ORCID iD: 0000-0003-4025-1448
Olabisi Onabanjo University
Ago-Iwoye, Ogun State, Nigeria
E-mail: oludeboradaniel@gmail.com
Sefiu A. Onitilo
ORCID iD: 0000-0002-4418-197X
Olabisi Onabanjo University
Ago Iwoye, Ogun State, Nigeria
E-mail: onitilo.sefiu@oouagoiwoye.edu.ng
Omolade B. Benjamin
ORCID iD: 0009-0004-0665-5558
Olabisi Onabanjo University
Ago Iwoye, Ogun State, Nigeria
E-mail: omoladebabajide342@gmali.com
Ayoola A. Olasunkanmi
ORCID iD: 0009-0006-3530-5391
Olabisi Onabanjo University
Ago Iwoye, Ogun State, Nigeria
E-mail: ayoolaolasunkanmi243@mail.com

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PROBLEM OF THE 8-TH EXPERIMENTAL PHYSICS OLYMPIAD, SKOPJE, 8 MAY 2021 DETERMINATION OF PLANCK CONSTANT BY LED

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INVESTIGATION OF

Bozhidar Slavchev, Elena Geleva, Blagorodka Veleva, Hristo Protohristov, Lyuben Dobrev, Desislava Dimitrova, Vladimir Bashev, Dimitar Tonev

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DEMONSTRATION OF DAMPED ELECTRICAL OSCILLATIONS

Elena Grebenakova, Stojan Manolev

2020 година
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ДОЦ. Д-Р МАРЧЕЛ КОСТОВ КОСТОВ ЖИВОТ И ТВОРЧЕСТВО

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CHEMISTRY: BULGARIAN JOURNAL OF SCIENCE EDUCATION ANNUAL CONTENTS / VOLUME 29, 2020 ПРИРОДНИТЕ НАУКИ В ОБРАЗОВАНИЕТО ГОДИШНО СЪДЪРЖАНИЕ / ГОДИНА XXIX, 2020

СТРАНИЦИ / PAGES КНИЖКА 1 / NUMBER 1: 1 – 140 КНИЖКА 2 / NUMBER 2: 141 – 276 КНИЖКА 3 / NUMBER 3: 277 – 432 КНИЖКА 4 / NUMBER 4: 433 – 548 КНИЖКА 5 / NUMBER 5: 549 – 660 КНИЖКА 6 / NUMBER 6: 661 – 764 EDUCATION: THEORY AND PRACTICE 11 – 18 Физиката – навсякъде около нас [Physics аround Us: The Honey] / Пенка Василева/ Penka Vasileva 19 – 26 Молекулите на удоволствието [The Pleasure Molecules: Introduc- tory Information] / Веселина Янкова, Снежана Демирова, Цветанка Мит

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JACOB’S LADDER FOR THE PHYSICS CLASSROOM

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COMPARATIVE PERFORMANCE AND DIGESTIBILITY OF NUTRIENTS IN AFSHARI AND GHEZEL RAM LAMBS

Morteza Karami, Fardis Fathizadeh, Arash Yadollahi, Mehran Aboozari, Yaser Rahimian, Reza Alipoor Filabadi

КАЛЦИЙ, ФОСФОР И ДРУГИ ФАКТОРИ ЗА КОСТНО ЗДРАВЕ

Радка Томова, Светла Асенова, Павлина Косева

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THE DAY OF THE INDUCTANCE: PROBLEMS OF THE 7

Todor M. Mishonov, Riste Popeski-Dimovski, Leonora Velkoska, Iglika M. Dimitrova, Vassil N. Gourev, Aleksander P. Petkov, Emil G. Petkov, Albert M. Varonov

MATHEMATICAL MODELING OF 2019 NOVEL CORONAVIRUS (2019 – NCOV) PANDEMIC IN NIGERIA

Sefiu A. Onitilo, Mustapha A. Usman, Olutunde S. Odetunde, Fatai A. Hammed, Zacheous O. Ogunwobi, Hammed A. Haruna, Deborah O. Daniel

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ВОДОРОДЪТ – НАЙ-ЕКОЛОГИЧНО ЧИСТОТО ГОРИВО

