Nonlocal reaction–diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction–diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.

We propose to study the wound healing in Zebrafish by using firstly a differential approach for modelling morphogens diffusion and cell chemotactic motion, and secondly a hybrid model of tissue regeneration, where cells are considered as individual objects and molecular concentrations are described by partial differential equations.

The study of the autonomic nervous system (ANS) function has shown to provide useful indicators for risk stratification and early detection on a variety of cardiovascular pathologies. However, data gathered during different tests of the ANS are difficult to analyse, mainly due to the complex mechanisms involved in the autonomic regulation of the cardiovascular system (CVS). Although model-based analysis of ANS data has been already proposed as a way to cope with this complexity, only a few models coupling the main (...) elements involved have been presented in the literature. In this paper, a new model of the CVS, representing the ventricles, the circulatory system and the regulation of the CVS activity by the ANS, is presented. The models of the vascular system and the ventricular activity have been developed using the Bond Graph formalism, as it proposes a unified representation for all energetic domains, facilitating the integration of mechanic and hydraulic phenomena. In order to take into account the electro-mechanical behaviour of both ventricles, an electrophysiologic model of the cardiac action potential, represented by a set of ordinary differential equations, has been integrated. The short-term ANS regulation of heart rate, cardiac contractility and peripheral vasoconstriction is represented by means of continuous transfer functions. These models, represented in different continuous formalisms, are coupled by using a multi-formalism simulation library. Results are presented for two different autonomic tests, namely the Tilt Test and the Valsalva Manoeuvre, by comparing real and simulated signals. (shrink)

The study of the autonomic nervous system (ANS) function has shown to provide useful indicators for risk stratification and early detection on a variety of cardiovascular pathologies. However, data gathered during different tests of the ANS are difficult to analyse, mainly due to the complex mechanisms involved in the autonomic regulation of the cardiovascular system (CVS). Although model-based analysis of ANS data has been already proposed as a way to cope with this complexity, only a few models coupling the main (...) elements involved have been presented in the literature. In this paper, a new model of the CVS, representing the ventricles, the circulatory system and the regulation of the CVS activity by the ANS, is presented. The models of the vascular system and the ventricular activity have been developed using the Bond Graph formalism, as it proposes a unified representation for all energetic domains, facilitating the integration of mechanic and hydraulic phenomena. In order to take into account the electro-mechanical behaviour of both ventricles, an electrophysiologic model of the cardiac action potential, represented by a set of ordinary differential equations, has been integrated. The short-term ANS regulation of heart rate, cardiac contractility and peripheral vasoconstriction is represented by means of continuous transfer functions. These models, represented in different continuous formalisms, are coupled by using a multi-formalism simulation library. Results are presented for two different autonomic tests, namely the Tilt Test and the Valsalva Manoeuvre, by comparing real and simulated signals. (shrink)

Radopholus Similis or burrowing nematode, is one of the most damaging and widespread nematodes attacking bananas, causing toppling or blackhead disease. A mathematical model for the population dynamics of R. Similis is considered, with the aim of investigating the impact of climatic factors on the growth of R. Similis. In this paper, based on the life cycle of R. Similis, we first propose a mathematical model to study and control the population dynamics of this banana pest. We show also how (...) control terms based on biological and chemical controls can be integrated to reduce the population of R. Similis within banana-plantain roots. Sensitivity analysis was performed to show the most important parameters of the model. We present the theoretical analysis of the model. More precisely, we derive a threshold parameter N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{N}}_0$$\end{document}, called the basic offspring number and show that the trivial equilibrium is globally asymptotically stable whenever N0≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{N}}_0\le 1$$\end{document}, while when N0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{N}}_0> 1$$\end{document}, the non trivial equilibrium is globally asymptotically stable. After, we extend the proposed model by taking account climatic factors that influence the growth of this pest. Biological and chemical controls are now introduced through impulsive equations. Threshold and equilibria are obtained and global stabilities have been studied. The theoretical results are supported by numerical simulations. Numerical results of model with biological and chemical controls reveal that biological methods are more effective than chemical methods. We also found that the month February is the best time to apply these controls. (shrink)

