The dynamics of two viral infections in a single host population with applications to hantavirus

Dynamic of a two-strain COVID-19 model with vaccination

COVID-19 is a respiratory illness caused by an ribonucleic acid (RNA) virus prone to mutations. In December 2020, variants with different characteristics that could affect transmissibility emerged around the world. To address this new dynamic of the disease, we formulate and analyze a mathematical model of a two-strain COVID-19 transmission dynamics with strain 1 vaccination. The model is theoretically analyzed and sufficient conditions for the stability of its equilibria are derived. In addition to the disease-free and endemic equilibria, the model also has single-strain 1 and strain 2 endemic equilibria. Using the center manifold theory, it is shown that the model does not exhibit the phenomenon of backward bifurcation, and global stability of the model equilibria are proved using various approaches. Simulations to support the model theoretical results are provided. We calculate the basic reproductive number R 1 and R 2 for both strains independently. Results indicate that - both strains will persist when R 1 > 1 and R 2 > 1 - Stain 2 could establish itself as the dominant strain if R 1 < 1 and R 2 > 1 , or when R 2 > R 1 > 1 . However, because of de novo herd immunity due to strain 1 vaccine efficacy and provided the initial stain 2 transmission threshold parameter R 2 is controlled to remain below unity, strain 2 will not establish itself/persist in the community.

The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function

In the history of the world, contagious diseases have been proved to pose serious threats to humanity that needs uttermost research in the field and its prompt implementations. With this motive, an attempt has been made to investigate the spread of such contagion by using a delayed stochastic epidemic model with general incidence rate, time-delay transmission, and the concept of cross immunity. It is proved that the system is mathematically and biologically well-posed by showing that there exist a positive and bounded global solution of the model. Necessary conditions are derived, which guarantees the permanence as well as extinction of the disease. The model is further investigated for the existence of an ergodic stationary distribution and established sufficient conditions. The non-zero periodic solution of the stochastic model is analyzed quantitatively. The analysis of optimality and time delay is used, and a proper strategy was presented for prevention of the disease. A scheme for the numerical simulations is developed and implemented in MATLAB, which reflects the long term behavior of the model. Simulation suggests that the noises play a vital role in controlling the spread of an epidemic following the proposed flow, and the case of disease extinction is directly proportional to the magnitude of the white noises. Since time delay reflects the dynamics of recurring epidemics, therefore, it is believed that this study will provide a robust basis for studying the behavior and mechanism of chronic infections.

Effect of density dependence on coinfection dynamics

In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural 'size' (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R 0 ≈ 1 . We show even more, that for the values R 0 > 1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

Hemorrhagic fever with renal syndrome in China: Mechanisms on two distinct annual peaks and control measures

Hemorrhagic fever with renal syndrome (HFRS) is a rodent-borne disease caused by several serotypes of hantavirus and 90% of all reported HFRS cases occur in China. However, the dynamics of such outbreak, particularly the characteristics of two distinct annual peaks in China, are not well understood. Here, we investigate several of the biologically plausible causes for the peaks in monthly HFRS cases, and find that the key factor is the interplay between periodic transmission rates and rodent periodic birth rate. Analysis of dynamical model reveals that vaccination plays a significant role in the control of HFRS in China. Sensitive analysis of different interventions demonstrates that integrating rodent culling and environmental management with the current vaccination program is effective for HFRS control. Our results suggest that for diseases from animals to human beings, the features of both animals and humans beings should be taken into account in the control and prevention strategies.

MONTHLY PERIODIC OUTBREAK OF HEMORRHAGIC FEVER WITH RENAL SYNDROME IN CHINA

Hemorrhagic fever with renal syndrome (HFRS) spreading from rodent to human beings is a major public health problem in China, which causes high mortality rate. Data obtained from the China Ministry of Health shows that cases of HFRS in China exhibited monthly periodic outbreak. To well reveal the mechanisms about the outbreak of HFRS, we established a dynamical model to explain the periodic behaviors of HFRS in China. We obtained the basic reproduction number [Formula: see text], analyzed the dynamical behavior of the model, and used the model to fit the monthly data of HFRS cases. Our results demonstrated that periodic transmission rates and rodent periodic birth rate of HFRS in China can give rise to the periodic outbreak of HFRS, hence providing insights into taking measures to control HFRS in China.

Coinfection Dynamics of Two Diseases in a Single Host Population

A susceptible-infectious-susceptible (SIS) epidemic model that describes the coinfection and cotransmission of two infectious diseases spreading through a single population is studied. The host population consists of two subclasses: susceptible and infectious, and the infectious individuals are further divided into three subgroups: those infected by the first agent/pathogen, the second agent/pathogen, and both. The basic reproduction numbers for all cases are derived which completely determine the global stability of the system if the presence of one agent/pathogen does not affect the transmission of the other. When the constraint on the transmissibility of the dually infected hosts is removed, we introduce the invasion reproduction number, compare it with two other types of reproduction number and show the uniform persistence of both diseases under certain conditions. Numerical simulations suggest that the system can display much richer dynamics such as backward bifurcation, bistability and Hopf bifurcation.