Миглена Славова, Радка Томова

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НАУЧНОТО ПРИЗНАНИЕ

Престижът на един унверситет не се определя от материалните ценнос- ти – сгради, лаборатории, аудитории, библиотеки, спортни съоръжения, които

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Веселина Янкова, Снежана Демирова, Цветанка Митева, Явор Князов, Христо Желев, Димитър Георгиев, Габриела Стоянова

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Lyubomir Lazov, Hristina Deneva, Galina Gencheva

ФИЗИКОХИМИЧНАТА ШКОЛА НА РОСТИСЛАВ КАИШЕВ В ПЕРИОДА 1950 – 1957 (СПОМЕНИ НА ЕДИН СВИДЕТЕЛ)

На 7 януари 2020 г. във Франция по- чина Боян Mутафчиев – учен с висока ре- путация във Франция и света. Той беше президент на Френската асоциация по кристален растеж, председател на Ко- мисията за изследвания в областта на микрогравитацията към Националния център за космически изследвания на Франция. Той беше ключова фигура в Ев- ропейската космическа агенция, в Евро-

2019 година
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LASER MICRO-PERFORATION AND FIELDS OF APPLICATION

Hristina Deneva, Lyubomir Lazov, Edmunds Teirumnieks

ПРИЛОЖНА ФОТОНИКА И АНТИОКСИДАНТНИ СВОЙСТВА НА ВИСОКООЛЕИНОВО СЛЪНЧОГЛЕДОВО МАСЛО С БИЛКОВИ ПРИМЕСИ

Кръстена Николова, Стефка Минкова, Поли Радушева, Георги Бошев, Еркан Фаридин, Нурал Джамбазов, Мариана Перифанова-Немска

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Берна Сабит, Джемиле Дервиш, Мая Никова, Йорданка Енева

IN VITRO EVALUATION OF THE ANTIOXIDANT PROPERTIES OF OLIVE LEAF EXTRACTS – CAPSULES VERSUS POWDER

Hugo Saint-James, Gergana Bekova, Zhanina Guberkova, Nadya Hristova-Avakumova, Liliya Atanasova, Svobodan Alexandrov, Trayko Traykov, Vera Hadjimitova

2019 − THE INTERNATIONAL YEAR OF THE PERIODIC TABLE OF CHEMICAL ELEMENTS

Maria Atanassova, Radoslav Angelov, Dessislava Gerginova, Alexander Zahariev

ТЕХНОЛОГИЯ

Б. В. Тошев

CHEMISTRY: BULGARIAN JOURNAL OF SCIENCE EDUCATION ANNUAL CONTENTS / VOLUME 28, 2019 ХИМИЯ. ПРИРОДНИТЕ НАУКИ В ОБРАЗОВАНИЕТО ГОДИШНО СЪДЪРЖАНИЕ / ГОДИНА XXVIII, 2019

СТРАНИЦИ / PAGES КНИЖКА 1 / NUMBER 1: 1 – 160 КНИЖКА 2 / NUMBER 2: 161 – 280 КНИЖКА 3 / NUMBER 3: 281 – 424 КНИЖКА 4 / NUMBER 4: 425 – 552 КНИЖКА 5 / NUMBER 5: 553 – 680 КНИЖКА 6 / NUMBER 6: 681 – 832

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ЗЕЛЕНА ХИМИЯ В УЧИЛИЩНАТА ЛАБОРАТОРИЯ

Александрия Генджова, Мая Тавлинова-Кирилова, Александра Камушева

ON THE GENETIC TIES BETWEEN EUROPEAN NATIONS

Jordan Tabov, Nevena Sabeva-Koleva, Georgi Gachev

SCIENCE CAN BRING PEOPLE TOGETHER

Nadya Kantareva-Baruh

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CHEMOMETRIC ANALYSIS OF SCHOOL LIFE IN VARNA

Radka Tomova, Petinka Galcheva, Ivajlo Trajkov, Antoaneta Hineva, Stela Grigorova, Rumyana Slavova, Miglena Slavova

ЦИКЛИТЕ НА КРЕБС

Ивелин Кулев

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ПРИНЦИПИТЕ НА КАРИЕРНОТО РАЗВИТИЕ НА МЛАДИЯ УЧЕН

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UTILISATION OF THE STATIC EVANS METHOD TO MEASURE MAGNETIC SUSCEPTIBILITIES OF TRANSITION METAL ACETYLACETONATE COMPLEXES AS PART OF AN UNDERGRADUATE INORGANIC LABORATORY CLASS

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THE 100

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A TALE OF SEVEN SCIENTISTS

Scerri, E.R. (2016). A Tale of Seven Scientists and a New Philosophy of Science.

