Invasion Reproduction Numbers Research Articles

Analysis of a diffusive two-strain malaria model with the carrying capacity of the environment for mosquitoes

We propose a malaria model involving the sensitive and resistant strains, which is described by reaction-diffusion equations. The model reflects the scenario that the vector and host populations disperse with distinct diffusion rates, susceptible individuals or vectors cannot be infected by both strains simultaneously, and the vector population satisfies the logistic growth. Our main purpose is to get a threshold type result on the model, especially the interaction effect of the two strains in the presence of spatial structure. To solve this issue, the basic reproduction number (BRN) R0i and invasion reproduction number (IRN) Rˆ0i of each strain (i = 1 and 2 are for the sensitive and resistant strains, respectively) are defined. Furthermore, we investigate the influence of the diffusion rates of populations and vectors on BRNs and IRNs.

Mycoloop: Modeling phytoplankton–chytrid–zooplankton interactions in aquatic food webs

A dynamic model is proposed to describe a mycoloop in aquatic food webs. The model consists of phytoplankton, chytrids and zooplankton. It characterizes that zooplankton consume both phytoplankton and free-living chytrid spores, and that chytrids infect phytoplankton. The dynamics of the model are investigated containing the dissipativity, existence and stability of equilibria, and persistence. The ecological reproductive indexes for phytoplankton or zooplankton invasion and basic reproduction numbers for chytrid transmission are derived. The parameter values of the model are estimated based on experimental data. Numerical simulations explore the effects of the mycoloop on phytoplankton blooms and chytrid transmission. This research reveals that the mycoloop structure increases or reduces phytoplankton blooms, and controls the spread of chytrids among phytoplankton.

Complex patterns in a space–time discrete mathematical model of antibiotic resistance in hospitals with self-diffusion

In this work, we extend an antibiotic resistance mathematical model in hospitals spatially on a two-dimensional coupled map lattice. The complex dynamics of the resultant space–time discrete model are investigated. An invasion reproduction number Rar and two control parameters Rsc and Rrc for sensitive bacteria and resistant bacteria, respectively, are defined. The three numbers play an important role in the existence and stability of fixed points. According to the normal form theorem and the center manifold theory, we show that the discrete model exhibits both the flip and the Turing bifurcations. We prove that the existence of the interior fixed point excludes the possibility of the occurrence of a Neimark–Sacker bifurcation. Numerical simulations, including phase diagrams, Lyapunov exponents, bifurcation diagrams, and complex patterns, are enrolled to verify the analyses. The complex patterns induced here are either a result of the pure Turing instability, or the flip instability (non-Turing), while no Turing-flip patterns are formed. The results of the current study show that the space–time discrete version of the model of antibiotic resistance in hospitals captures complex and rich nonlinear dynamics and adds to our understanding of the complicated pattern formation of the model.

INVASION AND SUPERINFECTION IN DETERMINISTIC AND STOCHASTIC TWO-STRAIN DENGUE MODELS WITH DEMOGRAPHIC AND SEASONAL VARIATION

The effects of seasonality on disease invasion and superinfection are investigated in dengue epidemic models with two viral strains. A two-strain host–vector nonautonomous ODE is formulated with seasonal periodicity in either vector recruitment or transmission. The basic reproduction number is derived and the dynamics of the single-strain periodic subsystem are summarized. Conditions for uniform persistence and existence of a single-strain periodic solution are verified and the invasion reproduction number is derived. A time-nonhomogeneous, continuous-time Markov chain (CTMC) is formulated from the ODE framework. The basic and invasion reproduction numbers apply to the ODE and CTMC. In the case of invasion, after one strain is established and the invasion reproduction number exceeds one, the ODE predicts a successful invasion, but in the CTMC, there is a periodic probability of invasion that depends on time and size of the invasion. A multitype branching process approximation of the CTMC provides an analytical method to predict the periodic probability of invasion. The numerical results show that the peak time of invasion is related to the current endemic strain and the dominant seasonal driver but it often precedes the peak time of the seasonal driver.

