Parameter Identification in a Tuberculosis Model for Cameroon

A deterministic model of tuberculosis in Cameroon is designed and analyzed with respect to its transmission dynamics. The model includes lack of access to treatment and weak diagnosis capacity as well as both frequency-and density-dependent transmissions. It is shown that the model is mathematically well-posed and epidemiologically reasonable. Solutions are non-negative and bounded whenever the initial values are non-negative. A sensitivity analysis of model parameters is performed and the most sensitive ones are identified by means of a state-of-the-art Gauss-Newton method. In particular, parameters representing the proportion of individuals having access to medical facilities are seen to have a large impact on the dynamics of the disease. The model predicts that a gradual increase of these parameters could significantly reduce the disease burden on the population within the next 15 years.IMU Berlin Einstein Foundation Progra

Cropping Practices and Effects on Soil Nutrient Adequacy Levels and Cassava Yield of Smallholder Farmers in Northern Zambia

Cassava is a staple food and a major source of income for many smallholder farmers. However, its yields are less than 6 t ha-1 compared to a potential yield of 20-25 t ha-1 in Zambia. Understanding cropping practices and constraints in cassava production systems is imperative for sustainable intensification. Therefore, a survey of 40 households each with three fields of cassava at 12, 24, and 36 months after planting (MAP) was conducted. Analyzed soil data, leaf area index (LAI), intercepted photosynthetically active radiation, and management practices from 120 fields were collected and subjected to descriptive statistics. To explain yield differences within the same cassava growth stage group, the data were grouped into low- and high-yield categories using the median, before applying a nonparametric test for one independent sample. Stepwise regressions were performed on each growth stage and the whole dataset to determine factors affecting tuber yield. Cassava intercropping and monocropping systems were the main cropping systems for the 12 and 24-36 MAP, respectively. Cassava yields declined by 209 and 633 kg ha-1 at 12 and 36 MAP due to soil nutrient depletion for each year of cultivation until field abandonment at 8-9 years. Fresh cassava yields ranged from 3.51-8.51, 13.52-25.84, and 16.92-30.98 t ha-1 at 12, 24, and 36 MAP, respectively. For every one unit increment in exchangeable K (cmol (+)/kg soil), cassava yield increased by 435, 268, and 406 kg ha-1 at 12, 24, and 36 MAP, respectively. One unit increment of magnesium (cmol (+)/kg soil) gave the highest yield increase of 525 kg ha-1 at 24 MAP. The low levels of soil organic carbon explained the deficient nitrogen in cassava fields, which limits the LAI growth and consequently reduced intercepted radiation and low yields. The effect of exchangeable K on growth was limited by the moderate availability of Mg and low N, thus the need for balanced fertilizer regimes. © 2021 Peter Kaluba et al