In this work a mechanistic explanation of the classical algae growth model built by M. R. Droop in the late sixties is proposed. We first recall the history of the construction of the “predictive” variable yield Droop model as well as the meaning of the introduced cell quota. We then introduce some theoretical hypotheses on the biological phenomena involved in nutrient storage by the algae that lead us to a “conceptual” model. Though more complex than Droop’s one, our model remains (...) accessible to a complete mathematical study: its confrontation to the Droop model shows both have the same asymptotic behavior. However, while Droop’s cell quota comes from experimental bio-chemical measurements not related to intra-cellular biological phenomena, its analogous in our model directly follows our theoretical hypotheses. This new model should then be looked at as a re-interpretation of Droop’s work from a theoretical biologist’s point of view. (shrink)

Modal pseudocomplemented De Morgan algebras were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1, pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ* = *))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is (...) determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18]. (shrink)

The study of the autonomic nervous system (ANS) function has shown to provide useful indicators for risk stratification and early detection on a variety of cardiovascular pathologies. However, data gathered during different tests of the ANS are difficult to analyse, mainly due to the complex mechanisms involved in the autonomic regulation of the cardiovascular system (CVS). Although model-based analysis of ANS data has been already proposed as a way to cope with this complexity, only a few models coupling the main (...) elements involved have been presented in the literature. In this paper, a new model of the CVS, representing the ventricles, the circulatory system and the regulation of the CVS activity by the ANS, is presented. The models of the vascular system and the ventricular activity have been developed using the Bond Graph formalism, as it proposes a unified representation for all energetic domains, facilitating the integration of mechanic and hydraulic phenomena. In order to take into account the electro-mechanical behaviour of both ventricles, an electrophysiologic model of the cardiac action potential, represented by a set of ordinary differential equations, has been integrated. The short-term ANS regulation of heart rate, cardiac contractility and peripheral vasoconstriction is represented by means of continuous transfer functions. These models, represented in different continuous formalisms, are coupled by using a multi-formalism simulation library. Results are presented for two different autonomic tests, namely the Tilt Test and the Valsalva Manoeuvre, by comparing real and simulated signals. (shrink)

Phylogenetic systematics is one of the most important analytical frameworks of modern Biology. It seems to be common knowledge that within phylogenetics, ‘groups’ must be defined based solely on the synapomorphies or on the “derived” characters that unite two or more taxa in a clade or monophyletic group. Thus, the idea of synapomorphy seems to be of fundamental influence and importance. Here I will show that the most common and straightforward understanding of synapomorphy as a shared derived character is not (...) sufficient and eventually must be rejected in favor of Nelson’s relational interpretation of such term. Arguing for this point and using three examples from previously published Apes’ genomic matrices, I explicitly demonstrate that the relationship )) with Hylobatidae as a sister taxon, may be successfully recovered by three-taxon statement analysis and three-taxon statement average consensus analysis even if all of the evident standard shared derived molecular characters of the relationship )) with Hylobatidae as a sister taxon, have been excluded from the molecular alignments. Neither conventional Maximum Parsimony nor Maximum Likelihood or Bayesian Inference can do this in such situation. Thus, our results show that the relationship )) with Hylobatidae as a sister taxon has appeared, in some way, behind standard shared derived characters: the last ones could be excluded, but the relationship remains the same. (shrink)

Recently the terms “codes” and “information” as used in the context of molecular biology have been the subject of much discussion. Here I propose that a variety of structural realism can assist us in rethinking the concepts of DNA codes and information apart from semantic criteria. Using the genetic code as a theoretical backdrop, a necessary distinction is made between codes qua symbolic representations and information qua structure that accords with data. Structural attractors are also shown to be entailed by (...) the mapping relation that any DNA code is a part of (as the domain). In this framework, these attractors are higher-order informational structures that obviate any “DNA-centric” reductionism. In addition to the implications that are discussed, this approach validates the array of coding systems now recognized in molecular biology. (shrink)

An analogy between the evolution of organisms and some complex computational problems (cryptosystem cracking, determination of the shortest path in a graph) is considered. It is shown that in the absence of a priori information about possible species of organisms such a problem is complex (is rated in the class NP) and cannot be solved in a polynomial number of steps. This conclusion suggests the need for re-examination of evolution mechanisms. Ideas of a deterministic approach to the evolution are discussed.