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AMPEROMETRIC NITRIC OXIDE SENSOR BASED ON MWCNT CHROMIUM(III) OXIDE NANOCOMPOSITE

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THE EFFECT OF AGING TIME ON Mg/Al HYDROTALCITES STRUCTURES

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2019: ДВЕ ВАЖНИ СЪБИТИЯ

На 20 декември 2017 г. Общото събрание на Организацията на обединени- те нации (ООН), на своята 72-ра сесия, прокламира годината 2019 за „Между- народна година на Периодичната таблица на химичните елементи“ (IYPT 2019). Съгласно представите на Томас Кун раз- витието на „нормалната наука“ става на основата на малък брой основополагащи научни резултати, наречени „научни пара- дигми“. За химията научните парадигми са три: (1) откриването на кислорода от Лавоазие; (

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Elena Karashtranova, Nikolay Karashtranov, Vladimir Vladimirov

SOME CONCEPTS FROM PROBABILITY AND STATISTICS AND OPPORTUNITIES TO INTEGRATE THEM IN TEACHING NATURAL SCIENCES

Elena Karashtranova, Nikolay Karashtranov, Nadezhda Borisova, Dafina Kostadinova

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2018 година
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ЗДРАВЕ И ОКОЛНА СРЕДА

Кадрие Шукри, Светлана Великова, Едис Мехмед

СЕМИНАР-ПРАКТИКУМЪТ В НЕФОРМАЛНОТО ОБУЧЕНИЕ – ВЪЗМОЖНОСТ И РАЗВИТИЕ НА УМЕНИЯ И ТВОРЧЕСТВО ПРЕЗ ПРИЗМАТА НА ФИЗИКАТА

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Даниела Узунова, Борис Велковски, Илко Симеонов Владислав Шабански, Димитър Колев

DESIGN AND DOCKING STUDIES OF HIS-LEU ANALOGUES AS POTENTIOAL ACE INHIBITORS

Rumen Georgiev, Tatyana Dzimbova, Atanas Chapkanov

X-RAY DIFFRACTION STUDY OF M Zn(TeО )

Kenzhebek T. Rustembekov, Mitko Stoev, Aitolkyn A. Toibek

ELECTROCHEMICAL IMPEDANCE STUDY OF BSCCO (2212) CUPRATE CERAMIC ADDITIVE TO THE ZINC ELECTRODE IN Ni-Zn BATTERIES

A. Vasev, P. Lilov, G. Ivanova, Y. Marinov, A. Stoyanova, V. Mikli, A. Stoyanova-Ivanova

CALIBRATION OF GC/MS METHOD FOR DETERMINATION OF PHTHALATES

N. Dineva, I. Givechev, D. Tanev, D. Danalev

SONOCHEMICAL SYNTHESIS OF 4-AMINOANTIPYRINE SCHIFF BASES A ND EVALUATION OF THEIR ANTIMICROBIAL, ANTI-TYROSINASE AND DPPH SCAVENGING ACTIVITIES

Maya Chochkova, Boyka Stoykova, Iva Romanova, Petranka Petrova, Iva Tsvetkova, Hristo Najdenski, Lubomira Nikolaeva- Glomb, Nadya Nikolova, Galya Ivanova, Atanas Chapkanov, Tsenka Milkova, Martin Štícha, Ivan Nemec