Ein mathematisches Modell für Tuberkulose in Kamerun

This thesis firstly presents a nonlinear extended deterministic model for the transmission dynamics of tuberculosis, based on realistic assumptions and data collected from the WHO. This model enables a comprehensive qualitative analysis of various aspects in the outbreak and control of tuberculosis in Sub-Saharan Africa countries and successfully reproduces the epidemiology of tuberculosis in Cameroon for the past (from 1994-2010). Some particular properties of the model and its solution have been presented using the comparison theorem applied to the theory of differential equations. The existence and the stability of a disease free equilibrium has been discussed using the Perron-Frobenius theorem and Metzler stable matrices. Furthermore, we computed the basic reproduction number, i.e. the number of cases that one case generates on average over the course of its infectious period. Rigorous qualitative analysis of the model reveals that, in contrast to the model without reinfections, the full model with reinfection exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the basic reproduction ratio ($\mathcal{R}_0$), is less than unity. The global stability of the disease-free equilibrium has been discussed using the concepts of Lyapunov stability and bifurcation theory. For a theoretical bifurcation analysis, rather than numerical computations, we have analyzed some polynomials using the Descartes sign rule. All these theoretical tools were successfully used within the study of endemic equilibria also besides the center manifold theory. The models incorporate the critical roles of health care workers, transmission heterogeneity and super-spreading events. With the help of a sensitivity analysis using data of Cameroon, we identified the relevant parameters which play a key role for the transmission and the control of the disease. This was possible applying sophisticated numerical methods (POEM) developed at ZIB. Using advanced approaches for optimal control considering the costs for chemoprophylaxis, treatment and educational campaigns should provide a framework for designing realistic cost effective strategies with different intervention methods. The forward-backward sweep method has been used to solve the numerical optimal control problem. The numerical result of the optimal control problem reveals that combined effort in education and chemoprophylaxis may lead to a reduction of 80\% in the number of infected people in 10 years. The mathematical and numerical approaches developed in this thesis could be similarly applied in many other Sub-Saharan countries where TB is a public health problem.In der vorliegendenen Arbeit wird ein nichtlineares deterministisches Modell für die Übertragungsdynamik der Tuberkulose basierend auf epidemiologischen Konzepten und Daten der Weltgesundheitsorganisation (WHO) entwickelt. Das Modell ermöglicht eine detaillierte qualitative Analyse des Ausbruchs, der Ausbreitung und der Kontrolle von Tuberkulose in subsaharischen afrikanischen Ländern und reproduziert den Verlauf der Tuberkulose-Epidemie in Kamerun von 1994 bis 2010. Spezielle Eigenschaften des Modells und seiner Lösungen werden mithilfe von Vergleichssätzen für Differentialgleichungen abgeleitet; Existenz und Stabilität eines krankheitsfreien Gleichgewichts werden unter Verwendung des Satzes von Perron-Frobenius und den Eigenschaften von Metzler-Matrizen analysiert. Die globale Stabilität des krankheitsfreien Gleichgewichts wird mittels der Konzepte der Lyapunov-Stabilität und der Bifurkationstheorie diskutiert. Die für das Studium des Verlaufs von Infektionsepidemien grundlegende Kennziffer ist die basisreproduktionszahl, d.h. die Zahl von weiteren Infektionen, die im Mittel von einem Infizierten während seiner infektiösen Periode verursacht wird. Die Analyse des Modells zeigt, dass die Berücksichtigung von Reinfektionen zu einer rückwärtsgerichteten Bifurkation führt, d.h. ein stabiles krankheitsfreies Gleichgewicht koexistiert mit einem stabilen endemischen Gleichgewicht, in dem die Basisreproduktionszahl kleiner als eins ist. Die theoretischen Methoden werden zur Untersuchung endemischer Gleichgewichtszustände verwendet, ebenso wie die Theorie der Zentrumsmannigfaltigkeiten. Die Modelle berücksichtigen auch die kritische Rolle des Gesundheitspersonals, die Übertragungsheterogenität und sogenannte “super-spreading Events”. Durch eine Sensitivitätsanalyse mit Hilfe von am ZIB entwickelter Verfahren (POEM/BioParkin) anhand realer Daten aus Kamerun lassen sich die Modellparameter identifizieren, die eine Schlüsselrolle für die Übertragung und Kontrolle der Tuberkulose innehaben. Für die Entwicklung wirksamer und kosteneffektiver Strategien zur Bekämpfung der Tuberkulose werden Methoden des Optimalsteuerung verwendet. Hierbei werden Kosten für Chemoprophylaxe, Behandlung und Aufklärungskampagnen berücksichtigt. Zur Lösung der Optimalsteuerungsprobleme wird ein Forward-Backward-Sweep-Ansatz eingesetzt. Die numerischen Ergebnisse zeigen, dass eine kombinierte Strategie in Aufklärung und Chemoprophylaxe zu einer Reduktion der Zahl infizierter Personen um 80\% in 10 Jahren führen könnte. Die mathematischen und numerischen Ansätze, die im Rahmen dieser Arbeit entwickelt wurden, könnten auf viele andere subsaharische Länder übertragen werden, in denen Tuberkulose eines der größten Gesundheitsproblem darstellt