This paper addresses the existence of Hopf bifurcations in a directed acyclic network of neurons, each of them being modeled by a Hindmarsh–Rose neuronal model. The bifurcation parameter is the small parameter corresponding to the ratio of time scales between the fast and the slow dynamics. We first prove that, under certain hypotheses, the single uncoupled neuron can undergo a Hopf bifurcation. Hopf bifurcation occurrences in a directed acyclic network of HR neurons are then discussed. Numerical simulations are carried out (...) to observe these bifurcations and to illustrate the theoretical results. (shrink)

Control of DNA supercoiling by the free-energy of hydrolysis of ATP that involves gene expression is analyzed in terms of three levels of unconnected metabolic pathways. These are synthesis and breakdown of topoisomerase mRNAs, synthesis and breakdown of topoisomerase proteins and supercoiling and relaxation of DNA. The so-called square-matrix method previously developed for the control of metabolic pathways, is extended to deal with this hierarchical control system. It turns out that also in this case, the matrix of control coefficients is (...) equal to the inverse of the so-called elasticity matrix, which contains all relevant elasticity coefficients as well as information about the structure and connectedness of the pathways involved. For a simpler case of a hierarchy of two systems, we demonstrate that the explicit matrix inversion method may be replaced by an implicit method in which the regulatory effects that run through the other level are described by an additional elasticy coefficient which may then be treated as if local. (shrink)

The concepts of regulation and homeostasis are of frequent use but lack a single universally accepted definition. Here we propose a definition of theregulatory strength andhomeostatic strength, which allow to assess the importance of a regulatory pathway in a quantitative fashion.

The fundamental Hennigian principle, grouping solely on synapomorphy, is seldom used in modern phylogenetics. In the submitted paper, we apply this principle in reanalyzing five datasets comprising 197 complete plastid genomes (plastomes). We focused on the latter because plastome-based DNA sequence data gained dramatic popularity in molecular systematics during the last decade. We show that pattern-cladistic analyses based on complete plastid genome sequences can successfully resolve affinities between plant taxa, simultaneously simplifying both the genomic and analytical frameworks of phylogenetic studies. (...) We developed “Matrix to Newick” (M2N), a program to represent the standard molecular alignment of plastid genomes in the form of trees or relationships directly. Thus, massive plastome-based DNA sequence data can be successfully represented in a relational form rather than as a standard molecular alignment. Application of methods of median supertree construction (the Average Consensus method has been used as an example in this study) or Maximum Parsimony analysis to relational representations of plastome sequence data may help systematist to avoid the complicated assumption-based frameworks of Maximum Likelihood or Bayesian phylogenetics that are most used today in massive plastid sequence data analyses. We also found that significant amounts of pure genomic information that typically accommodate the majority of current plastid phylogenomic studies can be effectively dropped by systematists if they focus on the pattern-cladistics or relational analyses of plastome-based molecular data. The proposed pattern-cladistic approach is a powerful and straightforward heuristic alternative to modern plastome-based phylogenetics. (shrink)