Allocation of hospital beds on the emergence of new infectious disease: A mathematical model.

This paper is concerned to a mathematical model for the management of hospital beds when a new infection emerges in the population with the existing infections. The study of this joint dynamics presents formidable mathematical challenges due to a limited number of hospital beds. We have derived the invasion reproduction number, which investigates the potential of a newly emerged infectious disease to persist when some infectious diseases are already invaded the host population. We have shown that the proposed system exhibits transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations under certain conditions. We have also shown that the total number of infected individuals may increase if the fraction of the total number of hospital beds is not properly allotted to the existing and a newly emerged infectious disease. The analytically obtained results are verified with the help of numerical simulations.

A mathematical model for the co-dynamics of COVID-19 and tuberculosis

In this study, we formulated and analyzed a deterministic mathematical model for the co-infection of COVID-19 and tuberculosis, to study the co-dynamics and impact of each disease in a given population. Using each disease’s corresponding reproduction number, the existence and stability of the disease-free equilibrium were established. When the respective threshold quantities RC, and RT are below unity, the COVID-19 and TB-free equilibrium are said to be locally asymptotically stable. The impact of vaccine (i.e., efficacy and vaccinated proportion) and the condition required for COVID-19 eradication was examined. Furthermore, the presence of the endemic equilibria of the sub-models is analyzed and the criteria for the phenomenon of backward bifurcation of the COVID-19 sub-model are presented. To better understand how each disease condition impacts the dynamics behavior of the other, we investigate the invasion criterion of each disease by computing the threshold quantity known as the invasion reproduction number. We perform a numerical simulation to investigate the impact of threshold quantities (RC,RT) with respect to their invasion reproduction number, co-infection transmission rate (βct), and each disease transmission rate (βc,βt) on disease dynamics. The outcomes established the necessity for the coexistence or elimination of both diseases from the communities. Overall, our findings imply that while COVID-19 incidence decreases with co-infection prevalence, the burden of tuberculosis on the human population increases.

A two-strain reaction–diffusion malaria model with seasonality and vector-bias

To investigate the combined effects of drug resistance, seasonality and vector-bias, we formulate a periodic two-strain reaction–diffusion model. It is a competitive system for sensitive and resistant strains, but the single-strain subsystem is cooperative. We derive the basic reproduction number \(\mathcal {R}_i\) and the invasion reproduction number \(\mathcal {\hat{R}}_i\) for strain \(i=1,2\), and establish the transmission dynamics in terms of these four quantities. More precisely, (i) if \(\mathcal {R}_1\le 1\) and \(\mathcal {R}_2\le 1\), then the disease is extinct; (ii) if \(\mathcal {R}_1>1\ge \mathcal {R}_2\) (\(\mathcal {R}_2>1\ge \mathcal {R}_1\)), then the sensitive (resistant) strains are persistent, while the resistant (sensitive) strains die out; (iii) if \(\hat{\mathcal {R}}_1>1\) and \(\mathcal {\hat{R}}_2>1\), then two strains are coexistent and periodic oscillation phenomenon is observed. We also study the asymptotic behavior of the basic reproduction number \(\mathcal {R}_0=\max \{\mathcal {R}_1, \mathcal {R}_2\}\) for our model regarding small and large diffusion coefficients. Numerically, we demonstrate the outcome of competition for two strains in different cases.

Modelling the potential influence of human migration and two strains on Ebola virus disease dynamics.

Migration of infected animals and humans, and mutation are considered as the source of the introduction of new pathogens and strains into a country. In this paper, we formulate a mathematical model of Ebola virus disease dynamics, that describes the introduction of a new strain of ebolavirus, through either mutation or immigration (which can be continuous or impulsive) of infectives. The mathematical analysis of the model shows that when the immigration of infectives is continuous, the new strain invades a country if its invasion reproduction number is greater than one. When the immigration is impulsive, a newly introduced strain is controllable when its reproduction number is less than the ratio of mortality to the population inflow and only locally stable equilibria exist. This ratio is one if the population size is constant. In case of mutation of the resident strain of ebolavirus, the coexistence of the resident and mutated strains is possible at least if their respective reproduction numbers are greater than one. Results indicate that the competition for the susceptible population is the immediate consequence of the coexistence of two different strains of ebolavirus in a country and this competition is favourable to the most infectious strain. Results also indicate that impulsive immigration of infectives when compared with continuous immigration of infectives gives time for the implementation of control measures. Our model results suggest controlled movements of people between countries that have had Ebola outbreaks despite the fact that closing boundaries is impossible.

