Multi-strain Model Research Articles

Planning and adjusting the COVID-19 booster vaccination campaign to reduce disease burden

As public health policies shifted in 2023 from emergency response to long-term COVID-19 disease management, immunization programs started to face the challenge of formulating routine booster campaigns in a still highly uncertain seasonal behavior of the COVID-19 epidemic. Mathematical models assessing past booster campaigns and integrating knowledge on waning of immunity can help better inform current and future vaccination programs. Focusing on the first booster campaign in the 2021/2022 winter in France, we used a multi-strain age-stratified transmission model to assess the effectiveness of the observed booster vaccination in controlling the succession of Delta, Omicron BA.1 and BA.2 waves. We explored counterfactual scenarios altering the eligibility criteria and inter-dose delay. Our study showed that the success of the immunization program in curtailing the Omicron BA.1 and BA.2 waves was largely dependent on the inclusion of adults among the eligible groups, and was highly sensitive to the inter-dose delay, which was changed over time. Shortening or prolonging this delay, even by only one month, would have required substantial social distancing interventions to curtail the hospitalization peak. Also, the time window for adjusting the delay was very short. Our findings highlight the importance of readiness and adaptation in the formulation of routine booster campaign in the current level of epidemiological uncertainty.

Validation of a Multi-Strain HIV Within-Host Model with AIDS Clinical Studies

Validation of a Multi-Strain HIV Within-Host Model with AIDS Clinical Studies

A fractional-order two-strain SVIR model with stability analysis

The outbreak and spread of infectious diseases has become a worldwide concern, and during the spread of infectious diseases, viruses have been found to evolve and diverge into different strains. Based on this phenomenon, the mathematical and biological properties of a multi-strain infectious disease model with vaccination have been studied. In this paper, a fractional-order two-strain SVIR model is proposed. Specifically, the model incorporates vaccine protection that follows a power-law tail distribution and where the vaccine affects the two-strain differently. And the fractional-order derivative form is given by derivation because its memory properties can better characterize the effect of vaccine protection. Meanwhile, we derive an explicit expression for the basic reproduction number of the two-strain and clarify its crucial role in governing the dynamics of the system. The stability results for the fractional-order model are given with the help of Routh–Hurwitz and Lyapunov methods. Moreover, it has been found that the spread of the two-strain of the infectious disease has different effects under different thresholds. Finally, the spread of COVID-19 in India is considered as a real case study to verify the rationality of the proposed model.

A mathematical model of COVID-19 with multiple variants of the virus under optimal control in Ghana.

In this paper, we suggest a mathematical model of COVID-19 with multiple variants of the virus under optimal control. Mathematical modeling has been used to gain deeper insights into the transmission of COVID-19, and various prevention and control strategies have been implemented to mitigate its spread. Our model is a SEIR-based model for multi-strains of COVID-19 with 7 compartments. We also consider the circulatory structure to account for the termination of immunity for COVID-19. The model is established in terms of the positivity and boundedness of the solution and the existence of equilibrium points, and the local stability of the solution. As a result of fitting data of COVID-19 in Ghana to the model, the basic reproduction number of the original virus and Delta variant was estimated to be 1.9396, and the basic reproduction number of the Omicron variant was estimated to be 3.4905, which is 1.8 times larger than that. We observe that even small differences in the incubation and recovery periods of two strains with the same initial transmission rate resulted in large differences in the number of infected individuals. In the case of COVID-19, infections caused by the Omicron variant occur 1.5 to 10 times more than those caused by the original virus. In terms of the optimal control strategy, we formulate three control strategies focusing on social distancing, vaccination, and testing-treatment. We have developed an optimal control model for the three strategies outlined above for the multi-strain model using the Pontryagin's Maximum Principle. Through numerical simulations, we analyze three optimal control strategies for each strain and also consider combinations of the two control strategies. As a result of the simulation, all control strategies are effective in reducing disease spread, in particular, vaccination strategies are more effective than the other two control strategies. In addition the combination of the two strategies also reduces the number of infected individuals by 1/10 compared to implementing one strategy, even when mild levels are implemented. Finally, we show that if the testing-treatment strategy is not properly implemented, the number of asymptomatic and unidentified infections may surge. These results could help guide the level of government intervention and prevention strategy formulation.