Environmental triggers for photosynthetic protein turnover determine the optimal nitrogen distribution and partitioning in the canopy

Plants continually adjust the photosynthetic functions in their leaves to fluctuating light, thereby optimizing the use of photosynthetic nitrogen (Nph) at the canopy level. To investigate the complex interplay between external signals during the acclimation processes, a mechanistic model based on the concept of protein turnover (synthesis and degradation) was proposed and parameterized using cucumber grown under nine combinations of nitrogen and light in growth chambers. Integrating this dynamic model into a multi-layer canopy model provided accurate predictions of photosynthetic acclimation of greenhouse cucumber canopies grown under high and low nitrogen supply in combination with day-to-day fluctuations in light at two different levels. This allowed us to quantify the degree of optimality in canopy nitrogen use for maximizing canopy carbon assimilation, which was influenced by Nph distribution along canopy depth or Nph partitioning between functional pools. Our analyses suggest that Nph distribution is close to optimum and Nph reallocation is more important under low nitrogen. Nph partitioning is only optimal under a light level similar to the average light intensity during acclimation, meaning that day-to-day light fluctuations inevitably result in suboptimal Nph partitioning. Our results provide insights into photoacclimation and can be applied to crop model improvement

Results of the sensitivity analysis at the final iterate, i.e. with the final set of parameter values from Table 2: (a) <i>l</i>₂ column norms of the scaled sensitivity matrix <i>J</i> and (b) subcondition numbers of parameters.

<p>In fact, all unknown parameters are identifiable for <i>ɛ</i> = 10⁻⁷. The magnitudes of the sensitivities alone, represented by the column norms, are an indicator for identifiability but generally do not provide information on the correct ordering of parameters.</p

Numbers for diagnosed infectious (<i>I</i>) and total population (<i>N</i>) in Cameroon over the period 1994–2010.

<p>Data published by WHO [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.ref027" target="_blank">27</a>].</p><p>Numbers for diagnosed infectious (<i>I</i>) and total population (<i>N</i>) in Cameroon over the period 1994–2010.</p

Evolution of model (2) showing the state trajectories for diagnosed infectious individuals (I) and total population (N).

<p>The dot plots represent the year-by-year trend in yearly case reports for Cameroon over the period 1994–2010. Parameter values are defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.t002" target="_blank">Table 2</a> and initial values are presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.t003" target="_blank">Table 3</a>.</p

Time series of model (2) showing the impact of a slow change on parameter values<i>θ</i>, <i>δ</i>, <i>p</i>₁ and <i>p</i>₂ with respect to time in order to reduce the TB burden by 20% within 15 years.

<p>Solid lines present the model predictions for TB dynamics using parameter values of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.t002" target="_blank">Table 2</a> and the dashed lines present the trajectories for parameters <i>θ</i>, <i>δ</i>, <i>p</i>₁ and<i>p</i>₂ set as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.e047" target="_blank">Equation (18)</a>. Parameter identification with artificial data gave<i>p</i>_<i>δ</i> = 8.56043⋅10⁷, <i>θ</i>_<i>δ</i> = 81.2807, <i>δ</i>_<i>δ</i> = 37.2240. All other parameters are defined as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.t002" target="_blank">Table 2</a>.</p

Evolution of model (2) showing the state trajectories for the relative amounts of susceptible population (<i>S</i>/<i>N</i>), latently infected population (<i>E</i>/<i>N</i>), infectious population ((<i>I</i>+<i>L</i>+<i>J</i>)/<i>N</i>), and recovered population (<i>R</i>/<i>N</i>).

<p>Parameter values are defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.t002" target="_blank">Table 2</a> and initial values are presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120607#pone.0120607.t003" target="_blank">Table 3</a>.</p

Numerical values of the TB model parameters.

<p>Numerical values of the TB model parameters.</p