Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China

A COVID-19 Infection Model Considering the Factors of Environmental Vectors and Re-Positives and Its Application to Data Fitting in Japan and Italy

COVID-19, which broke out globally in 2019, is an infectious disease caused by a novel strain of coronavirus, and its spread is highly contagious and concealed. Environmental vectors play an important role in viral infection and transmission, which brings new difficulties and challenges to disease prevention and control. In this paper, a type of differential equation model is constructed according to the spreading functions and characteristics of exposed individuals and environmental vectors during the virus infection process. In the proposed model, five compartments were considered, namely, susceptible individuals, exposed individuals, infected individuals, recovered individuals, and environmental vectors (contaminated with free virus particles). In particular, the re-positive factor was taken into account (i.e., recovered individuals who have lost sufficient immune protection may still return to the exposed class). With the basic reproduction number R0 of the model, the global stability of the disease-free equilibrium and uniform persistence of the model were completely analyzed. Furthermore, sufficient conditions for the global stability of the endemic equilibrium of the model were also given. Finally, the effective predictability of the model was tested by fitting COVID-19 data from Japan and Italy.

The blood-stage dynamics of malaria infection with immune response

This article is concerned with the dynamics of malaria infection model with diffusion and delay. The disease free threshold ℜ0 and the immune response threshold value ℜ1 of the malaria infection are obtained, which characterize the stability of the disease free equilibrium and infection equilibrium (with or without immune response). In addition, fluctuations occur when the model undergoes Hopf bifurcation as the delay passes through a certain critical value τ0. In this case, periodic oscillation appears near the positive steady state, which implies the recurrent attacks of disease. Finally, numerical simulations are provided to illustrate the theoretical results.

Solidification of Self-Emulsifying Drug Delivery Systems as a Novel Approach to the Management of Uncomplicated Malaria

Malaria affects millions of people annually, especially in third-world countries. The mainstay of treatment is oral anti-malarial drugs and vaccination. An increase in resistant strains of malaria parasites to most of the current anti-malarial drugs adds to the global burden. Moreover, existing and new anti-malarial drugs are hampered by significantly poor aqueous solubility and low permeability, resulting in low oral bioavailability and patient noncompliance. Lipid formulations are commonly used to increase solubility and efficacy and decrease toxicity. The present review discusses the findings from studies focusing on specialised oral lipophilic drug delivery systems, including self-emulsifying drug delivery systems (SEDDSs). SEDDSs facilitate the spontaneous formation of liquid emulsions that effectively solubilise the incorporated drugs into the gastrointestinal tract and thereby improve the absorption of poorly-soluble anti-malaria drugs. However, traditional SEDDSs are normally in liquid dosage forms, which are delivered orally to the site of absorption, and are hampered by poor stability. This paper discusses novel solidification techniques that can easily and economically be up-scaled due to already existing industrial equipment that could be utilised. This method could, furthermore, improve product stability and patient compliance. The possible impact that solid oral SEDDSs can play in the fight against malaria is highlighted.

Mathematical modeling of the impact of temperature variations and immigration on malaria prevalence in Nigeria

The study examines the population-level impact of temperature variability and immigration on malaria prevalence in Nigeria, using a novel deterministic model. The model incorporates disease transmission by immigrants into the community. In the absence of immigration, the model is shown to exhibit the phenomenon of backward bifurcation. The disease-free equilibrium of the autonomous version of the model was found to be locally asymptotically stable in the absence of infective immigrants. However, the model exhibits an endemic equilibrium point when the immigration parameter is greater than zero. The endemic equilibrium point is seen to be globally asymptotically stable in the absence of disease-induced mortality. Uncertainty and sensitivity analysis of the model, using parameter values and ranges relevant to malaria transmission dynamics in Nigeria, shows that the top three parameters that drive malaria prevalence (with respect to [Formula: see text]) are the mosquito natural death rate ([Formula: see text]), mosquito biting rate ([Formula: see text]) and the transmission rates between humans and mosquitoes ([Formula: see text]). Numerical simulations of the model show that in Nigeria, malaria burden increases with increasing mean monthly temperature in the range of 22–28[Formula: see text]. Thus, this study suggests that control strategies for malaria should be intensified during this period. It is further shown that the proportion of infective immigrants has marginal effect on the transmission dynamics of the disease. Therefore, the simulations suggest that a reduction in the fraction of infective immigrants, either exposed or infectious, would significantly reduce the malaria incidence in a population.

