is far away from the reference point and a convergence test is needed to estimate the error. however, this error is usually acceptable for data fitting. when fitting a set of data with a close time step, the error is relatively small because the reference point is nearby compared to when predicting a long period of time after a reference point. once the real-time r t is pulled out, one can compare it to the basic reproduction number r 0 . before the vaccination, r t gives the policy maker and general public a measure of the efficiency of social mitigation activities such as social distancing and face masking simply by dividing r t r 0 . under massive vaccination, the goal of disease control is to reduce the effective reproduction number r e = r t s n < 1 =s}<1} , where s is the number of susceptible population at the time and n is the total population. when r e < 1 <1} , the spreading decays and daily infected cases go down. the msir model for many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). this is called passive immunity. this added detail can be shown by including an m class (for maternally derived immunity) at the beginning of the model. msir compartmental modelto indicate this mathematically, an additional compartment is added, m(t). this results in the following differential equations: d m d t = λ − δ m − μ m d s d t = δ m − β s i n − μ s d i d t = β s i n − γ i − μ i d r d t = γ i − μ r &=\lambda -\delta m-\mu m\\&=\delta m--\mu s\\&=-\gamma i-\mu i\\&=\gamma i-\mu r\end carrier state some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. they may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. the most famous example of this is probably mary mallon, who infected 22 people with typhoid fever. the carrier compartment is labelled c. the seir model for many important infections, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. during this period the individual is in compartment e (for exposed). seir compartmental model assuming that the latency period is a random variable with exponential distribution with parameter a (i.e. the average latency period is a − 1 ), and also assuming the presence of vital dynamics with birth rate λ equal to death rate n μ (so that the total number n is constant), we have the model: d s d t = μ n − μ s − β i s n d e d t = β i s n − ( μ + a ) e d i d t = a e − ( γ + μ ) i d r d t = γ i − μ r . &=\mu n-\mu s-\\&=-(\mu +a)e\\&=ae-(\gamma +\mu )i\\&=\gamma i-\mu r.\end we have s + e + i + r = n , but this is only constant because of the simplifying assumption that birth and death rates are equal; in general n is a variable. for this model, the basic reproduction number is: r 0 = a μ + a β μ + γ . =.} similarly to the sir model, also, in this case, we have a disease-free-equilibrium (n,0,0,0) and an endemic equilibrium ee, and one can show that, independently from biologically meaningful initial conditions ( s ( 0 ) , e ( 0 ) , i ( 0 ) , r ( 0 ) ) ∈ { ( s , e , i , r ) ∈ [ 0 , n ] 4 : s ≥ 0 , e ≥ 0 , i ≥ 0 , r ≥ 0 , s + e + i + r = n } :s\geq 0,e\geq 0,i\geq 0,r\geq 0,s+e+i+r=n\right\}} it holds that: r 0 ≤ 1 ⇒ lim t → + ∞ ( s ( t ) , e ( t ) , i ( t ) , r ( t ) ) = d f e = ( n , 0 , 0 , 0 ) , \leq 1\rightarrow \lim _\left(s(t),e(t),i(t),r(t)\right)=dfe=(n,0,0,0),} r 0 > 1 , i ( 0 ) > 0 ⇒ lim t → + ∞ ( s ( t ) , e ( t ) , i ( t ) , r ( t ) ) = e e . >1,i(0)>0\rightarrow \lim _\left(s(t),e(t),i(t),r(t)\right)=ee.} in case of periodically varying contact rate β ( t ) the condition for the global attractiveness of dfe is that the following linear system with periodic coefficients: d e 1 d t = β ( t ) i 1 − ( γ + a ) e 1 d i 1 d t = a e 1 − ( γ + μ ) i 1 &=\beta (t)i_-(\gamma +a)e_\\&=ae_-(\gamma +\mu )i_\end is stable (i.e. it has its floquet's eigenvalues inside the unit circle in the complex plane). the seis model the seis model is like the seir model (above) except that no immunity is acquired at the end. s → e → i → s \to \to \to in this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the s(t) compartment. the following differential equations describe this model: d s d t = λ − β s i n − μ s + γ i d e d t = β s i n − ( ϵ + μ ) e d i d t = ε e − ( γ + μ ) i &=\lambda --\mu s+\gamma i\\&=-(\epsilon +\mu )e\\&=\varepsilon e-(\gamma +\mu )i\end the mseir model for the case of a disease, with the factors of passive immunity, and a latency period there