Mathematical model of plant-virus interactions mediated by RNA interference

Cross-protection, which refers to a process whereby artificially inoculating a plant with a mild strain provides protection against a more aggressive isolate of the virus, is known to be an effective tool of disease control in plants. In this paper we derive and analyse a new mathematical model of the interactions between two competing viruses with particular account for RNA interference. Our results show that co-infection of the host can either increase or decrease the potency of individual infections depending on the levels of cross-protection or cross-enhancement between different viruses. Analytical and numerical bifurcation analyses are employed to investigate the stability of all steady states of the model in order to identify parameter regions where the system exhibits synergistic or antagonistic behaviour between viral strains, as well as different types of host recovery. We show that not only viral attributes but also the propagating component of RNA-interference in plants can play an important role in determining the dynamics.

Competitive exclusion and coexistence in a two-strain pathogen model with diffusion

We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number R0 and show that when the model parameters are constant (spatially homogeneous), if R0 >1 then one strain will outcompete the other strain and drive it to extinction, but if R0 ≤ 1 then the disease-free equilibrium is globally attractive. When we assume that the diffusion rates are equal while the transmission and recovery rates are heterogeneous, then there are two possible outcomes under the condition R0 < 1: 1) Competitive exclusion where one strain dies out. 2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.

Mathematical Modeling of Viral Zoonoses in Wildlife

Zoonoses are a worldwide public health concern, accounting for approximately 75% of human infectious diseases. In addition, zoonoses adversely affect agricultural production and wildlife. We review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas where further modeling efforts are needed.

The role of adaptations in two-strain competition for sylvatic Trypanosoma cruzi transmission

This study presents a continuous-time model for the sylvatic transmission dynamics of two strains of Trypanosoma cruzi enzootic in North America, in order to study the role that adaptations of each strain to distinct modes of transmission (classical stercorarian transmission on the one hand, and vertical and oral transmission on the other) may play in the competition between the two strains. A deterministic model incorporating contact process saturation predicts competitive exclusion, and reproductive numbers for the infection provide a framework for evaluating the competition in terms of adaptive trade-off between distinct transmission modes. Results highlight the importance of oral transmission in mediating the competition between horizontal (stercorarian) and vertical transmission; its presence as a competing contact process advantages vertical transmission even without adaptation to oral transmission, but such adaptation appears necessary to explain the persistence of (vertically-adapted) T. cruzi IV in raccoons and woodrats in the southeastern United States.

MODELING THE EFFECTS OF SCHISTOSOMIASIS ON THE TRANSMISSION DYNAMICS OF HIV/AIDS

A schistosomiasis and HIV/AIDS co-infection model is presented as a system of nonlinear ordinary differential equations. Qualitative analysis (properties) of the model are presented. The disease-free equilibrium is shown to be locally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The Centre Manifold theory is used to show that the schistosomiasis only and HIV/AIDS only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. The model is numerically analyzed to assess the effects of schistosomiasis on the dynamics of HIV/AIDS. Analysis of the reproduction numbers and numerical simulations show that an increase of schistosomiasis cases result in an increase of HIV/AIDS cases, suggesting that schistosomiasis control have a positive impact in controlling the transmission dynamics of HIV/AIDS.

The effect of co-colonization with community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus strains on competitive exclusion

We investigate the in-hospital transmission dynamics of two methicillin-resistant Staphylococcus aureus (MRSA) strains: hospital-acquired methicillin resistant S. aureus (HA-MRSA) and community-acquired methicillin-resistant S. aureus (CA-MRSA). Under the assumption that patients can only be colonized with one strain of MRSA at a time, global results show that competitive exclusion occurs between HA-MRSA and CA-MRSA strains; the strain with the larger basic reproduction ratio will become endemic while the other is extinguished due to competition. Because new studies suggest that patients can be concurrently colonized with multiple strains of MRSA, we extend the model to allow patients to be co-colonized with HA-MRSA and CA-MRSA. Using the extended model, we explore the effect of co-colonization on competitive exclusion by determining the invasion reproduction ratios of the boundary equilibria. In contrast to results derived from the assumption that co-colonization does not occur, the extended model rarely exhibits competitive exclusion. More commonly, both strains become endemic in the hospital. When transmission rates are assumed equal and decolonization measures act equally on all strains, competitive exclusion never occurs. Other interesting phenomena are exhibited. For example, solutions can tend toward a co-existence equilibrium, even when the basic reproduction ratio of one of the strains is less than one.

A habitat-based model for the spread of hantavirus between reservoir and spillover species

New habitat-based models for spread of hantavirus are developed which account for interspecies interaction. Existing habitat-based models do not consider interspecies pathogen transmission, a primary route for emergence of new infectious diseases and reservoirs in wildlife and man. The modeling of interspecies transmission has the potential to provide more accurate predictions of disease persistence and emergence dynamics. The new models are motivated by our recent work on hantavirus in rodent communities in Paraguay. Our Paraguayan data illustrate the spatial and temporal overlaps among rodent species, one of which is the reservoir species for Jabora virus and others which are spillover species. Disease transmission occurs when their habitats overlap. Two mathematical models, a system of ordinary differential equations (ODE) and a continuous-time Markov chain (CTMC) model, are developed for spread of hantavirus between a reservoir and a spillover species. Analysis of a special case of the ODE model provides an explicit expression for the basic reproduction number, R(0), such that if R(0)<1, then the pathogen does not persist in either population but if R(0)>1, pathogen outbreaks or persistence may occur. Numerical simulations of the CTMC model display sporadic disease incidence, a new behavior of our habitat-based model, not present in other models, but which is a prominent feature of the seroprevalence data from Paraguay. Environmental changes that result in greater habitat overlap result in more encounters among various species that may lead to pathogen outbreaks and pathogen establishment in a new host.