Asymptotic analysis of a vector-borne disease model with the age of infection

Vector-borne infectious diseases may involve both horizontal transmission between hosts and transmission from infected vectors to susceptible hosts. In this paper, we incorporate these two transmission modes into a vector-borne disease model that includes general nonlinear incidence rates and the age of infection for both hosts and vectors. We show the existence, uniqueness, nonnegativity, and boundedness of solutions for the model. We study the existence and local stability of steady states, which is shown to be determined by the basic reproduction number. By showing the existence of a global compact attractor and uniform persistence of the system, we establish the threshold dynamics using the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable and otherwise the disease will be established when there is initial infection force for the hosts. We also study a model with the standard incidence rate and discuss the effect of different incidence rates on the disease dynamics.

Long-lasting insecticidal nets and the quest for malaria eradication: a mathematical modeling approach

Recent dramatic declines in global malaria burden and mortality can be largely attributed to the large-scale deployment of insecticidal-based measures, namely long-lasting insecticidal nets (LLINs) and indoor residual spraying. However, the sustainability of these gains, and the feasibility of global malaria eradication by 2040, may be affected by increasing insecticide resistance among the Anopheles malaria vector. We employ a new differential-equations based mathematical model, which incorporates the full, weather-dependent mosquito lifecycle, to assess the population-level impact of the large-scale use of LLINs, under different levels of Anopheles pyrethroid insecticide resistance, on malaria transmission dynamics and control in a community. Moreover, we describe the bednet-mosquito interaction using parameters that can be estimated from the large experimental hut trial literature under varying levels of effective pyrethroid resistance. An expression for the basic reproduction number, [Formula: see text], as a function of population-level bednet coverage, is derived. It is shown, owing to the phenomenon of backward bifurcation, that [Formula: see text] must be pushed appreciably below 1 to eliminate malaria in endemic areas, potentially complicating eradication efforts. Numerical simulations of the model suggest that, when the baseline [Formula: see text] is high (corresponding roughly to holoendemic malaria), very high bednet coverage with highly effective nets is necessary to approach conditions for malaria elimination. Further, while >50% bednet coverage is likely sufficient to strongly control or eliminate malaria from areas with a mesoendemic malaria baseline, pyrethroid resistance could undermine control and elimination efforts even in this setting. Our simulations show that pyrethroid resistance in mosquitoes appreciably reduces bednet effectiveness across parameter space. This modeling study also suggests that increasing pre-bloodmeal deterrence of mosquitoes (deterring them from entry into protected homes) actually hampers elimination efforts, as it may focus mosquito biting onto a smaller unprotected host subpopulation. Finally, we observe that temperature affects malaria potential independently of bednet coverage and pyrethroid-resistance levels, with both climate change and pyrethroid resistance posing future threats to malaria control.

Modelling the dynamics of direct and pathogens-induced dysentery diarrhoea epidemic with controls

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R0 is obtained such that R0≤ 1 indicates the possibility of dysentery eradication in the community while R0>1 represents uniform persistence of the disease. The Lyapunov-LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R0>1 . Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation.

Analysis of a vector-bias malaria transmission model with application to Mexico, Sudan and Democratic Republic of the Congo

Malaria is a deadly disease transmitted to human through the bite of infected female mosquitoes. The aim of this paper is to study the different vector-bias values between low and high transmission areas with the examples of Mexico (low) and Sudan, Democratic Republic of the Congo (Congo, DR) (high) during malaria transmission. We develop a malaria transmission model with vector-bias and investigate the basic reproduction number, the existence of equilibria and the corresponding globally asymptotically stable. Then, we simulate the reported cases of Mexico and Sudan, Democratic Republic of the Congo by World Health Organization (WHO) (WHO, 0000) and predict the direction of the disease. Our simulation results show that the most endemic country is Congo, DR with the highest vector-bias and R₀ values, followed by Sudan and Mexico with less, respectively and that the disease will die out in Mexico and persist in Sudan and Congo, DR. Furthermore, we perform sensitivity analysis of R₀ and give some useful comments on reducing the cases of the disease.