ELECTROSYNTHESIS OF CADMIUM SELENIDE NANOPARTICLES WITH SIMULTANEOUS EXTRACTION INTO P-XYLENE

S. S. Fomanyuk, V. O. Smilyk, G. Y. Kolbasov, I. A. Rusetskyi, T. A. Mirnaya

БИОЛОГИЧЕН АСПЕКТ НА РЕКАНАЛИЗАЦИЯ С ВЕНОЗНА ТРОМБОЛИЗА

Мариела Филипова, Даниела Попова, Стоян Везенков

CHEMISTRY: BULGARIAN JOURNAL OF SCIENCE EDUCATION ПРИРОДНИТЕ НАУКИ В ОБРАЗОВАНИЕТО VOLUME 27 / ГОДИНА XXVII, 2018 ГОДИШНО СЪДЪРЖАНИЕ СТРАНИЦИ / PAGES КНИЖКА 1 / NUMBER 1: 1 – 152 КНИЖКА 2 / NUMBER 2: 153 – 312 КНИЖКА 3 / NUMBER 3: 313 – 472 КНИЖКА 4 / NUMBER 4: 473 – 632 КНИЖКА 5 / NUMBER 5: 633 – 792 КНИЖКА 6 / NUMBER 6: 793 – 952 КНИЖКА 1 / NUMBER 1: 1 – 152 КНИЖКА 2 / NUMBER 2: 153 – 312 КНИЖКА

(South Africa), A. Ali, M. Bashir (Pakistan) 266 – 278: j-j Coupled Atomic Terms for Nonequivalent Electrons of (n-1)fx and nd1 Configurations and Correlation with L-S Terms / P. L. Meena (India) 760 – 770: Methyl, тhe Smallest Alkyl Group with Stunning Effects / S. Moulay 771 – 776: The Fourth State of Matter / R. Tsekov

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ИМОБИЛИЗИРАНЕ НА ФРУКТОЗИЛТРАНСФЕРАЗА ВЪРХУ КОМПОЗИТНИ ФИЛМИ ОТ ПОЛИМЛЕЧНА КИСЕЛИНА, КСАНТАН И ХИТОЗАН

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ELECTRICAL IMPEDANCE SPECTROSCOPY OF GRAPHENE-E7 LIQUID-CRYSTAL NANOCOMPOSITE

Todor Vlakhov, Yordan Marinov, Georgi. Hadjichristov, Alexander Petrov

ON THE POSSIBILITY TO ANALYZE AMBIENT NOISE RECORDED BY A MOBILE DEVICE THROUGH THE H/V SPECTRAL RATIO TECHNIQUE

Dragomir Gospodinov, Delko Zlatanski, Boyko Ranguelov, Alexander Kandilarov

RHEOLOGICAL PROPERTIES OF BATTER FOR GLUTEN FREE BREAD

G. Zsivanovits, D. Iserliyska, M. Momchilova, M. Marudova

ПОЛУЧАВАНЕ НА ПОЛИЕЛЕКТРОЛИТНИ КОМПЛЕКСИ ОТ ХИТОЗАН И КАЗЕИН

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CHEMILUMINESCENT AND PHOTOMETRIC DETERMINATION OF THE ANTIOXIDANT ACTIVITY OF COCOON EXTRACTS

Y. Evtimova, V. Mihailova, L. A. Atanasova, N. G. Hristova-Avakumova, M. V. Panayotov, V. A. Hadjimitova

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H O O H H H C H 104.45° 108.9° H A B

Figure 1. A – Water; B – Methanol Indeed, the literature survey reveals a trove of reports on methylation of such molecules and the effects of thus-tethered methyl groups were illustrated (Barreiro et al., 2011). An astounding fact is the link between the methyl group in methylated DNA and the cancer development (Newberne & Rogers, 1986; Wajed et al., 2001); methylation of biological molecules contributes to the regulation of gene expression and protein function, and RNA proces

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PARAMETRIC INTERACTION OF OPTICAL PULSES IN NONLINEAR ISOTROPIC MEDIUM

A. Dakova, V. Slavchev, D. Dakova, L. Kovachev

ДЕЙСТВИЕ НА ГАМА-ЛЪЧИТЕ ВЪРХУ ДЕЗОКСИРИБОНУКЛЕИНОВАТА КИСЕЛИНА

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RADIATION PROTECTION

Natasha Ivanova, Bistra Manusheva

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Carsten Tottmann, Valentin Hedderich, Poli Radusheva, Krastena Nikolova