Extinction of some strains and asymptotic profiles of coexistence endemic equilibria in a multi-strain epidemic model

This work studies the asymptotic dynamics of classical solutions of a multi-strain diffusive epidemic model in spatially heterogeneous environments. Results on the nonexistence of coexistence endemic equilibria are established. Moreover, explicit conditions on the parameters range which guarantee the extinctions of some strains of the disease are examined. In particular, we establish that the competition-exclusion principle occurs when the population diffuses very fast and one strain uniquely maximizes the ratio of the spatial average of transmission rate over that of the recovery rate. However, several strains of the disease may coexist when either: (i) the population diffuses very slowly, (ii) or a subgroup diffuses very slowly and the other subgroup diffuses very fast. In these two scenarios, the asymptotic profiles of the coexistence endemic equilibria with respect to the diffusion rates of the population are presented. In particular, we observe a spatial segregation of the infected populations when they diffuse very slowly.

Influence of waning immunity on vaccination decision-making: A multi-strain epidemic model with an evolutionary approach analyzing cost and efficacy

In this research, we introduce a comprehensive epidemiological model that accounts for multiple strains of an infectious disease and two distinct vaccination options. Vaccination stands out as the most effective means to prevent and manage infectious diseases. However, when there are various vaccines available, each with its costs and effectiveness, the decision-making process for individuals becomes paramount. Furthermore, the factor of waning immunity following vaccination also plays a significant role in influencing these choices. To understand how individuals make decisions in the context of multiple strains and waning immunity, we employ a behavioral model, allowing an epidemiological model to be coupled with the dynamics of a decision-making process. Individuals base their choice of vaccination on factors such as the total number of infected individuals and the cost-effectiveness of the vaccine. Our findings indicate that as waning immunity increases, people tend to prioritize vaccines with higher costs and greater efficacy. Moreover, when more contagious strains are present, the equilibrium in vaccine adoption is reached more rapidly. Finally, we delve into the social dilemma inherent in our model by quantifying the social efficiency deficit (SED) under various parameter combinations.

Impact of vaccinations, boosters and lockdowns on COVID-19 waves in French Polynesia

Estimating the impact of vaccination and non-pharmaceutical interventions on COVID-19 incidence is complicated by several factors, including successive emergence of SARS-CoV-2 variants of concern and changing population immunity from vaccination and infection. We develop an age-structured multi-strain COVID-19 transmission model and inference framework to estimate vaccination and non-pharmaceutical intervention impact accounting for these factors. We apply this framework to COVID-19 waves in French Polynesia and estimate that the vaccination programme averted 34.8% (95% credible interval: 34.5–35.2%) of 223,000 symptomatic cases, 49.6% (48.7–50.5%) of 5830 hospitalisations and 64.2% (63.1–65.3%) of 1540 hospital deaths that would have occurred in a scenario without vaccination up to May 2022. We estimate the booster campaign contributed 4.5%, 1.9%, and 0.4% to overall reductions in cases, hospitalisations, and deaths. Our results suggest that removing lockdowns during the first two waves would have had non-linear effects on incidence by altering accumulation of population immunity. Our estimates of vaccination and booster impact differ from those for other countries due to differences in age structure, previous exposure levels and timing of variant introduction relative to vaccination, emphasising the importance of detailed analysis that accounts for these factors.

Symmetry in a multi-strain epidemiological model with distributed delay as a general cross-protection period and disease enhancement factor

Important biological features of viral infectious diseases caused by multiple agents with interacting strain dynamics continue to pose challenges for mathematical modelling development. Motivated by dengue fever epidemiology, we study a system of Integro-Differential Equations (IDE) considering strain structure of pathogens. Knowing that complex dynamics observed in dengue models are driven by the combination of two biological features, the temporary cross-immunity (TCI) and disease enhancement via the antibody-dependent enhancement process (ADE), our IDE system incorporates the TCI with a general time delay term, and the ADE effect by a constant factor to differentiate the susceptibility of individuals experiencing a primary or a secondary infection. Aiming at analysing the effect of the symmetry on dengue serotypes in the IDE framework, a detailed qualitative analysis of the model is performed and the instability of the coexistence steady state is shown using the perturbation theory approach. Numerical simulations identify the bifurcation structures and confirm the stability analysis. Results for the symmetric and asymmetric models are discussed.

