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Modelling Covid-19 Dynamics With Periodic Contact Rates, Waning Vaccine Immunity and Temporary Cross Immunity

This paper presents a novel delay differential equation model incorporating periodic contact rates, waning vaccine immunity, and temporary cross-immunity to analyze the transmission dynamics of COVID-19, demonstrating that disease eradication is achievable when the basic reproduction number falls below unity under specific parameter regimes.

Original authors: FAROUK SAAD, Shehu Maitama, Mubarak Zurmi, Ali Althobaiti, Chanakarn Kiataramkul, Usman Salihu, Musa Danyaro, Idris Ahmed

Published 2026-09-29

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CC BY 4.0

Original authors: FAROUK SAAD, Shehu Maitama, Mubarak Zurmi, Ali Althobaiti, Chanakarn Kiataramkul, Usman Salihu, Musa Danyaro, Idris Ahmed

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of how diseases move through populations, scientists often look for the hidden rhythms that drive an outbreak. Just as the seasons change the way people interact, the spread of an infectious virus is rarely a straight line; it pulses and shifts with time. A critical factor in this dance is the concept of immunity. When a person recovers from an illness or receives a vaccine, their body builds a shield against future infection. However, this shield is not always permanent. Over time, the protection can fade, a process known as waning immunity, leaving individuals vulnerable once again. Furthermore, vaccines are rarely perfect; they may prevent severe sickness but still allow the virus to pass through, and the virus itself can change, creating new strains that might bypass existing defenses. Understanding how these factors—changing contact patterns, fading protection, and imperfect vaccines—interact is essential for predicting whether a disease will vanish or continue to circulate.

A team of researchers has built a new mathematical model to explore these exact dynamics in the context of the coronavirus pandemic. Rather than assuming that people's behavior and their immunity remain constant, the team created a system that accounts for the fact that contact rates between people fluctuate over time, much like the changing seasons. They also incorporated the reality that vaccine protection does not last forever and that immunity gained from a previous infection might only be temporary. By weaving these elements together, the researchers aimed to see how a disease behaves when it faces a population that is partially vaccinated, partially immune, and constantly changing its exposure to the virus.

The researchers constructed a detailed simulation that divides the population into groups: those who are susceptible, those who are vaccinated, those infected with the first strain of the virus, those infected with a second strain, and those who have recovered. A key feature of their work is the inclusion of a time delay, which represents the period of temporary cross-immunity. This is the window of time after recovering from one strain during which a person is protected from catching a different strain, before that protection eventually wears off. The model also tracks how the rate at which people meet each other changes periodically, reflecting real-world patterns where social interaction rises and falls.

Through their analysis, the team discovered a specific condition under which the disease can be completely wiped out. They found that if the basic reproduction number—a measure of how many new infections one sick person causes on average—stays below a critical threshold, the disease will eventually die away. This threshold depends heavily on how effective the vaccine is, how quickly immunity fades, and how often people come into contact with one another. The study shows that even an imperfect vaccine can be powerful enough to stop an outbreak, but only if the coverage is high enough and the protection lasts long enough. If the immunity fades too quickly, or if the contact rates spike too high, the disease can find a way to persist, leading to recurring waves of infection.

The researchers ran computer simulations to test these ideas, adjusting the speed at which immunity wanes and the rate at which people get vaccinated. In scenarios where the vaccination rate was high and immunity faded slowly, the number of infected people dropped steadily over time. However, when the protection from the vaccine or previous infection faded rapidly, the system became unstable. The simulations showed that rapid waning could lead to sustained oscillations, where the number of cases rises and falls in a predictable but dangerous pattern, preventing the disease from ever fully disappearing. The inclusion of the time delay for cross-immunity added another layer of complexity, causing peaks in infection to arrive later than they would in a simpler model, which suggests that the timing of outbreaks can be harder to predict when temporary immunity is involved.

A sensitivity analysis performed by the authors revealed which factors matter most. The results indicated that the baseline rate at which people contact one another is the most influential factor in driving the spread of the virus. While the effectiveness of the vaccine and the speed of immunity loss are also critical, the frequency of human interaction plays the dominant role. This implies that public health strategies must focus not only on vaccination but also on managing how people interact, especially during periods when the virus is likely to spread more easily. The study suggests that relying solely on vaccination without addressing contact rates may not be enough to control the disease if the protection provided by the vaccine is not strong or long-lasting.

The researchers acknowledge that their model is a simplification of a complex reality. It assumes that new people enter the population at a constant rate and does not account for differences in age or location, which can significantly affect how a disease spreads. Additionally, the model focuses on vaccination for the first strain and does not fully explore the dynamics of booster shots or the competition between multiple evolving strains. Despite these limitations, the work provides a clear framework for understanding how periodic changes in behavior and the temporary nature of immunity can shape the long-term future of an epidemic. The findings suggest that to keep a disease at bay, societies must maintain high vaccination coverage and be prepared to adjust their social behaviors as the seasons change, recognizing that the shield of immunity is not a permanent wall but a barrier that requires constant reinforcement.