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Рая Грозданова-Узунова, Тодор Узунов, Пепа Узунова

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PHYSICS IS AN EVER YOUNG SCIENCE

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ПРОБЛЕМИ ПРИ ОБУЧЕНИЕТО ПО ФИЗИКА ВЪВ ВВМУ „Н. Й. ВАПЦАРОВ“

А. Христова, Г. Вангелов, И. Ташев, М. Димидов

ИЗГРАЖДАНЕ НА СИСТЕМА ОТ УЧЕБНИ ИНТЕРНЕТ РЕСУРСИ ПО ФИЗИКА И ОЦЕНКА НА ДИДАКТИЧЕСКАТА ИМ СТОЙНОСТ

Желязка Райкова, Георги Вулджев, Наталия Монева, Нели Комсалова, Айше Наби

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Георги Върбанов, Радостин Михайлов, Деница Симеонова, Йорданка Енева

NATURAL RADIONUCLIDES IN DRINKING WATER

Natasha Ivanova, Bistra Manusheva

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АДАПТИРАНЕ НА ОБРАЗОВАНИЕТО ДНЕС ЗА УТРЕШНИЯ ДЕН

И. Панчева, М. Недялкова, П. Петков, Х. Александров, В. Симеонов

STRUCTURAL ELUCIDATION OF UNKNOWNS: A SPECTROSCOPIC INVESTIGATION WITH AN EMPHASIS ON 1D AND 2D

Vittorio Caprio, Andrew S. McLachlan, Oliver B. Sutcliffe, David C. Williamson, Ryan E. Mewis

INTEGRATED ENGINEERING EDUCATION: THE ROLE OF ANALYSIS OF STUDENTS’ NEEDS

Veselina Kolarski, Dancho Danalev, Senia Terzieva

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ZAGREB CONNECTION INDICES OF TiO NANOTUBES

Sohaib Khalid, Johan Kok, Akbar Ali, Mohsin Bashir

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GEOECOLOGICAL ANALYSIS OF INDUSTRIAL CITIES: ON THE EXAMPLE OF AKTOBE AGGLOMERATION

Zharas Berdenov, Erbolat Mendibaev, Talgat Salihov, Kazhmurat Akhmedenov, Gulshat Ataeva

TECHNOGENESIS OF GEOECOLOGICAL SYSTEMS OF NORTHEN KAZAKHSTAN: PROGRESS, DEVELOPMENT AND EVOLUTION

Kulchichan Dzhanaleyeva, Gulnur Mazhitova, Altyn Zhanguzhina, Zharas Berdenov, Tursynkul Bazarbayeva, Emin Atasoy

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В последните години тези традиционни за българското учителство конфе- ренции се организират от Българското дружество по химическо образование и история и философия на химията. То е асоцииран член на Съюза на химици- те в България, който пък е член на Европейската асоциация на химическите и

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SYNTHESIS OF TiO -M (Cd, Co, Mn)

Candra Purnawan, Sayekti Wahyuningsih, Dwita Nur Aisyah

EFFECT OF DIFFERENT CADMIUM CONCENTRATION ON SOME BIOCHEMICAL PARAMETERS IN ‘ISA BROWN’ HYBRID CHICKEN

Imer Haziri, Adem Rama, Fatgzim Latifi, Dorjana Beqiraj-Kalamishi, Ibrahim Mehmeti, Arben Haziri

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Baldomero Herrera-González Universidad Autónoma del Estado de México, Mexico

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4/2013: сп. „Философия“ Портфолио на преподавателя и студента по „Философия на oбразованието“ (за академич- ни цели) / Яна Рашева-Мерджанова „Философия на oбразованието“ има своите хилядолетни опити и резултати. Изследването на тази духовна вселена води до структура и до обособяването на активности, които са съответни, от една страна, на философскообразователното мислене и култура, а от друга страна – са въвеж- дащи в тяхното овладяване. За ролята на етиката в ученията на П. Рикьор и Е. Лев