Resistin-like Molecule α and Pulmonary Vascular Remodeling: A Multi-Strain Murine Model of Antigen and Urban Ambient Particulate Matter Co-Exposure.

Pulmonary hypertension (PH) has a high mortality and few treatment options. Adaptive immune mediators of PH in mice challenged with antigen/particulate matter (antigen/PM) has been the focus of our prior work. We identified key roles of type-2- and type-17 responses in C57BL/6 mice. Here, we focused on type-2-response-related cytokines, specifically resistin-like molecule (RELM)α, a critical mediator of hypoxia-induced PH. Because of strain differences in the immune responses to type 2 stimuli, we compared C57BL/6J and BALB/c mice. A model of intraperitoneal antigen sensitization with subsequent, intranasal challenges with antigen/PM (ovalbumin and urban ambient PM2.5) or saline was used in C57BL/6 and BALB/c wild-type or RELMα-/- mice. Vascular remodeling was assessed with histology; right ventricular (RV) pressure, RV weights and cytokines were quantified. Upon challenge with antigen/PM, both C57BL/6 and BALB/c mice developed pulmonary vascular remodeling; these changes were much more prominent in the C57BL/6 strain. Compared to wild-type mice, RELMα-/- had significantly reduced pulmonary vascular remodeling in BALB/c, but not in C57BL/6 mice. RV weights, RV IL-33 and RV IL-33-receptor were significantly increased in BALB/c wild-type mice, but not in BALB/c-RELMα-/- or in C57BL/6-wild-type or C57BL/6-RELMα-/- mice in response to antigen/PM2.5. RV systolic pressures (RVSP) were higher in BALB/c compared to C57BL/6J mice, and RELMα-/- mice were not different from their respective wild-type controls. The RELMα-/- animals demonstrated significantly decreased expression of RELMβ and RELMγ, which makes these mice comparable to a situation where human RELMβ levels would be significantly modified, as only humans have this single RELM molecule. In BALB/c mice, RELMα was a key contributor to pulmonary vascular remodeling, increase in RV weight and RV cytokine responses induced by exposure to antigen/PM2.5, highlighting the significance of the genetic background for the biological role of RELMα.

Modeling the dynamics of COVID-19 in the presence of Delta and Omicron variants with vaccination and non-pharmaceutical interventions

Since its inception in December 2019, many safe and effective vaccines have been invented and approved for use against COVID-19 along with various non-pharmaceutical interventions. But the emergence of numerous SARS-CoV-2 variants has put the effectiveness of these vaccines, and other intervention measures under threat. So it is important to understand the dynamics of COVID-19 in the presence of its variants of concern (VOC) in controlling the spread of the disease. To address these situations and to find a way out of this problem, a new mathematical model consisting of a system of non-linear differential equations considering the original COVID-19 strain with its two variants of concern (Delta and Omicron) has been proposed and formulated in this paper. We then analyzed the proposed model to study the transmission dynamics of this multi-strain model and to investigate the consequences of the emergence of multiple new SARS-CoV-2 variants which are more transmissible than the previous ones. The control reproduction number, an important threshold parameter, is then calculated using the next-generation matrix method. Further, we presented the discussion about the stability of the model equilibrium. It is shown that the disease-free equilibrium (DFE) of the model is locally asymptotic stable when the control reproduction is less than unity. It is also shown that the model has a unique endemic equilibrium (EEP) which is locally asymptotic stable when the control reproduction number is greater than unity. Using the Center Manifold theory it is shown that the model also exhibits the backward bifurcation phenomenon when the control reproduction number is less than unity. Again without considering the re-infection of the recovered individuals, it is proved that the disease-free equilibrium is globally asymptotically stable when the reproduction threshold is less than unity. Finally, numerical simulations are performed to verify the analytic results and to show the impact of multiple new SARS-CoV-2 variants in the population which are more contagious than the previous variants. Global uncertainty and sensitivity analysis has been done to identify which parameters have a greater impact on disease dynamics and control disease transmission. Numerical simulation suggests that the emergence of new variants of concern increases COVID-19 infection and related deaths. It also reveals that a combination of non-pharmaceutical interventions with vaccination programs of new more effective vaccines should be continued to control the disease outbreak. This study also suggests that more doses of vaccine should provide to combat new and deadly variants like Delta and Omicron.

