A mathematical modelling framework for linked within-host and between-host dynamics for infections with free-living pathogens in the environment

Application of the replication-transmission relativity theory in the development of multiscale models of infectious disease dynamics

Despite the existence of a powerful theoretical foundation for the development of multiscale models of infectious disease dynamics in the form of the replication-transmission relativity theory, the majority of current modelling studies focus more on single-scale modelling. The explicit aim of this study is to change the current predominantly single-scale modelling landscape in the design of planning frameworks for the control, elimination and even eradication of infectious disease systems through the exploitation of multiscale modelling methods based on the application of the replication-transmission relativity theory. We first present a structured roadmap for the development of multiscale models of infectious disease systems. The roadmap is tested on hookworm infection. The testing of the feasibility of the roadmap established a fundamental result which can be generalized to confirm that the complexity of an infectious disease system is encapsulated with a level of organization spanning a microscale and a macroscale.

An immuno-epidemiological model linking between-host and within-host dynamics of cholera

Cholera, a severe gastrointestinal infection caused by the bacterium Vibrio cholerae, remains a major threat to public health, with a yearly estimated global burden of 2.9 million cases. Although most existing models for the disease focus on its population dynamics, the disease evolves from within-host processes to the population, making it imperative to link the multiple scales of the disease to gain better perspectives on its spread and control. In this study, we propose an immuno-epidemiological model that links the between-host and within-host dynamics of cholera. The immunological (within-host) model depicts the interaction of the cholera pathogen with the adaptive immune response. We distinguish pathogen dynamics from immune response dynamics by assigning different time scales. Through a time-scale analysis, we characterise a single infected person by their immune response. Contrary to other within-host models, this modelling approach allows for recovery through pathogen clearance after a finite time. Then, we scale up the dynamics of the infected person to construct an epidemic model, where the infected population is structured by individual immunological dynamics. We derive the basic reproduction number ($ \mathcal{R}_0 $) and analyse the stability of the equilibrium points. At the disease-free equilibrium, the disease will either be eradicated if $ \mathcal{R}_0 < 1 $ or otherwise persists. A unique endemic equilibrium exists when $ \mathcal{R}_0 > 1 $ and is locally asymptotically stable without a loss of immunity.

COVID-19 dynamics and immune response: Linking within-host and between-host dynamics

The global impact of COVID-19 has led to the development of numerous mathematical models to understand and control the pandemic. However, these models have not fully captured how the disease's dynamics are influenced by both within-host and between-host factors. To address this, a new mathematical model is proposed that links these dynamics and incorporates immune response. The model is compartmentalized with a fractional derivative in the sense of Caputo-Fabrizio, and its properties are studied to show a unique solution. Parameter estimation is carried out by fitting real-life data, and sensitivity analysis is conducted using various methods. The model is then numerically implemented to demonstrate how the dynamics within infected hosts drive human-to-human transmission, and various intervention strategies are compared based on the percentage of averted deaths. The simulations suggest that a combination of medication to boost the immune system, prevent infected cells from producing the virus, and adherence to COVID-19 protocols is necessary to control the spread of the virus since no single intervention strategy is sufficient.

Modeling COVID-19 infection in high-risk settings and low-risk settings

In this research paper we present a mathematical model for COVID-19 in high-risk settings and low-risk settings which might be infection dynamics between hotspots and less risky communities. The main idea was to couple the SIR model with alternating risk levels from the two different settings high and low-risk settings. Therefore, building from this model we partition the infected class into two categories, the symptomatic and the asymptomatic. Using this approach we simulated COVID-19 dynamics in low and high-risk settings with auto-switching risk settings. Again, the model was analyzed using both analytic methods and numerical methods. The results of this study suggest that switching risk levels in different settings plays a pivotal role in COVID-19 progression dynamics. Hence, population reaction time to adhere to preventative measures and interventions ought to be implemented with flash speed targeting first the high-risk setting while containing the dynamics in low-risk settings.