2013 година
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DIFFERENTIAL TEACHING IN SCHOOL SCIENCE EDUCATION: CONCEPTUAL PRINCIPLES

G. Yuzbasheva Kherson Academy of Continuing Education, Ukraine

АНАЛИЗ НА ПОСТИЖЕНИЯТА НА УЧЕНИЦИТЕ ОТ ШЕСТИ КЛАС ВЪРХУ РАЗДЕЛ „ВЕЩЕСТВА И ТЕХНИТЕ СВОЙСТВА“ ПО „ЧОВЕКЪТ И ПРИРОДАТА“

Иваничка Буровска, Стефан Цаковски Регионален инспекторат по образованието – Ловеч

HISTORY AND PHILOSOPHY OF SCIENCE: SOME RECENT PERIODICALS (2013)

Chemistry: Bulgarian Journal of Science Education

45. НАЦИОНАЛНА КОНФЕРЕНЦИЯ НА УЧИТЕЛИТЕ ПО ХИМИЯ

„Образователни стандарти и природонаучна грамотност“ – това е темата на състоялата се от 25 до 27 октомври 2013 г. в Габрово 45. Национална конфе- ренция на учителите по химия с международно участие, която по традиция се проведе комбинирано с Годишната конференция на Българското дружество за химическо образование и история и философия на химията. Изборът на темата е предизвикан от факта, че развиването на природонаучна грамотност е обща тенденция на реформите на учебните програми и главна

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ЗА ХИМИЯТА НА БИРАТА

Ивелин Кулев

МЕТЕОРИТЪТ ОТ БЕЛОГРАДЧИК

Б. В. Тошев Българско дружество за химическо образование и история и философия на химията

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RECASTING THE DERIVATION OF THE CLAPEYRON EQUATION INTO A CONCEPTUALLY SIMPLER FORM

Srihari Murthy Meenakshi Sundararajan Engineering College, India

CHEMICAL REACTIONS DO NOT ALWAYS MODERATE CHANGES IN CONCENTRATION OF AN ACTIVE COMPONENT

Joan J. Solaz-Portolés, Vicent Sanjosé Universitat de Valènciа, Spain

POLYMETALLIC COMPEXES: CV. SYNTHESIS, SPECTRAL, THERMOGRAVIMETRIC, XRD, MOLECULAR MODELLING AND POTENTIAL ANTIBACTERIAL PROPERTIES OF TETRAMERIC COMPLEXES OF Co(II), Ni(II), Cu(II), Zn(II), Cd(II) AND Hg(II) WITH OCTADENTATE AZODYE LIGANDS

Bipin B. Mahapatra, S. N. Dehury, A. K. Sarangi, S. N. Chaulia G. M. Autonomous College, India Covt. College of Engineering Kalahandi, India DAV Junior College, India

ПРОФЕСОР ЕЛЕНА КИРКОВА НАВЪРШИ 90 ГОДИНИ

CELEBRATING 90TH ANNIVERSARY OF PROFESSOR ELENA KIRKOVA

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SIMULATION OF THE FATTY ACID SYNTHASE COMPLEX MECHANISM OF ACTION

M.E.A. Mohammed, Ali Abeer, Fatima Elsamani, O.M. Elsheikh, Abdulrizak Hodow, O. Khamis Haji

FORMING OF CONTENT OF DIFFERENTIAL TEACHING OF CHEMISTRY IN SCHOOL EDUCATION OF UKRAINE

G. Yuzbasheva Kherson Academy of Continuing Education, Ukraine

ИЗСЛЕДВАНЕ НА РАДИКАЛ-УЛАВЯЩА СПОСОБНОСТ

Станислав Станимиров, Живко Велков

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Книжка 1
COLORFUL EXPERIMENTS FOR STUDENTS: SYNTHESIS OF INDIGO AND DERIVATIVES