Gil R, Arroyo-Anllo EM, Ingrand P, Gil M, Neau JP, Ornon C, Bonnaud V. Self-consciousness and Alzheimer’s disease. Acta Neurol Scand 2001: 104: 296–300. # Munksgaard 2001. Objectives – To propose a neuropsychological study of the various aspects of self-consciousness (SC) in Alzheimer’s disease. Methods – Forty-five patients with probable mild or moderate AD were included in the study. Severity of their dementia was assessed by the Mini Mental State (MMS). Fourteen questions were prepared to evaluate SC. Results – (...) No significant correlations were found between SC score and educational level, age, and duration of disease. A significant correlation was found between SC score and the severity of dementia, whereas frontal disturbances were just short of the significance threshold. The various aspects of SC were not impaired to the same degree. The most disturbed ones were awareness of cognitive deficiencies, moral judgements and prospective memory. The least disturbed aspects were awareness of identity and of mental representation of the body. Items relating to anosognosia and moral judgements were significantly correlated with the MMS score, whereas affective state, body representation disorders, prospective memory, and capacities for introspection were not related to the severity of the dementia. Consciousness of identity was sound, regardless of MMS score. Conclusions – AD clearly induces an heterogeneous impairment of SC. SC requires a convergence of many neural networks. In AD, neuronal alterations involve many cortical areas and information sent to the associative frontal cortex from memory, language and visuospatial areas is lacking or disturbed. Thus, the sequential order of successive stimuli cannot be maintained by the heteromodal associative cortex (dorsal convexity of the prefrontal cortex), and the supramodal associative cortex (located rostrally in the frontal lobes) is unable to provide reliable monitoring and assessment of simultaneous neural cognitive networks carrying insufficient and inadequate input. The core deficiency in AD patients might be impaired SC equated with the disability to maintain sequential and simultaneous ‘‘attention to life’’. The Self-Consciousness Questionnaire, a clinical scale providing multidimensional measurement, indicates that different aspects of consciousness are not correlated with overall cognitive deficiency as determined by the MMSE. (shrink)

The theory of chemical symbiosis suggests that biological systems started with the collaboration of two polymeric molecules existing in early Earth: nucleic acids and peptides. Chemical symbiosis emerged when RNA-like nucleic acid polymers happened to fold into 3D structures capable to bind amino acids together, forming a proto peptidyl-transferase center. This folding catalyzed the formation of quasi-random small peptides, some of them capable to bind this ribozyme structure back and starting to form an initial layer that would produce the larger (...) subunit of the ribosome by accretion. TCS suggests that there is no chicken-and-egg problem into the emergence of biological systems as RNAs and peptides were of equal importance to the origin of life. Life has initially emerged when these two macromolecules started to interact in molecular symbiosis. Further, we suggest that life evolved into progenotes and cells due to the emergence of new layers of symbiosis. Mutualism is the strongest force in biology, capable to create novelties by emergent principles; on which the whole is bigger than the sum of the parts. TCS aims to apply the Margulian view of biology into the origins of life field. (shrink)

The rôle of mathematical models in growth studies is discussed and a distinction made between the structural and descriptive aspects of a mathematical formulation. A simple scheme is set up to represent in broad outline the simultaneous growth and interaction of larval and adult systems inMyrmica rubra and is matched against observed results.

This paper is devoted to computer modelling of the development and regeneration of multicellular biological structures. Some species are able to regenerate parts of their body after amputation damage, but the global rules governing cooperative cell behaviour during morphogenesis are not known. Here, we consider a simplified model organism, which consists of tissues formed around special cells that can be interpreted as stem cells. We assume that stem cells communicate with each other by a set of signals, and that the (...) values of these signals depend on the distance between cells. Thus the signal distribution characterizes location of stem cells. If the signal distribution is changed, then the difference between the initial and the current signal distribution affects the behaviour of stem cells—e.g. as a result of an amputation of a part of tissue the signal distribution changes which stimulates stem cells to migrate to new locations, appropriate for regeneration of the proper pattern. Moreover, as stem cells divide and form tissues around them, they control the form and the size of regenerating tissues. This two-level organization of the model organism, with global regulation of stem cells and local regulation of tissues, allows its reproducible development and regeneration. (shrink)