An Embedded Multiscale Modelling to Guide Control and Elimination of Paratuberculosis in Ruminants

In recent years, multiscale modelling approach has begun to receive an overwhelming appreciation as an appropriate technique to characterize the complexity of infectious disease systems. In this study, we develop an embedded multiscale model of paratuberculosis in ruminants at host level that integrates the within-host scale and the between-host. A key feature of embedded multiscale models developed at host level of organization of an infectious disease system is that the within-host scale and the between-host scale influence each other in a reciprocal (i.e., both) way through superinfection, that is, through repeated infection before the host recovers from the initial infectious episode. This key feature is demonstrated in this study through a multiscale model of paratuberculosis in ruminants. The results of this study, through numerical analysis of the multiscale model, show that superinfection influences the dynamics of paratuberculosis only at the start of the infection, while the MAP bacteria replication continuously influences paratuberculosis dynamics throughout the infection until the host recovers from the initial infectious episode. This is largely because the replication of MAP bacteria at the within-host scale sustains the dynamics of paratuberculosis at this scale domain. We further use the embedded multiscale model developed in this study to evaluate the comparative effectiveness of paratuberculosis health interventions that influence the disease dynamics at different scales from efficacy data.

Estimation and optimal control of the multiscale dynamics of Covid-19: a case study from Cameroon

This work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). A multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism is proposed first. Indeed, there are human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self-infection by susceptible individuals. The global stability of the disease-free equilibrium is shown when a given threshold T 0 is less or equal to 1 and the basic reproduction number R 0 is calculated. A convergence index T 1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of infectious extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. Using Partial Rank Correlation Coefficient with a three levels fractional experimental design, the sensitivity of  R 0 , T 0 and T 1 to control parameters is evaluated. Following this study, the most significant parameter is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into account economic impacts of SARS-CoV-2, optimal fighting strategies are determined and discussed. The study is applied to real and available data from Cameroon with a model fitting. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy against SARS-CoV-2.

Rationale and Criteria for a COVID-19 Model Framework

Complex systems are inherently multilevel and multiscale systems. The infectious disease system is considered a complex system resulting from the interaction between three sub-systems (host, pathogen, and environment) organized into a hierarchical structure, ranging from the cellular to the macro-ecosystem level, with multiscales. Therefore, to describe infectious disease phenomena that change through time and space and at different scales, we built a model framework where infectious disease must be considered the set of biological responses of human hosts to pathogens, with biological pathways shared with other pathologies in an ecological interaction context. In this paper, we aimed to design a framework for building a disease model for COVID-19 based on current literature evidence. The model was set up by identifying the molecular pathophysiology related to the COVID-19 phenotypes, collecting the mechanistic knowledge scattered across scientific literature and bioinformatic databases, and integrating it using a logical/conceptual model systems biology. The model framework building process began from the results of a domain-based literature review regarding a multiomics approach to COVID-19. This evidence allowed us to define a framework of COVID-19 conceptual model and to report all concepts in a multilevel and multiscale structure. The same interdisciplinary working groups that carried out the scoping review were involved. The conclusive result is a conceptual method to design multiscale models of infectious diseases. The methodology, applied in this paper, is a set of partially ordered research and development activities that result in a COVID-19 multiscale model.

The research and development process for multiscale models of infectious disease systems

Multiscale modelling of infectious disease systems falls within the domain of complexity science—the study of complex systems. However, what should be made clear is that current progress in multiscale modelling of infectious disease dynamics is still as yet insufficient to present it as a mature sub-discipline of complexity science. In this article we present a methodology for development of multiscale models of infectious disease systems. This methodology is a set of partially ordered research and development activities that result in multiscale models of infectious disease systems built from different scientific approaches. Therefore, the conclusive result of this article is a methodology to design multiscale models of infectious diseases. Although this research and development process for multiscale models cannot be claimed to be unique and final, it constitutes a good starting point, which may be found useful as a basis for further refinement in the discourse for multiscale modelling of infectious disease dynamics.