The impact of COVID-19 on a Malaria dominated region: A mathematical analysis and simulations

One of society’s major concerns that have continued for a long time is infectious diseases. It has been demonstrated that certain disease infections, in particular multiple disease infections, make it more challenging to identify and treat infected individuals, thus deteriorating human health. As a result, a COVID-19-malaria co-infection model is developed and analyzed to study the effects of threshold quantities and co-infection transmission rate on the two diseases’ synergistic relationship. This allowed us to better understand the co-dynamics of the two diseases in the population. The existence and stability of the disease-free equilibrium of each single infection were first investigated by using their respective reproduction number. The COVID-19 and malaria-free equilibrium are locally asymptotically stable when the individual threshold quantities RC and RM are below unity. Additionally, the occurrence of the malaria prevalent equilibrium is examined, and the requirements for the backward bifurcation’s existence are provided. Sensitivity analysis reveals that the two main parameters that influence the spread of COVID-19 infection are the disease transmission rate (βc) and the fraction of the exposed individuals becoming symptomatic (ψ), while malaria transmission is influenced by the abundance of vector population, which is driven by recruitment rate (πv) with an increase in the effective biting rate (b), probability of malaria transmission per mosquito bite (βm), and probability of malaria transmission from infected humans to vectors (βv). The findings from the numerical simulation of the model show that COVID-19 will predominate in the populace and drives malaria to extinction when RM<1<RC, whereas malaria will dominate in the population and drives COVID-19 into extinction when RC<1<RM. At the disease’s endemic equilibrium, the two diseases will coexist with the one with the highest reproduction number predominating but not eradicating the other. It was demonstrated in particular that COVID-19 will invade a population where malaria is endemic if the invasion reproduction number exceeds unity. The findings also demonstrate that when the two diseases are at endemic equilibrium, the prevalence of co-infection increases COVID-19’s burden on the population while decreasing malaria incidence.

Dynamics of an immune-epidemiological model with virus evolution and superinfection

To get a better understanding how virus evolution influences on the disease spread, in this paper, we develop a nested immune-epidemiological model with superinfection. The interaction between virokins of within exposed host is described by an ordinary differential equation model. For this within host model, besides the infect-free equilibrium, there are two boundary equilibria and a unique coexistence equilibrium. Virokins coexistence within exposed host leads to the superinfection occurs between hosts. To consider the superinfection between host, we proposed an age structured epidemic model with latency age, in which the transfer rate and the superinfection rate of the exposed individuals both depend on the latency age. For this between host model, the local stabilities of the two boundary steady states and the existence of coexistence steady state are totally determined by the invasion reproduction numbers, R12, R21. Especially, uniform persistence of the between host model is strictly proved by use of integral semigroup theory when R12>1 and R21>1. Some numerical simulations are carried out to verify the main theoretical results and to illustrate the two concerns ((i) the effect of virokins coexistence within exposed hosts on the superinfection between hosts, (ii) the influence of the virus evolution on the dynamics of the age structured model). Finally, a brief conclusion is given in Section 6.

Competitive exclusion and coexistence phenomena of a two-strain SIS model on complex networks from global perspectives.