Vanessa BIANDA, Jos-Antonio CONSTENLA, Rolf HAUBRICHS, Pierre-Lonard ZAFFALON

OBSERVING CHANGE IN POTASSIUM ABUNDANCE IN A SOIL EROSION EXPERIMENT WITH FIELD INFRARED SPECTROSCOPY

Mila Ivanova Luleva, Harald van der Werff, Freek van der Meer, Victor Jetten

ЦАРСКАТА ПЕЩЕРА

Рафаил ПОПОВ

УЧИЛИЩНИ ЛАБОРАТОРИИ И ОБОРУДВАНЕ SCHOOL LABORATORIES AND EQUIPMENT

Учебни лаборатории Илюстрации от каталог на Franz Hugershoff, Лайциг, притежаван от бъдещия

2012 година
Книжка 6
ADDRESING STUDENTS’ MISCONCEPTIONS CONCERNING CHEMICAL REACTIONS AND SYMBOLIC REPRESENTATIONS

Marina I. Stojanovska, Vladimir M. Petruševski, Bojan T. Šoptrajanov

АНАЛИЗ НА ПОСТИЖЕНИЯТА НА УЧЕНИЦИТЕ ОТ ПЕТИ КЛАС ВЪРХУ РАЗДЕЛ „ВЕЩЕСТВА И ТЕХНИТЕ СВОЙСТВА“ ПО ЧОВЕКЪТ И ПРИРОДАТА

Иваничка Буровска, Стефан Цаковски Регионален инспекторат по образованието – Ловеч

ЕКОТОКСИКОЛОГИЯ

Васил Симеонов

ПРОФ. МЕДОДИЙ ПОПОВ ЗА НАУКАТА И НАУЧНАТА ДЕЙНОСТ (1920 Г.)

Проф. Методий Попов (1881-1954) Госпожици и Господа студенти,

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КОНЦЕПТУАЛНА СХЕМА НА УЧИЛИЩНИЯ КУРС П О ХИМИЯ – МАКР О СКОПСКИ ПОДХОД

Б. В. Тошев Българско дружество за химическо образование и история и философия на химията

ROLE OF ULTRASONIC WAVES TO STUDY MOLECULAR INTERACTIONS IN AQUEOUS SOLUTION OF DICLOFENAC SODIUM

Sunanda S. Aswale, Shashikant R. Aswale, Aparna B. Dhote Lokmanya Tilak Mahavidyalaya, INDIA Nilkanthrao Shinde College, INDIA

SIMULTANEOUS ESTIMATION OF IBUPROFEN AND RANITIDINE HYDROCHLORIDE USING UV SPECTROPHOT O METRIC METHOD

Jadupati Malakar, Amit Kumar Nayak Bengal College of Pharmaceutical Sciences and Research, INDIA

GAPS AND OPPORTUNITIES IN THE USE OF REMOTE SENSING FOR SOIL EROSION ASSESSMENT

Mila Ivanova Luleva, Harald van der Werff, Freek van der Meer, Victor Jetten

РАДИОХИМИЯ И АРХЕОМЕТРИЯ: ПРО Ф. ДХН ИВЕЛИН КУЛЕВ RADIOCHEMISTRY AND ARCHEOMETRY: PROF. IVELIN KULEFF, DSc

Б. В. Тошев Българско дружество за химическо образование и история и философия на химията

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TEACHING THE CONSTITUTION OF MATTER

Małgorzata Nodzyńska, Jan Rajmund Paśko

СЪСИРВАЩА СИСТЕМА НА КРЪВТА

Маша Радославова, Ася Драгоева

CATALITIC VOLCANO

Emil Lozanov

43-ТА МЕЖДУНАРОДНА ОЛИМПИАДА ПО ХИМИЯ

Донка ТАШЕВА, Пенка ЦАНОВА

ЮБИЛЕЙ: ПРОФ. ДХН БОРИС ГЪЛЪБОВ JUBILEE: PROF. DR. BORIS GALABOV

Б. В. Тошев Българско дружество за химическо образование и история и философия на химията

ПЪРВИЯТ ПРАВИЛНИК ЗА УЧЕБНИЦИТЕ (1897 Г.)

Чл. 1. Съставянето и издаване на учебници се предоставя на частната инициа- тива. Забележка: На учителите – съставители на учебници се запрещава сами да разпродават своите учебници. Чл. 2. Министерството на народното просвещение може да определя премии по конкурс за съставяне на учебници за горните класове на гимназиите и специ- алните училища. Чл. 3. Никой учебник не може да бъде въведен в училищата, ако предварително не е прегледан и одобрен от Министерството на народното просвещение. Чл.