Complex systems such as infectious disease systems are inherently multilevel and multiscale systems. The study of such complex systems is called complexity science. In this article we present a methodology to design multiscale models of infectious disease systems from a complex systems perspective. Within this perspective we define complexity science as the study of the interconnected relationships of the levels and scales of organization of a complex system. We therefore, define the degree of complexity of a complex system as the number of levels and scales of organization of the complex system needed to describe it. In this work we first present a common multiscale vision of the multilevel and multiscale structure of infectious disease systems as complex systems in which the levels and scales of organization of an infectious disease system interact through different self-sustained multiscale cycles/loops (primary multiscale loops, or secondary multiscale loops, or tertiary multiscale loops) formed at different levels of organization of an infectious disease system due to ongoing reciprocal influence between the microscale and the macroscale. Guided by this multiscale vision, we propose a four-stage research and development process that result in multiscale models of infectious disease systems built from different scientific approaches.

A general method for multiscale modelling of vector-borne disease systems

The inability to develop multiscale models which can describe vector-borne disease systems in terms of the complete pathogen life cycle which represents multiple targets for control has hindered progress in our efforts to control, eliminate and even eradicate these multi-host infections. This is because it is currently not easy to determine precisely where and how in the life cycles of vector-borne disease systems the key constrains which are regarded as crucial in regulating pathogen population dynamics in both the vertebrate host and vector host operate. In this article, we present a general method for development of multiscale models of vector-borne disease systems which integrate the within-host and between-host scales for the two hosts (a vertebrate host and a vector host) that are implicated in vector-borne disease dynamics. The general multiscale modelling method is an extension of our previous work on multiscale models of infectious disease systems which established a basic science and accompanying theory of how pathogen population dynamics at within-host scale scales up to between-host scale and in turn how it scales down from between-host scale to within-host scale. Further, the general method is applied to multiscale modelling of human onchocerciasis-a vector-borne disease system which is sometimes called river blindness as a case study.

The Replication-Transmission Relativity Theory for Multiscale Modelling of Infectious Disease Systems

It is our contention that for multiscale modelling of infectious disease systems to evolve and expand in scope, it needs to be founded on a theory. Such a theory would improve our ability to describe infectious disease systems in terms of their scales and levels of organization, and their inter-relationships. In this article we present a relativistic theory for multiscale modelling of infectious disease systems, that can be considered as an extension of the relativity principle in physics, called the replication-transmission relativity theory. This replication-transmission relativity theory states that at any level of organization of an infectious disease system there is no privileged/absolute scale which would determine, disease dynamics, only interactions between the microscale and macroscale. Such a relativistic theory provides a scientific basis for a systems level description of infectious disease systems using multiscale modelling methods. The central idea of this relativistic theory is that at every level of organization of an infectious disease system, the reciprocal influence between the microscale and the macroscale establishes a pathogen replication-transmission multiscale cycle. We distinguish two kinds of reciprocal influence between the microscale and the macroscale based on systematic differences in their conditions of relevancy. Evidence for the validity of the replication-transmission relativity theory is presented using a multiscale model of hookworm infection that is developed at host level when the relationship between the microscale and the macroscale is described by one of the forms of reciprocal influence.

Linked within-host and between-host models and data for infectious diseases: a systematic review

The observed dynamics of infectious diseases are driven by processes across multiple scales. Here we focus on two: within-host, that is, how an infection progresses inside a single individual (for instance viral and immune dynamics), and between-host, that is, how the infection is transmitted between multiple individuals of a host population. The dynamics of each of these may be influenced by the other, particularly across evolutionary time. Thus understanding each of these scales, and the links between them, is necessary for a holistic understanding of the spread of infectious diseases. One approach to combining these scales is through mathematical modeling. We conducted a systematic review of the published literature on multi-scale mathematical models of disease transmission (as defined by combining within-host and between-host scales) to determine the extent to which mathematical models are being used to understand across-scale transmission, and the extent to which these models are being confronted with data. Following the PRISMA guidelines for systematic reviews, we identified 24 of 197 qualifying papers across 30 years that include both linked models at the within and between host scales and that used data to parameterize/calibrate models. We find that the approach that incorporates both modeling with data is under-utilized, if increasing. This highlights the need for better communication and collaboration between modelers and empiricists to build well-calibrated models that both improve understanding and may be used for prediction.