Genetic heterogeneity plays an important role in exploring the interplays of microorganisms. Competitive exclusion principle is the main principle that governs causative agentries of diseases competition. Identifying coexistence mechanisms is a core issue for studying the interactions of multi-strains. In this paper, we are concerned with the dynamics of a two-strain SIS epidemic model with general incidence rate on complex networks. We derive the basic reproduction numbers and the invasion reproduction numbers associated with each strain, which determine the competitive, exclusion and coexistence of the two strains. We strictly prove that the competitive exclusion principle holds in a global sense and the endemic equilibrium coexists uniquely and globally. Numerical examples visibly illustrate the evolution of the two strains.

Effect of General Cross-Immunity Protection and Antibody-Dependent Enhancement in Dengue Dynamics

A mathematical model to describe the dynamic of a multiserotype infectious disease at the population level is studied. Applied to dengue fever epidemiology, we analyse a mathematical model with time delay to describe the cross-immunity protection period, including a key parameter for the antibody-dependent enhancement (ADE) effect, the well-known features of dengue fever infection. Numerical experiments are performed to show the stability of the coexistence equilibrium, which is completely determined by the basic reproduction number and by the invasion reproduction number, as well as the bifurcation structures for different scenarios of dengue fever transmission in a population. The model shows a rich dynamical behavior, from fixed points and periodic oscillations up to chaotic behaviour with complex attractors.

Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model.

Dengue fever is endemic in tropical and subtropical countries, and certain important features of the spread of dengue fever continue to pose challenges for mathematical modelling. Here we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involve multi-serotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody-dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that are completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics by constructing a suitable Lyapunov functional. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for a specific range of values of a key parameter representing ADE.

A simple two-strain HSV epidemic model with palliative treatment

A two-strain model of the transmission dynamics of herpes simplex virus (HSV) with treatment is formulated as a deterministic system of nonlinear ordinary differential equations. The model is then analyzed qualitatively, with numerical simulations provided to support the theoretical results. The basic reproduction number \(R_0\) is computed with \(R_0=\text{max}\lbrace R_1, R_2 \rbrace \) where \(R_1\) and \(R_2\) represent respectively the reproduction number for HSV1 and HSV2. We also compute the invasion reproductive numbers \(\tilde{R}_1\) for strain 1 when strain 2 is at endemic equilibrium and \(\tilde{R}_2\) for strain 2 when strain 1 is at endemic equilibrium. To determine the relative importance of model parameters to disease transmission, sensitivity analysis is carried out. The reproduction number is most sensitive respectively to the contact rates \(\beta_1\), \(\beta_2\) and the recruitment rate \(\pi\). Numerical simulations indicate the co-existence of the two strains, with HSV1, dominating but not driving out HSV2 whenever \(R_1 &gt; R_2 &gt; 1\) and vice versa.

Modeling and Research on an Immuno-Epidemiological Coupled System with Coinfection.

In this paper, a two-strain model with coinfection that links immunological and epidemiological dynamics across scales is formulated. On the with-in host scale, the two strains eliminate each other with the strain having the larger immunological reproduction number persisting. However, on the population scale coinfection is a common occurrence. Individuals infected with strain one can become coinfected with strain two and similarly for individuals originally infected with strain two. The immunological reproduction numbers R_{j}, the epidemiological reproduction numbers {mathcal {R}}_{j} and invasion reproduction numbers {mathcal {R}}_{j}^{i} are computed. Besides the disease-free equilibrium, there are strain one and strain two dominance equilibria. The disease-free equilibrium is locally asymptotically stable when the epidemiological reproduction numbers {mathcal {R}}_{j} are smaller than one. In addition, each strain dominance equilibrium is locally asymptotically stable if the corresponding epidemiological reproduction number is larger than one and the invasion reproduction number of the other strain is smaller than one. The coexistence equilibrium exists when all the reproduction numbers are greater than one. Simulations suggest that when both invasion reproduction numbers are smaller than one, bistability occurs with one of the strains persisting or the other, depending on initial conditions.

Analysis of a two-strain malaria transmission model with spatial heterogeneity and vector-bias.