JOHN DEWEY: HOW WE THINK (1910)

John Dewey (1859 – 1952)

ИНФОРМАЦИЯ ЗА СПЕЦИАЛНОСТИТЕ В ОБЛАСТТА НА ПРИРОДНИТЕ НАУКИ В СОФИЙСКИЯ УНИВЕРСИТЕТ „СВ. КЛИМЕНТ ОХРИДСКИ“ БИОЛОГИЧЕСКИ ФАКУЛТЕТ

1. Биология Студентите от специалност Биология придобиват знания и практически умения в областта на биологическите науки, като акцентът е поставен на организмово равнище. Те се подготвят да изследват биологията на организмите на клетъчно- организмово, популационно и екосистемно ниво в научно-функционален и прило- жен аспект, с оглед на провеждане на научно-изследователска, научно-приложна, производствена и педагогическа дейност. Чрез широк набор избираеми и факул- тативни курсове студентите

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УЧИТЕЛИТЕ ПО ПРИРОДНИ НАУКИ – ЗА КОНСТРУКТИВИСТКАТА УЧЕБНА СРЕДА В БЪЛГАРСКОТО УЧИЛИЩЕ

Адриана Тафрова-Григорова, Милена Кирова, Елена Бояджиева

ПОВИШАВАНЕ ИНТЕРЕСА КЪМ ИСТОРИЯТА НА ХИМИЧНИТЕ ЗНАНИЯ И ПРАКТИКИ ПО БЪЛГАРСКИТЕ ЗЕМИ

Людмила Генкова, Свобода Бенева Българско дружество за химическо образование и история и философия на химията

НАЧАЛО НА ПРЕПОДАВАНЕТО НА УЧЕБЕН ПРЕДМЕТ ХИМИЯ В АПРИЛОВОТО УЧИЛИЩЕ В ГАБРОВО

Мария Николова Национална Априловска гимназия – Габрово

ПРИРОДОНАУЧНОТО ОБРАЗОВАНИЕ В БЪЛГАРИЯ – ФОТОАРХИВ

В един дълъг период от време гимназиалните учители по математика, физика, химия и естествена

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„МАГИЯТА НА ХИМИЯТА“ – ВЕЧЕР НА ХИМИЯТА В ЕЗИКОВА ГИМНАЗИЯ „АКАД. Л. СТОЯНОВ“ БЛАГОЕВГРАД

Стефка Михайлова Езикова гимназия „Акад. Людмил Стоянов“ – Благоевград

МЕЖДУНАРОДНАТА ГОДИНА НА ХИМИЯТА 2011 В ПОЩЕНСКИ МАРКИ

Б. В. Тошев Българско дружество за химическо образование и история и философия на химията

ЗА ПРИРОДНИТЕ НАУКИ И ЗА ПРАКТИКУМА ПО ФИЗИКА (Иванов, 1926)

Бурният развой на естествознанието във всичките му клонове през XIX –ия век предизвика дълбоки промени в мирогледа на културния свят, в техниката и в индустрията, в социалните отношения и в държавните интереси. Можем ли днес да си представим един философ, един държавен мъж, един обществен деец, един индустриалец, просто един културен човек, който би могъл да игнорира придобив- ките на природните науки през последния век. Какви ужасни катастрофи, какви социални сътресения би сполетяло съвре

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MURPHY’S LAW IN CHEMISTRY

Milan D. Stojković

42

Донка Ташева, Пенка Цанова

СЕМЕЙНИ УЧЕНИЧЕСКИ ВЕЧЕРИНКИ

Семейството трябва да познава училишето и училишето трябва да познава семейството. Взаимното познанство се налага от обстоятелството, че те, макар и да са два различни по природата си фактори на възпитанието, преследват една и съща проста цел – младото поколение да бъде по-умно, по-нравствено, физически по-здраво и по-щастливо от старото – децата да бъдат по-щастливи от родителите