A coupled multiscale model to guide malaria control and elimination

In this paper, we share with the biomathematics community a new coupled multiscale model which has the potential to inform policy and guide malaria control and elimination. The formulation of this multiscale model is based on integrating four submodels which are: (i) a sub-model for the mosquito-to-human transmission of malaria parasite, (ii) a sub-model for the human-to-mosquito transmission of malaria parasite, (iii) a within-mosquito malaria parasite population dynamics sub-model and (iv) a within-human malaria parasite population dynamics sub-model. The integration of the four submodels is achieved by assuming that the transmission parameters of the sub-model for the mosquito-to-human transmission of malaria at the epidemiological scale are functions of the dependent variables of the within-mosquito sporozoite population dynamics while the transmission parameters of the sub-model for the human-to-mosquito transmission of malaria are functions of the dependent variables of the within-human gametocyte population dynamics. This establishes a unidirectionally coupled multiscale model where the within-human and within-mosquito submodels are unidirectionally coupled to the human-to-mosquito and mosquito-to-human submodels. A fast and slow time scale analysis is performed on this system. The result is a simple multiscale model which describes the mechanics of malaria transmission in terms of the major components of the complete malaria parasite life-cycle. This multiscale modelling approach may be found useful in guiding malaria control and elimination.

FROM INDIVIDUAL HEALTH TO COMMUNITY HEALTH: TOWARDS MULTISCALE MODELING OF DIRECTLY TRANSMITTED INFECTIOUS DISEASE SYSTEMS

In this paper, we present a new method for developing a class of nested multiscale models for directly transmitted infectious disease systems that integrates within-host scale and between-host scale using community pathogen load (CPL) as a new public health measure of a community’s level of infectiousness and as an indicator of the effectiveness of health interventions. The approach develops a multiscale modeling science base for directly transmitted infectious disease systems (where the inside-host environment’s biological entities such as cells, tissues, organs, body fluids, whole body are the reservoir of infective pathogen in the community) that is comparable to an existing multiscale modeling science base for environmentally transmitted infectious diseases (where the outside-host geographical environment’s physical entities such as soil, air, formites/contact surfaces, food and water are the reservoir of infective pathogen in the community) where pathogen load in the environment is explicitly incorporated into the model. This is achieved by assuming that infected hosts in the community are homogeneous and unevenly distributed microbial habitats. We illustrate the utility of this multiscale modeling methodology by evaluating the comparative effectiveness of HIV/AIDS preventive and treatment interventions as a case study.

Control Strategies in Multigroup Models: The Case of the Star Network Topology

In this paper, we propose control strategies for multigroup epidemic models. We use compartmental [Formula: see text] models to study the dynamics of n host groups sharing the same source of infection in addition to the transmission among members of the same group. In particular, we consider a model for infectious diseases with free-living pathogens in the environment and a metapopulation model with a central patch. We give the detailed derivation of the target reproduction number under three public health interventions and provide the corresponding biological insights. Moreover, using the next-generation approach, we calculate the basic reproduction numbers associated with subsystems of our models and determine algebraic connections to the target reproduction number of the complete model. The analysis presented here illustrates that understanding the topological structure of the infection process and partitioning it into simple cycles is useful to design and evaluate the control strategies.

