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In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction model that includes the following: (i) fear effect incorporated in prey birth rate; (ii) group defence of prey against predators; and (iii) prey refuge. We provide comprehensive mathematical analysis of extinction and persistence scenarios for both prey and predator species. To better explore the dynamics of the system, a thorough investigation of bifurcation analysis has been performed using fear level, prey birth rate, and prey death rate caused by intra-prey competition as bifurcation parameter. All potential occurrences of bi-stability dynamics have also been investigated for some relevant sets of parametric values. Our numerical evaluations show that high levels of fear can stabilize the prey-predator system by ruling out the possibility of periodic solutions. Also, our model Hopf bifurcation is subcritical in contrast to traditional prey-predator models, which ignore the cost of fear and have supercritical Hopf bifurcations in general. In contrast to the general trend, predator species go extinct at higher values of prey birth rates. We have also found that, contrary to the typical tendency for prey species to go extinct, both prey and predator populations may coexist in the system as intra-prey competition level grows noticeably. The stability and Turing instability of associated spatial model have also been investigated analytically. We also perform the numerical simulation to observe the effect of different parameters on the density distribution of species. Different types of spatiotemporal patterns like spot, mixture of spots and stripes have been observed via variation of time evolution, diffusion coefficient of predator population, level of fear factor and prey refuge. The fear level parameter (k) has a great impact on the spatial dynamics of model system.

A deterministic model with testing of infected individuals has been proposed to investigate the potential consequences of the impact of testing strategy. The model exhibits global dynamics concerning the disease-free and a unique endemic equilibrium depending on the basic reproduction number when the recruitment of infected individuals is zero; otherwise, the model does not have a disease-free equilibrium, and disease never dies out in the community. Model parameters have been estimated using the maximum likelihood method with respect to the data of early COVID-19 outbreak in India. The practical identifiability analysis shows that the model parameters are estimated uniquely. The consequences of the testing rate for the weekly new cases of early COVID-19 data in India tell that if the testing rate is increased by 20% and 30% from its baseline value, the weekly new cases at the peak are decreased by 37.63% and 52.90%; and it also delayed the peak time by four and fourteen weeks, respectively. Similar findings are obtained for the testing efficacy that if it is increased by 12.67% from its baseline value, the weekly new cases at the peak are decreased by 59.05% and delayed the peak by 15 weeks. Therefore, a higher testing rate and efficacy reduce the disease burden by tumbling the new cases, representing a real scenario. It is also obtained that the testing rate and efficacy reduce the epidemic's severity by increasing the final size of the susceptible population. The testing rate is found more significant if testing efficacy is high. Global sensitivity analysis using partial rank correlation coefficients (PRCCs) and Latin hypercube sampling (LHS) determine the key parameters that must be targeted to worsen/contain the epidemic.

Infectious disease transmission dynamics are particularly sensitive to social contact patterns, and the precautions people take to limit disease transmission. It depends on the age distribution of the community. Thus, knowing the age$-$specific prevalence and incidence of infectious diseases is critical for predicting future disease burden and the efficacy of interventions like immunization. This study uses an SEIR age$-$structured multi-group epidemic model to understand how social contact affects disease control. We created location$-$specific social contact matrices in the community to see how social mixing has affected illness spread. We estimated the basic reproduction number $(R_0)$ of the system and plotted its global behavior in terms of $R_0$. Optimal control for the problem has also been established quantitatively. The proposed model's transmission rate for India from September 1, 2020, to December 31, 2020, has also been estimated. We replicated the lifting of non-pharmaceutical therapies by allowing participants to return to work in phases and studied the impact of this. Our findings imply that identifying symptomatic sick people aged $20-49$ can help lower the number of infected people when schools are closed. When some schools are partially open, awareness of symptomatic infected persons helps reduce cases. The simulation results also suggest that limiting contact at school and other meeting areas could significantly lower the number of instances. Using the least square approach, it was discovered that the time$-$dependent transmission rate is more realistic than the constant spread rate for COVID$-19$ in India. To reduce the COVID$-19$ in burden, we found that gradually loosening control methods could flatten and lower other peaks. Our findings may help health policymakers decide on timely age$-$based immunization distribution strategies and hence control the disease.

Since December 2020, variants of COVID-19 (especially Delta and Omicron) appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number for each strain independently and take the maximum for $R_0$. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when the basic reproduction number $R_0<1. An intermediate mutation rate $\nu_1$ leads to oscillations. When $\nu_1$ increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy to native strain as 90\%, the cumulative cases and deaths would be reduced by 3.27\% and 5.2\%, respectively; and by considering the vaccine efficacy to mutant strain as 90\%, the cumulative cases and deaths would be reduced by 0.9\% and 2.5\%, respectively. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates but better due to stability at a lower infection level with a larger mutation rate.