A model representing isotope separation during water evaporation in plants was constructed. The model accounts for substance diffusion, convective transfer and evaporation from the surface of the leaves. The dependence of the system’s separation and enrichment coefficients on various parameters was determined. A comparison was made between the enrichment coefficients calculated from experimental data from different plants and those based on the model. Qualitative agreement between the experimental and theoretical values was obtained for the case of $$\frac{uh}{D} {\gg} 1$$ uh (...) D ≫ 1, where u is the average velocity of water in the plant, h is the height of the plant, and D is the diffusion coefficient of the substance. (shrink)

The state-of-art research in the field of life’s organization confronts the need to investigate a number of interacting components, their properties and conditions of sustainable behaviour within a natural system. In biology, ecology and life sciences, the performance of such stable system is usually related to homeostasis, a property of the system to actively regulate its state within a certain allowable limits. In our previous work, we proposed a deterministic model for systems’ homeostasis. The model was based on dynamical system’s (...) theory and pairwise relationships of competition, amensalism and antagonism taken from theoretical biology and ecology. However, the present paper proposes a different dimension to our previous results based on the same model. In this paper, we introduce the influence of inter-component relationships in a system, wherein the impact is characterized by direction (neutral, positive, or negative) as well as its (absolute) value, or strength. This makes the model stochastic which, in our opinion, is more consistent with real-world elements affected by various random factors. The case study includes two examples from areas of hydrobiology and medicine. The models acquired for these cases enabled us to propose a convincing explanation for corresponding phenomena identified by different types of natural systems. (shrink)

Correct interpretation of the data from integral laboratory tests, including Thrombin Generation Test, requires biochemistry-based mathematical models of blood coagulation. The purpose of this study is to describe the experimental TGT data from healthy donors and hemophilia A and B patients. We derive a simplified ODE model and apply it to analyze the TGT data from healthy donors and HA/HB patients with in vitro added tissue factor pathway inhibitor antibody. This model allows the characterization of hemophilia patients in the space (...) of three most important model parameters. The proposed approach may provide a new quantitative tool for the analysis of experimental TGT. Also, it gives a reduced model of coagulation verified against clinical data to be used in future theoretical large-scale modeling of thrombosis in flow. (shrink)

Theoretical and experimental studies on prey–predator systems where predator is supplied with alternate sources of food have received significant attention over the years due to their relevance in achieving biological conservation and biological control. Some of the outcomes of these studies suggest that with appropriate quality and quantity of additional food, the system can be steered towards any desired state eventually with time. One of the limitations of previous studies is that the desired state is reached asymptotically, which makes the (...) outcomes not easily applicable in practical scenarios. To overcome this limitation, in this work, we formulate and study optimal control problems to achieve the desired outcomes in minimum (finite) time. We consider two different models of additional food provided prey–predator systems involving Holling type IV functional response (with inhibitory effect of prey). In the first scenario, additional food is incorporated implicitly into the predator’s functional response with a possibility of achieving biological conservation through co-existence of species and biological control by maintaining prey at a level that is least harmful to the system. In the second, the effect of additional food is incorporated explicitly into the predator’s compartment with the goal of pest management by maintaining prey density at a very minimal damaging level. For both cases, appropriate optimal control strategies are derived and the theoretical findings are illustrated by numerical simulations. We also discuss the ecological significance of the theoretical findings for both models. (shrink)

Mathematical modeling is a convenient way for characterization of complex ecosystems. This approach was applied to study the dynamics of zooplankton in Lake Sevan (Armenia) at different stages of anthropogenic eutrophication with the use of a novel method called discrete modeling of dynamical systems with feedback (DMDS). Simulation demonstrated that the application of this method helps in characterization of inter- and intra-component relationships in a natural ecosystem. This method describes all possible pairwise inter-component relationships like “plus–plus,” “minus–minus,” “plus–minus,” “plus–zero,” “minus–zero,” (...) and “zero–zero” that occur in most ecosystems. Based on the results, a working hypothesis was formulated. It was found that the sensitivity to weak external influence in zooplanktons was the greatest during the mid period of eutrophication in Lake Sevan, whereas in the final stages of eutrophication, an outbreak in the biomass production of cyanobacteria was evident. To support this approach, a weak external disturbance in the form of magnetic storm was used to see its effect on species Daphnia longispina sevanica. A statistically significant correlation between the frequency of magnetic storms and the number of this species was revealed and an increase in the number of toxic cyanobacteria species as a consequence of eutrophication. This paper, for the first time, suggests a DMDS method, to diagnose impact of anthropogenic eutrophication on environment. (shrink)