Spreading processes with mutations over multilayer networks

A key scientific challenge during the outbreak of novel infectious diseases is to predict how the course of the epidemic changes under countermeasures that limit interaction in the population. Most epidemiological models do not consider the role of mutations and heterogeneity in the type of contact events. However, pathogens have the capacity to mutate in response to changing environments, especially caused by the increase in population immunity to existing strains, and the emergence of new pathogen strains poses a continued threat to public health. Further, in the light of differing transmission risks in different congregate settings (e.g., schools and offices), different mitigation strategies may need to be adopted to control the spread of infection. We analyze a multilayer multistrain model by simultaneously accounting for i) pathways for mutations in the pathogen leading to the emergence of new pathogen strains, and ii) differing transmission risks in different settings, modeled as network layers. Assuming complete cross-immunity among strains, namely, recovery from any infection prevents infection with any other (an assumption that will need to be relaxed to deal with COVID-19 or influenza), we derive the key epidemiological parameters for the multilayer multistrain framework. We demonstrate that reductions to existing models that discount heterogeneity in either the strain or the network layers may lead to incorrect predictions. Our results highlight that the impact of imposing/lifting mitigation measures concerning different contact network layers (e.g., school closures or work-from-home policies) should be evaluated in connection with their effect on the likelihood of the emergence of new strains.

The effects of public health measures on severe dengue cases: An optimal control approach

Dengue fever is the most important viral mosquito-borne disease worldwide, with approximately 3.9 billion people at risk of acquiring dengue infection. Measures against mosquito bite combined with vector control programs to reduce mosquito population have been used in endemic countries for several years. Most recently, vaccines have become an important ally to prevent and control disease transmission. Economic costs of dengue control programs vary from region to region and therefore designing an optimal control strategy must be evaluated at different epidemiological contexts. Using a multi-strain vector-host mathematical model, we investigate the impact of different control measures to reduce dengue prevalence. A detailed sensitivity analysis to identify the key parameters influencing disease transmission is followed by an exploratory analysis of the possible solutions for the optimal control problem considering preventive measures to avoid mosquito bites, reduce mosquito population and vaccinate human hosts. The proposed cost functional includes a weighted sum of several efforts (not necessarily quantified as economic costs) for the controls which are evaluated alone and combined. The control system is analyzed using the Pontryagin‘s Principle for optimal control where different strategies are compared. Our results have shown that the simultaneous use of intervention measures are highly effective to reduce disease cases, however, the use of a single control measure can be as effective as the use of two or more controls combined. A careful evaluation of the epidemiological scenario is advised before designing strategies for disease prevention and control, allowing an optimal allocation of the public health resources.

Stability analysis and optimal control on a multi-strain coinfection model with amplification and vaccination

In this paper, a multi-strain coinfection model with amplification (or mutation) is established to characterize the interaction between common strain and amplified strain, as well as vaccination. The basic reproduction number ℛ0 is derived, from which the criteria on the existence and local (or global) stability of equilibria (including disease-free, dominant-strain and coexistence-strain) are established. By analyzing the effectiveness of vaccination, we find that a critical inoculation level could make the disease eliminate when ℛ0<1, while inefficient vaccines could cause backward bifurcation when ℛ0<1. Based on sensitivity analysis and realistic control policy, the optimal strategy of disease control is obtained. The theoretical results are illustrated by numerical simulation and clinical data of COVID-19 in Morocco.