Hantavirus transmission in sylvan and peridomestic environments

We developed a compartmental model for hantavirus infection in deer mice (Peromyscus maniculatus) with the goal of comparing relative importance of direct and indirect transmission in sylvan and peridomestic environments. A direct transmission occurs when the infection is mediated by the contact of an infected and an uninfected mouse, while an indirect transmission occurs when the infection is mediated by the contact of an uninfected mouse with, for instance, infected soil. Based on population dynamics data and estimates of hantavirus decay in the two types of environments, our model predicts that direct transmission dominates in the sylvan environment, while both pathways are important in peridomestic environments. The model allows us to compute a basic reproduction number R(0), which indicates whether the virus will be endemic or eradicated from the mouse population, in both an autonomous and a time-periodic model. Our analysis can be used to evaluate various eradication strategies.

Mathematical analysis of a two strain HIV/AIDS model with antiretroviral treatment

A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use.

Modeling HIV/AIDS and tuberculosis coinfection

An HIV/AIDS and TB coinfection model which considers antiretroviral therapy for the AIDS cases and treatment of all forms of TB, i.e., latent and active forms of TB, is presented. We begin by presenting an HIV/AIDS-TB coinfection model and analyze the TB and HIV/AIDS submodels separately without any intervention strategy. The TB-only model is shown to exhibit backward bifurcation when its corresponding reproduction number is less than unity. On the other hand, the HIV/AIDS-only model has a globally asymptotically stable disease-free equilibrium when its corresponding reproduction number is less than unity. We proceed to analyze the full HIV-TB coinfection model and extend the model to incorporate antiretroviral therapy for the AIDS cases and treatment of active and latent forms of TB. The thresholds and equilibria quantities for the models are determined and stabilities analyzed. From the study we conclude that treatment of AIDS cases results in a significant reductions of numbers of individuals progressing to active TB. Further, treatment of latent and active forms of TB results in delayed onset of the AIDS stage of HIV infection.

Decreased overall virulence in coinfected hosts leads to the persistence of virulent parasites

Multiple infections are known to affect virulence evolution. Some studies even show that coinfections may decrease the overall virulence (the disease-induced mortality of a coinfected host). Yet, epidemiological studies tend to overlook the overall virulence, and within-host models tend to ignore epidemiological processes. Here, I develop an epidemiological model where overall virulence is an explicit function of the virulence of the coinfecting strains. I show that in most cases, a unique strain is evolutionarily stable (in accordance with the model I use here). However, when the overall virulence is lower than the virulence of each of the coinfecting strains (i.e., when coinfections decrease virulence), the evolutionary equilibrium may be invaded by highly virulent strains, leading to the coexistence of two strains on an evolutionary timescale. This model has theoretical and experimental implications: it underlines the importance of overall virulence and of epidemiological feedbacks on virulence evolution.

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ℝ(0) can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ℝ(j), j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.

Interference and the persistence of vertically transmitted parasites

Given their ubiquity in nature, understanding the factors that allow the persistence of multiple enemies and in particular vertically transmitted parasites (VTPs) is of considerable importance. Here a model that allows a virulent VTP to be maintained in a system containing a host and a horizontally transmitted parasite (HTP) is analysed. The method of persistence relies on the VTP offering the host a level of protection from the HTP. The VTP is assumed to reduce the HTPs ability to transmit to the host through ecological interference. We show that VTPs are more likely to persist with HTPs that prevent host reproduction than with those that allow it. The VTP persists more easily in r-selected hosts and with highly transmittable HTPs. As the level of protection through interference increases the densities of the host also increase. We also show that VTPs when they do persist tend to stabilise the host population cycles produced by free-living HTPs. The study raised questions about persistence of diseases through interactions with others, and also the stabilising effects of VTPs on dynamical systems in a biological control context.

Mathematical models for hantavirus infection in rodents

Hantavirus pulmonary syndrome is an emerging disease of humans that is carried by wild rodents. Humans are usually exposed to the virus through geographically isolated outbreaks. The driving forces behind these outbreaks is poorly understood. Certainly, one key driver of the emergence of these viruses is the virus population dynamics within the rodent population. Two new mathematical models for hantavirus infection in rodents are formulated and studied. The new models include the dynamics of susceptible, exposed, infective, and recovered male and female rodents. The first model is a system of ordinary differential equations while the second model is a system of stochastic differential equations. These new models capture some of the realistic dynamics of the male/female rodent hantavirus interaction: higher seroprevalence in males and variability in seroprevalence levels.