The risk index for an SIR epidemic model and spatial spreading of the infectious disease

In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number RDA0 for an associated model with Dirichlet boundary condition, we introduce the risk index RF0(t) for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if RF0(t0) ≤ 1 for some t0 and the disease is vanishing if RF0(∞) < 1, while if RF0 (0) < 1, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.

Mathematical assessment of the role of Dengvaxia vaccine on the transmission dynamics of dengue serotypes

A new mathematical model is designed and used to assess the impact of the newly-released Dengvaxia vaccine on the transmission dynamics of two co-circulating dengue strains (where strain 1 consists of dengue serotypes 1, 3 and 4; and strain 2 consists of dengue serotype 2). It is shown that the model exhibits the phenomenon of backward bifurcation when the disease-induced mortality in the host population exceeds a certain threshold value or if the vaccine does not provide perfect protection against infection with the two strains. In the absence of backward bifurcation, the disease-free equilibrium of the model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. It is shown that the community-wide use of the vaccine could induce positive, negative or no population-level impact, depending on the sign of a certain epidemiological threshold quantity (known as the vaccine impact factor). Simulations of the model, using data from Oaxaca, Mexico, show that, although the community-wide use of the vaccine will significantly reduce dengue burden in the community, it is unable to lead to the elimination of the two dengue strains. It is further shown that the use of Dengvaxia vaccine in dengue-naive populations may induce increased risk of severe disease in these populations.

The backward bifurcation of a model for malaria infection

In this paper, an epidemic model of a vector-borne disease, namely, malaria, is considered. The explicit expression of the basic reproduction number is obtained, the local and global asymptotical stability of the disease-free equilibrium is proved under certain conditions. It is shown that the model exhibits the phenomenon of backward bifurcation where the stable disease-free equilibrium coexists with a stable endemic equilibrium. Further, it is proved that the unique endemic equilibrium is globally asymptotically stable under certain conditions.

Comments on "A Mathematical Study to Control Visceral Leishmaniasis: An Application to South Sudan"

Deterministic (ordinary differential equation) models for the transmission dynamics of vector-borne diseases that incorporate disease-induced death in the host(s) population(s) are generally known to exhibit the phenomenon of backward bifurcation (where a stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number of the model is less than unity). Further, it is well known that, in these models, the phenomenon of backward bifurcation does not occur when the disease-induced death rate is negligible (e.g., if the disease-induced death rate is set to zero). In a recent paper on the transmission dynamics of visceral leishmaniasis (a disease vectored by sandflies), titled "A Mathematical Study to Control Visceral Leishmaniasis: An Application to South Sudan," published in Bulletin of Mathematical Biology, Vol. 79, Pages 1110-1134, 2017, Ghosh et al. (2017) stated that their deterministic model undergoes a backward bifurcation even when the disease-induced mortality in the host population is set to zero. This result is contrary to the well-established theory on the dynamics of vector-borne diseases. In this short note, we illustrate some of the key errors in the Ghosh et al. (2017) study.

The dynamics of vector-borne relapsing diseases

In this paper, we describe the dynamics of a vector-borne relapsing disease, such as tick-borne relapsing fever, using the methods of compartmental models. After some motivation and model description we provide a proof of a conjectured general form of the reproductive ratio R₀, which is the average number of new infections produced by a single infected individual. A disease free equilibrium undergoes a bifurcation at R₀=1 and we show that for an arbitrary number of relapses it is a transcritical bifurcation with a single branch of endemic equilibria that is locally asymptotically stable for R₀ sufficiently close to 1. Furthermore, we show there is no backwards bifurcation. We then show that these results can be extended to variants of the model with an example that allows for variation in the number of relapses before recovery. Finally, we discuss implications of our results and directions for future research.

Optimal control of a malaria model with asymptomatic class and superinfection

In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R₀, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R₀<1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard incidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. Simulations suggest that total prevalence of malaria is the highest if all individuals show symptoms upon infection, but then undergoes an incomplete treatment and the lowest when all the individuals first move to the symptomatic class then treated successfully. Total prevalence is average if more individuals upon infection move to the asymptomatic class. We study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals. Simulations suggest that the optimal control strategies are very dynamic. Although they always lead to decrease in the symptomatic infectious individuals, they may lead to increase in the number of asymptomatic infectious individuals. This last scenario occurs if a large portion of newly infected individuals move to the symptomatic class but many of them do not complete treatment or if they all complete treatment but the superinfection rate of asymptomatic individuals is average.