Applying a multi-strain dengue model to epidemics data

Dengue disease transmission is a complex vector-borne disease, mainly due to the co-circulation of four serotypes of the virus. Mathematical models have proved to be a useful tool to understand the complexity of this disease. In this work, we extend the model studied by Esteva et al., 2003, originally proposed for two serotypes, to four circulating serotypes. Using epidemic data of dengue fever in Iquitos (Peru) and San Juan (Puerto Rico), we estimate numerically the co-circulation parameter values for selected outbreaks using a bootstrap method, and we also obtained the Basic Reproduction Number, R0, for each serotype, using both analytical calculations and numerical simulations. Our results indicate that the impact of co-circulation of serotypes in population dynamics of dengue infection is such that there is a reduced effect from DENV-3 to DENV-4 in comparison to no-cross effect for epidemics in Iquitos. Concerning San Juan epidemics, also comparing to no-cross effect, we also observed a reduced effect from the predominant serotype DENV-3 to both DENV-2 and DENV-1 epidemics neglecting the very small number of cases of DENV-4.

Estimating time-dependent contact: a multi-strain epidemiological model of SARS-CoV-2 on the island of Ireland

Mathematical modelling plays a key role in understanding and predicting the epidemiological dynamics of infectious diseases. We construct a flexible discrete-time model that incorporates multiple viral strains with different transmissibilities to estimate the changing patterns of human contact that generates new infections. Using a Bayesian approach, we fit the model to longitudinal data on hospitalisation with COVID-19 from the Republic of Ireland and Northern Ireland during the first year of the pandemic. We describe the estimated change in human contact in the context of government-mandated non-pharmaceutical interventions in the two jurisdictions on the island of Ireland. We take advantage of the fitted model to conduct counterfactual analyses exploring the impact of lockdown timing and introducing a novel, more transmissible variant. We found substantial differences in human contact between the two jurisdictions during periods of varied restriction easing and December holidays. Our counterfactual analyses reveal that implementing lockdowns earlier would have decreased subsequent hospitalisation substantially in most, but not all cases, and that an introduction of a more transmissible variant - without necessarily being more severe - can cause a large impact on the health care burden.

Machine learning aided multiscale modelling of the HIV-1 infection in the presence of NRTI therapy.

Human Immunodeficiency Virus (HIV) is one of the most common chronic infectious diseases in humans. Extending the expected lifetime of patients depends on the use of optimal antiretroviral therapies. Emergence of the drug-resistant strains can reduce the effectiveness of treatments and lead to Acquired Immunodeficiency Syndrome (AIDS), even with antiretroviral therapy. Investigating the genotype-phenotype relationship is a crucial process for optimizing the therapy protocols of the patients. Here, a mathematical modelling framework is proposed to address the impact of existing mutations, timing of initiation, and adherence levels of nucleotide reverse transcriptase inhibitors (NRTIs) on the evolutionary dynamics of the virus strains. For the first time, the existing Stanford HIV drug resistance data have been combined with a multi-strain within-host ordinary differential equation (ODE) model to track the dynamics of the most common NRTI-resistant strains. Overall, the D4T-3TC, D4T-AZT and TDF-D4T drug combinations have been shown to provide higher success rates in preventing treatment failure and further drug resistance. The results are in line with the genotype-phenotype data and pharmacokinetic parameters of the NRTI inhibitors. Moreover, we show that the undetectable mutant strains at the diagnosis have a significant effect on the success/failure rates of the NRTI treatments. Predictions on undetectable strains through our multi-strain within-host model yielded the possible role of viral evolution on the treatment outcomes. It has been recognized that the improvement of multi-scale models can contribute to the understanding of the evolutionary dynamics, and treatment options, and potentially increase the reliability of genotype-phenotype models.

