time, they can follow the restrictions again. but during this pause the second wave can come and become even stronger than the first one. social dynamics should be considered. the social physics models of social stress complement the classical epidemics models. an example of using the numerical sir model to fit the covid-19 data (from u.s. cdc) in the state of ohio, u.s.a. r t fitted using the sir and the sirv model are both shown. note that although the sir model can model an individual wave, a more complex model like sirss would better model multiple waves. the simplest sir-social stress (sirss) model is organised as follows. the susceptible individuals (s) can be split in three subgroups by the types of behavior: ignorant or unaware of the epidemic (sign), rationally resistant (sres), and exhausted (sexh) that do not react on the external stimuli (this is a sort of refractory period). in other words: s(t) = sign(t) + sres(t) + sexh(t). symbolically, the social stress model can be presented by the "reaction scheme" (where i denotes the infected individuals): s i g n + 2 i → s r e s + 2 i +2\to +2 – mobilization reaction (the autocatalytic form here means that the transition rate is proportional to the square of the infected fraction i); s r e s → s e x h \to – exhaustion process due to fatigue from anti-epidemic restrictions; s e x h → s i g n \to – slow relaxation to the initial state (end of the refractory period). the main sir epidemic reaction s . . . + i → 2 i +\to has different reaction rate constants β for sign, sres, and sexh. presumably, for sres, β is lower than for sign and sign. the differences between countries are concentrated in two kinetic constants: the rate of mobilization and the rate of exhaustion calculated for covid-19 epidemic in 13 countries. these constants for this epidemic in all countries can be extracted by the fitting of the sirss model to publicly available data the kdv-sir equation based on the classical sir model, a korteweg-de vries (kdv)–sir equation and its analytical solution have been proposed to illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions. the kdv-sir equation is written as follows: d 2 i d t − σ o 2 i + 3 2 σ o 2 i m a x i 2 = 0 i}-\sigma _^i+^i^=0} . here, σ o = γ ( r o − 1 ) =\gamma (r_-1)} , r o = β γ s o n = , and i m a x = s o 2 ( r o − 1 ) 2 r o 2 =-1)^^ . s o indicates the initial value of the state variable s . parameters σ o (σ-naught) and r o (r-naught) are the time-independent relative growth rate and basic reproduction number, respectively. i m a x presents the maximum of the state variables i (for the number of infected persons). an analytical solution to the kdv-sir equation is written as follows: i = i m a x s e c h 2 ( σ o 2 t ) sech^\left(t\right)} , which represents a solitary wave solution. modelling vaccination the sir model can be modified to model vaccination. typically these introduce an additional compartment to the sir model, v , for vaccinated individuals. below are some examples. vaccinating newborns in presence of a communicable diseases, one of the main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate p ∈ ( 0 , 1 ) : d s d t = ν n ( 1 − p ) − μ s − β i n s d i d t = β i n s − ( μ + γ ) i d v d t = ν n p − μ v &=\nu n(1-p)-\mu s-\beta s\\&=\beta s-(\mu +\gamma )i\\&=\nu np-\mu v\end where v is the class of vaccinated subjects. it is immediate to show that: lim t → + ∞ v ( t ) = n p , v(t)=np,} thus we shall deal with the long term behavior of s and i , for which it holds that: r 0 ( 1 − p ) ≤ 1 ⇒ lim t → + ∞ ( s ( t ) , i ( t ) ) = d f e = ( n ( 1 − p ) , 0 ) (1-p)\leq 1\rightarrow \lim _\left(s(t),i(t)\right)=dfe=\left(n\left(1-p\right),0\right)} r 0 ( 1 − p ) > 1 , i ( 0 ) > 0 ⇒ lim t → + ∞ ( s ( t ) , i ( t ) ) = e e = ( n r 0 ( 1 − p ) , n ( r 0 ( 1 − p ) − 1 ) ) . (1-p)>1,\quad i(0)>0\rightarrow \lim _\left(s(t),i(t)\right)=ee=\left((1-p)}},n\left(r_(1-p)-1\right)\right).