In this paper, we introduce a reaction-diffusion malaria model which incorporates vector-bias, spatial heterogeneity, sensitive and resistant strains. The main question that we study is the threshold dynamics of the model, in particular, whether the existence of spatial structure would allow two strains to coexist. In order to achieve this goal, we define the basic reproduction number [Formula: see text] and introduce the invasion reproduction number [Formula: see text] for strain [Formula: see text]. A quantitative analysis shows that if [Formula: see text], then disease-free steady state is globally asymptotically stable, while competitive exclusion, where strain i persists and strain j dies out, is a possible outcome when [Formula: see text] [Formula: see text], and a unique solution with two strains coexist to the model is globally asymptotically stable if [Formula: see text], [Formula: see text]. Numerical simulations reinforce these analytical results and demonstrate epidemiological interaction between two strains, discuss the influence of resistant strains and study the effects of vector-bias on the transmission of malaria.

Dynamics of a competing two-strain SIS epidemic model with general infection force on complex networks

Most diseases have multiple pathogenic strains, which may impose difficulty in combatting the disease and lead to rich dynamics. However, their dynamical properties are not well understood. For this purpose, we formulate and analyze a two-strain SIS epidemic model with a competing mechanism and general infection force on complex networks. We derive the basic reproduction number and introduce the invasion reproduction numbers for each strain. We demonstrate that if R0<1, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out. If R0>1, the conditions of the existence and global asymptotical stability of dominant equilibria are further studied. The persistence of the system is also addressed. Numerical simulations are given to illustrate the theoretical results.

On synergistic co-infection in crop diseases. The case of the Maize Lethal Necrosis Disease

Crops are often subject to intense attacks by pests and diseases. Among them, Maize Lethal Necrosis (MLN) is a serious disease that impact maize crops in many Southern countries. It results from the synergistic interaction of two plant viruses, transmitted by two vectors. In this paper, we develop a general deterministic epidemic model of crop-vector-borne disease for synergistic co-infection, with a particular focus on MLN disease. A theoretical analysis shows that different thresholds exists that drive the dynamics of the system: the well known basic reproduction numbers and also invasion reproduction numbers. The latter are essential for the emergence or not of the MLN disease. After a global sensitivity analysis, we illustrate our results through numerical simulations and discuss potential control methods such as vector control and roguing.

Effects of Coinfection on the Dynamics of Two Pathogens in a Tick-Host Infection Model

As both ticks and hosts may carry one or more pathogens, the phenomenon of coinfection of multiple tick-borne diseases becomes highly relevant and plays a key role in tick-borne disease transmission. In this paper, we propose a coinfection model involving two tick-borne diseases in a tick-host population and calculate the basic reproduction numbers at the disease-free equilibrium and two boundary equilibria. To explore the impact of coinfection, we also derive the invasion reproduction numbers which indicate the potential of a pathogen to persist when another pathogen already exists in tick and host populations. Then, we obtain the global stability of the system at the disease-free equilibrium and the boundary equilibrium, respectively, and further demonstrate the existence conditions for uniform persistence of the two diseases. The final numerical simulations mainly verify the theoretical results of coinfection.

Infection severity across scales in multi-strain immuno-epidemiological Dengue model structured by host antibody level

Infection by distinct Dengue virus serotypes and host immunity are intricately linked. In particular, certain levels of cross-reactive antibodies in the host may actually enhance infection severity leading to Dengue hemorrhagic fever (DHF). The coupled immunological and epidemiological dynamics of Dengue calls for a multi-scale modeling approach. In this work, we formulate a within-host model which mechanistically recapitulates characteristics of antibody dependent enhancement in Dengue infection. The within-host scale is then linked to epidemiological spread by a vector-host partial differential equation model structured by host antibody level. The coupling allows for dynamic population-wide antibody levels to be tracked through primary and secondary infections by distinct Dengue strains, along with waning of cross-protective immunity after primary infection. Analysis of both the within-host and between-host systems are conducted. Stability results in the epidemic model are formulated via basic and invasion reproduction numbers as a function of immunological variables. Additionally, we develop numerical methods in order to simulate the multi-scale model and assess the influence of parameters on disease spread and DHF prevalence in the population.