A primer on multiscale modelling of infectious disease systems

The development of multiscale models of infectious disease systems is a scientific endeavour whose progress depends on advances on three main frontiers: (a) the conceptual framework frontier, (b) the mathematical technology or technical frontier, and (c) the scientific applications frontier. The objective of this primer is to introduce foundational concepts in multiscale modelling of infectious disease systems focused on these three main frontiers. On the conceptual framework frontier we propose a three-level hierarchical framework as a foundational idea which enables the discussion of the structure of multiscale models of infectious disease systems in a general way. On the scientific applications frontier we suggest ways in which the different structures of multiscale models can serve as infrastructure to provide new knowledge on the control, elimination and even eradication of infectious disease systems, while on the mathematical technology or technical frontier we present some challenges that modelers face in developing appropriate multiscale models of infectious disease systems. We anticipate that the foundational concepts presented in this primer will be central in articulating an integrated and more refined disease control theory based on multiscale modelling - the all-encompassing quantitative representation of an infectious disease system.

A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China

Schistosomiasis, a parasitic disease caused by \textit{ Schistosoma Japonicum}, is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan, and Yunnan. The monthly data of human schistosomiasis cases in Hubei, Hunan, and Anhui provinces (lake and marshland regions) released by the Chinese Center for Disease Control and Prevention (China CDC) display a periodic pattern with more cases in late summer and early autumn. Based on this observation, we construct a deterministic model with periodic transmission rates to study the seasonal transmission dynamics of schistosomiasis in these lake and marshland regions in China. We calculate the basic reproduction number R0, discuss the dynamical behavior of solutions to the model, and use the model to fit the monthly data of human schistosomiasis cases in Hubei. We also perform some sensitivity analysis of the basic reproduction number R0 in terms of model parameters. Our results indicate that treatment of at-risk population groups, improving sanitation, hygiene education, and snail control are effective measures in controlling human schistosomiasis in these lakes and marshland regions.

PREVENTING THE SPREAD OF SCHISTOSOMIASIS IN GHANA: POSSIBLE OUTCOMES OF INTEGRATED OPTIMAL CONTROL STRATEGIES

The goal of a future free from schistosomiasis in Ghana can be achieved through integrated strategies, targeting simultaneously several stages of the life cycle of the schistosome parasite. In this paper, the transmission of schistosomiasis is modeled as a multi-scale 12-dimensional system of ODEs that includes vector-host and within-host dynamics of infection. An explicit expression for the basic reproduction number [Formula: see text] is obtained via the next generation method, this expression being interpreted in biological terms, as well as in terms of reproductive numbers for each type of interaction involved. After discussing the stability of the disease-free equilibrium and the existence and uniqueness of the endemic equilibrium, the Center Manifold Theory is used to show that for values of [Formula: see text] larger than 1, but close to 1, the unique endemic equilibrium is locally asymptotically stable. A sensitivity analysis indicates that [Formula: see text] is most sensitive to the natural death rate of the vector population, while numerical simulations of optimal control strategies reveal that the most effective strategy for the control and possible elimination of schistosomiasis should combine sanitary measures (access to safe water, improved sanitation and hygiene education), large-scale treatment of infected population and vector control measures (via the use of molluscicides), for a significant amount of time.

A complete categorization of multiscale models of infectious disease systems

Modelling of infectious disease systems has entered a new era in which disease modellers are increasingly turning to multiscale modelling to extend traditional modelling frameworks into new application areas and to achieve higher levels of detail and accuracy in characterizing infectious disease systems. In this paper we present a categorization framework for categorizing multiscale models of infectious disease systems. The categorization framework consists of five integration frameworks and five criteria. We use the categorization framework to give a complete categorization of host-level immuno-epidemiological models (HL-IEMs). This categorization framework is also shown to be applicable in categorizing other types of multiscale models of infectious diseases beyond HL-IEMs through modifying the initial categorization framework presented in this study. Categorization of multiscale models of infectious disease systems in this way is useful in bringing some order to the discussion on the structure of these multiscale models.