A multi-strain model with asymptomatic transmission: Application to COVID-19 in the US

COVID-19, induced by the SARS-CoV-2 infection, has caused an unprecedented pandemic in the world. New variants of the virus have emerged and dominated the virus population. In this paper, we develop a multi-strain model with asymptomatic transmission to study how the asymptomatic or pre-symptomatic infection influences the transmission between different strains and control strategies that aim to mitigate the pandemic. Both analytical and numerical results reveal that the competitive exclusion principle still holds for the model with the asymptomatic transmission. By fitting the model to the COVID-19 case and viral variant data in the US, we show that the omicron variants are more transmissible but less fatal than the previously circulating variants. The basic reproduction number for the omicron variants is estimated to be 11.15, larger than that for the previous variants. Using mask mandate as an example of non-pharmaceutical interventions, we show that implementing it before the prevalence peak can significantly lower and postpone the peak. The time of lifting the mask mandate can affect the emergence and frequency of subsequent waves. Lifting before the peak will result in an earlier and much higher subsequent wave. Caution should also be taken to lift the restriction when a large portion of the population remains susceptible. The methods and results obtained her e may be applied to the study of the dynamics of other infectious diseases with asymptomatic transmission using other control measures.

Vaccination and variants: A COVID-19 multi-strain model evolution for the Philippines

Coronavirus disease 2019 (COVID-19) management and response is a challenging task due to the uncertainty and complexity of the nature surrounding the virus. In particular, the emergence of new variants and the polarizing response from the populace complicate government efforts to control the pandemic. In this study, we developed a compartmental model that includes (1) a vaccinated compartment, (2) reinfection after a particular time, and (3) COVID-19 variants dominant in the Philippines. Furthermore, we incorporated stochastic terms to capture uncertainty brought about by the further evolution of the new variants and changing control measures via parametric perturbation. Results show the importance of booster shots that increase the vaccine-induced immunity duration. Without booster shots, simulations showed that the dominant strain would still cause significant infection until 31 December 2023. Moreover, our stochastic model output showed significant variability in this case, implying greater uncertainty with future predictions. All these adverse effects, fortunately, can be effectively countered by increasing the vaccine-induced immunity duration that can be done through booster shots.

Global Stability of Multi-Strain SEIR Epidemic Model with Vaccination Strategy

A three-strain SEIR epidemic model with a vaccination strategy is suggested and studied in this work. This model is represented by a system of nine nonlinear ordinary differential equations that describe the interaction between susceptible individuals, strain-1-vaccinated individuals, strain-1-exposed individuals, strain-2-exposed individuals, strain-3-exposed individuals, strain-1-infected individuals, strain-2-infected individuals, strain-3-infected individuals, and recovered individuals. We start our analysis of this model by establishing the existence, positivity, and boundedness of all the solutions. In order to show global stability, the model has five equilibrium points: The first one stands for the disease-free equilibrium, the second stands for the strain-1 endemic equilibrium, the third one describes the strain-2 equilibrium, the fourth one represents the strain-3 equilibrium point, and the last one is called the total endemic equilibrium. We establish the global stability of each equilibrium point using some suitable Lyapunov function. This stability depends on the strain-1 reproduction number R01, the strain-2 basic reproduction number R02, and the strain-3 reproduction number R03. Numerical simulations are given to confirm our theoretical results. It is shown that in order to eradicate the infection, the basic reproduction numbers of all the strains must be less than unity.

Dynamics of a multi-strain malaria model with diffusion in a periodic environment

This paper mainly explores the complex impacts of spatial heterogeneity, vector-bias effect, multiple strains, temperature-dependent extrinsic incubation period (EIP) and seasonality on malaria transmission. We propose a multi-strain malaria transmission model with diffusion and periodic delays and define the reproduction numbers and (i = 1, 2). Quantitative analysis indicates that the disease-free ω-periodic solution is globally attractive when , while if (), then strain i persists and strain j dies out. More interestingly, when and are greater than 1, the competitive exclusion of the two strains also occurs. Additionally, in a heterogeneous environment, the coexistence conditions of the two strains are and . Numerical simulations verify the analytical results and reveal that ignoring vector-bias effect or seasonality when studying malaria transmission will underestimate the risk of disease transmission.