Technical Summary: Modelling Covid-19 Dynamics With Periodic Contact Rates, Waning Vaccine Immunity and Temporary Cross Immunity

Problem Statement
The paper addresses the complex transmission dynamics of COVID-19, specifically focusing on the interplay between periodic contact rates, the waning efficacy of vaccines over time, and temporary cross-immunity between viral strains. The authors note that while vaccines are effective, they are imperfect and their protection diminishes, while the virus exhibits seasonal transmission patterns and potential for reinfection. Existing models often fail to simultaneously account for time delays in immunity, periodic forcing of contact rates, and the specific dynamics of waning vaccine-induced immunity within a multi-strain context. The study aims to develop a mathematical framework to characterize these dynamics and determine conditions under which the disease can be eradicated.

Methodology
The authors formulate a novel Delay Differential Equation (DDE) model extending the standard SIR framework. The total population $N$ is divided into five compartments: Susceptible ( $S$ ), Vaccinated ( $V$ ), Infected with Strain 1 ( $I_1$ ), Infected with Strain 2 ( $I_2$ ), and Recovered ( $R_l$ ).

Key features of the model formulation include:

Periodic Contact Rates: The transmission coefficients $\beta_1(t)$ and $\beta_2(t)$ are modeled as time-dependent functions incorporating sinusoidal fluctuations to represent seasonal variations: $\beta(t) = \beta_0[1 + \beta_{1}\sin(\omega t)]$ .
Waning Immunity: The model incorporates a waning rate ( $\delta$ ) for vaccine-induced immunity, allowing vaccinated individuals to return to the susceptible class.
Time Delay: A time delay ( $\tau$ ) is introduced in the recovery equation to model the duration of temporary cross-immunity, reflecting the lag between infection and the loss of protective immunity.
Vaccination: A fraction ( $p$ ) of the recruited population is vaccinated immediately, with a rate $\alpha$ for vaccinating susceptible individuals.

The mathematical analysis proceeds through the following steps:

Existence and Uniqueness: The authors prove that the system satisfies the Lipschitz condition, ensuring a unique solution exists.
Positivity and Boundedness: It is demonstrated that all state variables remain non-negative and bounded for all $t \geq 0$ , ensuring biological feasibility.
Equilibrium Analysis: The Disease-Free Equilibrium (DFE) is derived.
Stability Analysis: Using Floquet theory, the stability of the periodic system is analyzed by evaluating the eigenvalues of the time-dependent Jacobian matrix. The basic reproduction numbers ( $R_1$ and $R_2$ ) for Strain 1 and Strain 2 are derived.
Global Stability: A proof is provided showing that if $\max\{R_1, R_2\} < 1$ , the DFE is globally asymptotically stable.
Numerical Simulation: MATLAB is used to simulate the system, comparing scenarios with and without time delays and varying vaccination ( $\alpha$ ) and waning ( $\delta$ ) rates.

Key Contributions and Results

Derivation of Reproduction Numbers: The study derives explicit expressions for the basic reproduction numbers $R_1$ and $R_2$ . These expressions incorporate vaccination coverage, vaccine efficacy ( $\sigma$ ), waning rates, and periodic contact amplitudes.
Stability Conditions: The analytical results establish that the disease-free equilibrium is locally and globally asymptotically stable if $\max\{R_1, R_2\} < 1$ . This implies that under specific parameter regimes, the disease can be eradicated.
Impact of Waning and Periodicity: Numerical simulations reveal that while balanced vaccination and waning rates lead to a decline in infections, high waning rates ( $\delta$ ) relative to vaccination rates ( $\alpha$ ) can cause sustained oscillations or higher infection peaks.
Role of Time Delay: The inclusion of time delay (DDE) introduces delayed peaks and potential for periodic outbreaks compared to Ordinary Differential Equation (ODE) counterparts. The simulations show that the system exhibits oscillatory behavior consistent with the periodic contact rates.
Sensitivity Analysis: The study identifies that the basic reproduction number is most sensitive to baseline contact rates, followed by vaccine efficacy and waning rates.

Significance and Claims
The paper claims that the proposed DDE model provides a robust framework for understanding COVID-19 dynamics where immunity is imperfect and waning, and transmission is seasonal. The authors assert that their findings offer actionable insights for public health policy:

Vaccination Strategy: High vaccination coverage is essential, but the model highlights that waning immunity necessitates booster campaigns, particularly when the waning rate is high.
Timing of Interventions: The presence of periodic contact rates suggests that public health measures should be reinstated ahead of predicted seasonal peaks, as relaxing measures during low-transmission periods may lead to resurgence.
Cross-Immunity: The model underscores that post-infection protection is temporary; recovered populations may become susceptible again, reinforcing the need for continued vaccination even for those previously infected.

The authors acknowledge limitations, including the assumption of constant recruitment, the lack of age structure or spatial heterogeneity, and the use of assumed parameter values due to a lack of region-specific empirical data. They also note that the model currently focuses on vaccination for Strain 1 and treats the delay as constant. Despite these simplifications, the paper concludes that periodic forcing and waning immunity can drive oscillations or resurgence even under moderate vaccination coverage, emphasizing the importance of sustained non-pharmaceutical interventions and the timing of booster doses.

Technical Summary: Modelling Covid-19 Dynamics With Periodic Contact Rates, Waning Vaccine Immunity and Temporary Cross Immunity

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