Modelling coupled within host and population dynamics of [Formula: see text] and [Formula: see text] HIV infection

Most existing models have considered the immunological processes occurring within the host and the epidemiological processes occurring at population level as decoupled systems. We present a new model using continuous systems of non linear ordinary differential equations by directly linking the within host dynamics capturing the interactions between Langerhans cells, CD4[Formula: see text] T-cells, R5 HIV and X4 HIV and the without host dynamics of a basic compartmental HIV/AIDS model. The model captures the biological theories of the cells that take part in HIV transmission. The study incorporates in its analysis the differences in time scales of the fast within host dynamics and the slow without host dynamics. In the mathematical analysis, important thresholds, the reproduction numbers, were computed which are useful in predicting the progression of the infection both within the host and without the host. The study results showed that the model exhibits four within host equilibrium points inclusive of three endemic equilibria whose effects translate into different scenarios at the population level. All the endemic equilibria were shown to be globally stable using Lyapunov functions and this is an important result in linking the within host dynamics to the population dynamics, because the disease free equilibrium point ceases to exist. The effects of linking were observed on the endemic equilibrium points of both the within host and population dynamics. Linking the two dynamics was shown to increase in the viral load within the host and increase in the epidemic levels in the population dynamics.

A Multiscale Model for the World's First Parasitic Disease Targeted for Eradication: Guinea Worm Disease

Guinea worm disease (GWD) is both a neglected tropical disease and an environmentally driven infectious disease. Environmentally driven infectious diseases remain one of the biggest health threats for human welfare in developing countries and the threat is increased by the looming danger of climate change. In this paper we present a multiscale model of GWD that integrates the within-host scale and the between-host scale. The model is used to concurrently examine the interactions between the three organisms that are implicated in natural cases of GWD transmission, the copepod vector, the human host, and the protozoan worm parasite (Dracunculus medinensis), and identify their epidemiological roles. The results of the study (through sensitivity analysis of R₀) show that the most efficient elimination strategy for GWD at between-host scale is to give highest priority to copepod vector control by killing the copepods in drinking water (the intermediate host) by applying chemical treatments (e.g., temephos, an organophosphate). This strategy should be complemented by health education to ensure that greater numbers of individuals and communities adopt behavioural practices such as voluntary reporting of GWD cases, prevention of GWD patients from entering drinking water bodies, regular use of water from safe water sources, and, in the absence of such water sources, filtering or boiling water before drinking. Taking into account the fact that there is no drug or vaccine for GWD (interventions which operate at within-host scale), the results of our study show that the development of a drug that kills female worms at within-host scale would have the highest impact at this scale domain with possible population level benefits that include prevention of morbidity and prevention of transmission.

Disease dynamics in a coupled cholera model linking within-host and between-host interactions

A new modelling framework is proposed to study the within-host and between-host dynamics of cholera, a severe intestinal infection caused by the bacterium Vibrio cholerae. The within-host dynamics are characterized by the growth of highly infectious vibrios inside the human body. These vibrios shed from humans contribute to the environmental bacterial growth and the transmission of the disease among humans, providing a link from the within-host dynamics at the individual level to the between-host dynamics at the population and environmental level. A fast-slow analysis is conducted based on the two different time scales in our model. In particular, a bifurcation study is performed, and sufficient and necessary conditions are derived that lead to a backward bifurcation in cholera epidemics. Our result regarding the backward bifurcation highlights the challenges in the prevention and control of cholera.

Environmentally transmitted parasites: Host-jumping in a heterogeneous environment

Groups of chronically infected reservoir-hosts contaminate resource patches by shedding a parasite׳s free-living stage. Novel-host groups visit the same patches, where they are exposed to infection. We treat arrival at patches, levels of parasite deposition, and infection of the novel host as stochastic processes, and derive the expected time elapsing until a host-jump (initial infection of a novel host) occurs. At stationarity, mean parasite densities are independent of reservoir-host group size. But within-patch parasite-density variances increase with reservoir group size. The probability of infecting a novel host declines with parasite-density variance; consequently larger reservoir groups extend the mean waiting time for host-jumping. Larger novel-host groups increase the probability of a host-jump during any single patch visit, but also reduce the total number of visits per unit time. Interaction of these effects implies that the waiting time for the first infection increases with the novel-host group size. If the reservoir-host uses resource patches in any non-uniform manner, reduced spatial overlap between host species increases the waiting time for host-jumping.