The paradox of biological diversity is the key problem of theoretical ecology. The paradox consists in the contradiction between the competitive exclusion principle and the observed biodiversity. The principle is important as the basis for ecological theory. On a relatively simple model we show a mechanism of indefinite coexistence of complete competitors which violates the known formulations of the competitive exclusion principle. This mechanism is based on timely recovery of limiting resources and their spatio-temporal allocation between competitors. Because of limitations (...) of the black-box modeling there was a problem to formulate the exclusion principle correctly. Our white-box multiscale model of two-species competition is based on logical deterministic individual-based cellular automata. This approach provides an automatic deductive inference on the basis of a system of axioms, and gives a direct insight into mechanisms of the studied system. It is one of the most promising methods of artificial intelligence. We reformulate and generalize the competitive exclusion principle and explain why this formulation provides a solution of the biodiversity paradox. In addition, we propose a principle of competitive coexistence. (shrink)

Developmental models that account for the metabolic effect of temperature variability on poikilotherms, such as degree-day models, have been widely used to study organism emergence, range and development, particularly in agricultural and vector-borne disease contexts. Though simple and easy to use, structural and parametric issues can influence the outputs of such models, often substantially. Because the underlying assumptions and limitations of these models have rarely been considered, this paper reviews the structural, parametric, and experimental issues that arise when using degree-day (...) models, including the implications of particular structural or parametric choices, as well as assumptions that underlie commonly used models. Linear and non-linear developmental functions are compared, as are common methods used to incorporate temperature thresholds and calculate daily degree-days. Substantial differences in predicted emergence time arose when using linear versus non-linear developmental functions to model the emergence time in a model organism. The optimal method for calculating degree-days depends upon where key temperature threshold parameters fall relative to the daily minimum and maximum temperatures, as well as the shape of the daily temperature curve. No method is shown to be universally superior, though one commonly used method, the daily average method, consistently provides accurate results. The sensitivity of model projections to these methodological issues highlights the need to make structural and parametric selections based on a careful consideration of the specific biological response of the organism under study, and the specific temperature conditions of the geographic regions of interest. When degree-day model limitations are considered and model assumptions met, the models can be a powerful tool for studying temperature-dependent development. (shrink)

The COVID-19 pandemic has resulted in more than 524 million cases and 6 million deaths worldwide. Various drug interventions targeting multiple stages of COVID-19 pathogenesis can significantly reduce infection-related mortality. The current within-host mathematical modeling study addresses the optimal drug regimen and efficacy of combination therapies in the treatment of COVID-19. The drugs/interventions considered include Arbidol, Remdesivir, Interferon and Lopinavir/ritonavir. It is concluded that these drugs, when administered singly or in combination, reduce the number of infected cells and viral load. (...) Four scenarios dealing with the administration of a single drug, two drugs, three drugs and all four are discussed. In all these scenarios, the optimal drug regimen is proposed based on two methods. In the first method, these medical interventions are modeled as control interventions and a corresponding objective function and optimal control problem are formulated. In this framework, the optimal drug regimen is derived. Later, using the comparative effectiveness method, the optimal drug regimen is derived based on the basic reproduction number and viral load. The average number of infected cells and viral load decreased the most when all four drugs were used together. On the other hand, the average number of susceptible cells decreased the most when Arbidol was administered alone. The basic reproduction number and viral load decreased the most when all four interventions were used together, confirming the previously obtained finding of the optimal control problem. The results of this study can help physicians make decisions about the treatment of the life-threatening COVID-19 infection. (shrink)