} in other words, if p < p ∗ = 1 − 1 r 0 =1- the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. this means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease. vaccination and information modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. in order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects: p = p ( i ) , p ′ ( i ) > 0. in such a case the eradication condition becomes: p ( 0 ) ≥ p ∗ , ,} i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline. vaccination of non-newborns in case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows: d s d t = μ n ( 1 − p ) − μ s − ρ s − β i n s d v d t = μ n p + ρ s − μ v &=\mu n(1-p)-\mu s-\rho s-\beta s\\&=\mu np+\rho s-\mu v\end leading to the following eradication condition: p ≥ 1 − ( 1 + ρ μ ) 1 r 0 \right) pulse vaccination strategy this strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. every t time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. this leads to the following impulsive differential equations for the susceptible and vaccinated subjects: d s d t = μ n − μ s − β i n s , s ( n t + ) = ( 1 − p ) s ( n t − ) , n = 0 , 1 , 2 , … d v d t = − μ v , v ( n t + ) = v ( n t − ) + p s ( n t − ) , n = 0 , 1 , 2 , … &=\mu n-\mu s-\beta s,\quad s(nt^)=(1-p)s(nt^),&&n=0,1,2,\ldots \\&=-\mu v,\quad v(nt^)=v(nt^)+ps(nt^),&&n=0,1,2,\ldots \end it is easy to see that by setting i = 0 one obtains that the dynamics of the susceptible subjects is given by: s ∗ ( t ) = 1 − p 1 − ( 1 − p ) e − μ t e − μ m o d ( t , t ) (t)=1-e^ and that the eradication condition is: r 0 ∫ 0 t s ∗ ( t ) d t < 1 \int _^s^(t)\,dt<1} the influence of age: age-structured models age has a deep influence on the disease spread rate in a population, especially the contact rate. this rate summarizes the effectiveness of contacts between susceptible and infectious subjects. taking into account the ages of the epidemic classes s ( t , a ) , i ( t , a ) , r ( t , a ) (to limit ourselves to the susceptible-infectious-removed scheme) such that: s ( t ) = ∫ 0 a m s ( t , a ) d a ^s(t,a)\,da} i ( t ) = ∫ 0 a m i ( t , a ) d a ^i(t,a)\,da} r ( t ) = ∫ 0 a m r ( t , a ) d a ^r(t,a)\,da} (where a m ≤ + ∞ \leq +\infty } is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations: ∂ t s ( t , a ) + ∂ a s ( t , a ) = − μ ( a ) s ( a , t ) − s ( a , t ) ∫ 0 a m k ( a , a 1 ; t ) i ( a 1 , t ) d a 1 s(t,a)+\partial _s(t,a)=-\mu (a)s(a,t)-s(a,t)\int _^k(a,a_;t)i(a_,t)\,da_ ∂ t i ( t , a ) + ∂ a i ( t , a ) = s ( a , t ) ∫ 0 a m k ( a , a 1 ; t ) i ( a 1 , t ) d a 1 − μ ( a ) i ( a , t ) − γ ( a ) i ( a , t ) i(t,a)+\partial _i(t,a)=s(a,t)\int _^;t)i(a_,t)da_-\mu (a)i(a,t)-\gamma (a)i(a,t)} ∂ t r ( t , a ) + ∂ a r ( t , a ) = − μ ( a ) r ( a , t ) + γ ( a ) i ( a , t ) r(t,a)+\partial _r(t,a)=-\mu (a)r(a,t)+\gamma (a)i(a,t)} where: f ( a , t , i ( ⋅ , ⋅ ) ) = ∫ 0 a m k ( a , a 1 ; t ) i ( a 1 , t ) d a 1 ^k(a,a_;t)i(a_,t)\,da_ is the force of infection, which, of course, will depend, though the contact kernel k ( a , a 1 ; t ) ;t)} on the interactions between the ages. complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed: i ( t , 0 ) = r ( t , 0 ) = 0 but that are nonlocal for the density of susceptible newborns: s ( t , 0 ) = ∫ 0 a m ( φ s ( a ) s ( a , t ) + φ i ( a ) i ( a , t ) + φ r ( a ) r ( a , t ) ) d a ^\left(\varphi _(a)s(a,t)+\varphi _(a)i(a,t)+\varphi _(a)r(a,t)\right)\,da} where φ j ( a ) , j = s , i , r (a),j=s,i,r} are the fertilities of the adults. moreover, defining now the density of the total population n ( t , a ) = s ( t , a ) + i ( t , a ) + r ( t , a ) one obtains: ∂ t n ( t , a ) + ∂ a n ( t , a ) = − μ ( a ) n ( a ,