is the mseir model. m → s → e → i → r \to \to \to \to d m d t = λ − δ m − μ m d s d t = δ m − β s i n − μ s d e d t = β s i n − ( ε + μ ) e d i d t = ε e − ( γ + μ ) i d r d t = γ i − μ r &=\lambda -\delta m-\mu m\\&=\delta m--\mu s\\&=-(\varepsilon +\mu )e\\&=\varepsilon e-(\gamma +\mu )i\\&=\gamma i-\mu r\end the mseirs model an mseirs model is similar to the mseir, but the immunity in the r class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended. m → s → e → i → r → s \to \to \to \to \to variable contact rates it is well known that the probability of getting a disease is not constant in time. as a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic. in addition, some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. with childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. as a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate f = β ( t ) i n , β ( t + t ) = β ( t ) ,\quad \beta (t+t)=\beta (t)} with period t equal to one year. thus, our model becomes d s d t = μ n − μ s − β ( t ) i n s d i d t = β ( t ) i n s − ( γ + μ ) i &=\mu n-\mu s-\beta (t)s\\&=\beta (t)s-(\gamma +\mu )i\end (the dynamics of recovered easily follows from r = n − s − i ), i.e. a nonlinear set of differential equations with periodically varying parameters. it is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. it is easy to see that if: 1 t ∫ 0 t β ( t ) μ + γ d t < 1 ⇒ lim t → + ∞ ( s ( t ) , i ( t ) ) = d f e = ( n , 0 ) , \int _^\,dt<1\rightarrow \lim _(s(t),i(t))=dfe=(n,0),} whereas if the integral is greater than one the disease will not die out and there may be such resonances. for example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input. this allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic. sir model with diffusion spatiotemporal compartmental models describe not the total number, but the density of susceptible/infective/recovered persons. consequently, they also allow to model the distribution of infected persons in space. in most cases, this is done by combining the sir model with a diffusion equation ∂ t s = d s ∇ 2 s − β i s n , ∂ t i = d i ∇ 2 i + β i s n − γ i , ∂ t r = d r ∇ 2 r + γ i , &\partial _s=d_\nabla ^s-,\\&\partial _i=d_\nabla ^i+-\gamma i,\\&\partial _r=d_\nabla ^r+\gamma i,\end where d s , d i and d r are diffusion constants. thereby, one obtains a reaction-diffusion equation. (note that, for dimensional reasons, the parameter β has to be changed compared to the simple sir model.) early models of this type have been used to model the spread of the black death in europe. extensions of this model have been used to incorporate, e.g., effects of nonpharmaceutical interventions such as social distancing. interacting subpopulation seir model as social contacts, disease severity and lethality, as well as the efficacy of prophylactic measures may differ substantially between interacting subpopulations, e.g., the elderly versus the young, separate seir models for each subgroup may be used that are mutually connected through interaction links. such interacting subpopulation seir models have been used for modeling the covid-19 pandemic at continent scale to develop personalized, accelerated, subpopulation-targeted vaccination strategies that promise a shortening of the pandemic and a reduction of case and death counts in the setting of limited access to vaccines during a wave of virus variants of concern. sir model on networks the sir model has been studied on networks of various kinds in order to model a more realistic form of connection than the homogeneous mixing condition which is usually required. a simple model for epidemics on networks in which an individual has a probability p of being infected by each of his infected neighbors in a given time step leads to results similar to giant component formation on erdos renyi random graphs. sirss model - combination of sir with modelling of social stress dynamics of epidemics depend on how people's behavior changes in time. for example, at the beginning of the epidemic, people are ignorant and careless, then, after the outbreak of epidemics and alarm, they begin to comply with the various restrictions and the spreading of epidemics may decline. over time, some people get tired/frustrated by the restrictions and stop following them (exhaustion), especially if the number of new cases drops down. after resting for some